Research ArticlePower Forecasting of Combined Heating and Cooling SystemsBased on Chaotic Time Series

Liu Hai12 Song Yong2 and Du Qingfu2

1School of Energy and Power Engineering Shandong University Jinan 250061 China2School of Mechanical Electrical amp Information Engineering Shandong University at Weihai Weihai 264209 China

Correspondence should be addressed to Liu Hai sdulh8691163com

Received 21 October 2014 Revised 2 May 2015 Accepted 4 May 2015

Academic Editor Kalyana C Veluvolu

Copyright copy 2015 Liu Hai et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Theoretic analysis shows that the output power of the distributed generation system is nonlinear and chaotic And it is coupledwith the microenvironment meteorological data Chaos is an inherent property of nonlinear dynamic system A predicator of theoutput power of the distributed generation system is to establish a nonlinear model of the dynamic system based on real time seriesin the reconstructed phase space Firstly chaos should be detected and quantified for the intensive studies of nonlinear systems Ifthe largest Lyapunov exponent is positive the dynamical system must be chaotic Then the embedding dimension and the delaytime are chosen based on the improved C-C method The attractor of chaotic power time series can be reconstructed based on theembedding dimension and delay time in the phase space By now the neural network can be trained based on the training sampleswhich are observed from the distributed generation systemThe neural network model will approximate the curve of output poweradequately Experimental results show that the maximum power point of the distributed generation system will be predicted basedon the meteorological data The system can be controlled effectively based on the prediction

1 Introduction

As the energy and environmental stresses increased the dis-tributed generation systemhas become the best energy supplyoptionsThe distributed generation system has been noted bymany countries and regions as one of the first elements whichsaves primary energy and reduces greenhouse gas emissions[1] It is a well-established technologywith high efficiency andvery low pollutant emissions It achieves small installationspace lowmaintenance and long life serviceThe system canallow a wide range of operating conditions to match thermaland electric end-user requirements The building owner caninvest in an on-site system to supply power using nonrenew-able or renewable technologies The on-site systems can alsosupply the heating and cooling integrated with heat exchang-ers solar thermal collectors and absorption chillers [2]

The operation and control of distributed generationsystem are facing a stringent challenge to keep the efficiencyand stability of the system [3] The major breakthrough hasnot been made in the effective solutions for optimal dispatchand control strategy of distributed generation system On

the other hand the existence of the uncertainty in thesupply available from renewable generators has caused severedifficulty for the control and optimization of the system Thedistributed generation system is a very complicated nonlinearsystem with time varying Chaos is an inherent property ofnonlinear dynamic system Therefore the kinetic forms ofdistributed generation system must be chaotic The forecast-ing of chaotic time series is widely applied in many fields [4]The intention of chaotic time series forecasting is to establisha nonlinear model of the dynamic system based on real timeseries in the reconstructed phase space The most commonmethods of chaotic time series forecasting mainly includeglobal forecasting method [5] local-region forecastingmethod [6] self-adapting forecasting method [7] and Lya-punov exponent forecasting method [8]

The output power of the distributed generation systemand the environmentalmeteorology are strongly coupledThechaotic time series of the microenvironment meteorologicaldata and the output power are obtained by analysis of theprevious meteorology rules in this region In this paper webuild the model of the microenvironment meteorology and

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2015 Article ID 174203 7 pageshttpdxdoiorg1011552015174203

2 Journal of Control Science and Engineering

(a) The cooling subsystem (b) The heating subsystem

Figure 1 The combined heating and cooling system

Fieldbus

Plate heatexchanger

Heat dissipating

Exhaustfumes

Electricdevice

Electricitygrid

Controlsystem

Energymanagement

Gas engine

Heatingload

CoolingloadElectric

load

Monitoringsystem

Biogas

Gas

Lithiumbromide

absorptionrefrigeration

V1

V2

V3

V4

Figure 2 The block diagram of distributed generation system

the output power of the distributed generation system basedon neural network and chaos time series forecasting Theneural network is trained based on the training sampleswhich are chaotic time series obtained from the distributedgeneration system The neural network approximates thecurve of output power adequately The functional relation-ships between the output power and the meteorologicaldata are fitted to build the chaos forecasting model of theoutput power The maximum power point is tracked by thechaos forecasting model in the conversion process betweenthe renewable energy sources and the conventional energysourcesThen different kinds of energy system are controlledharmoniously and banded together closely The model of thesystem ismostly established based on accurate dataThe erroris general among experimental data owing to the disturbanceThe precision of forecasting model based on neural networkdepends on the generalization of neural network In orderto improve the neural network predictor the structure andparameters of the model should be optimized

2 The Distributed Generation System

With the increasing demand for energy the shortage ofenergy sources in the world is becoming more and more

serious At the same time the consumption of the tradi-tional energy sources aggravates the environmental pollutionTherefore the utilization and the critical technology ofrenewable energy source have been paid more and moreattentionThe combined cooling heating and power systemshave been considered as the best solution of future energysupply [9 10] This paper deals with the power forecasting ofrenewable distributed energy system as depicted in Figure 1The distributed generation systems operate based on the ideaof the comprehensive step utilization of different grades heatenergy [11]The operation of the system is the conversion pro-cess of different grades energy The different subsystems aremutually interrelated and constrained during the operationof the system So the operation of the distributed generationsystem is a complex process with many factors

The distributed generation system can be conceptuallyseen as being composed of the combination of local subsys-tems producing electricity heat cooling and so on Figure 2shows the basic layout with the relevant energy flows Thecore of the system is represented by one chemical block andtwomain physical blocks the biomass gas generating systemthe lithium bromide absorption refrigeration cycle and theplate heat exchanger

Journal of Control Science and Engineering 3

The chemical reaction is the process in which biomassgas is converted to electrical energy in the biomass gasgenerating systemThe available solid biomass resources withlow calorific value are convertible to gas Small electricitygenerators are powered by biogas or natural gas As onebioenergy technology biomass gas generating technology canprovide clean energyThe biomass gas generating system con-sumes large amounts of solid reject and generates enormousamounts of thermal and electrical energy The waste heatof the generating unit is used to drive the lithium bromideabsorption refrigeration cycle and the plate heat exchanger

The heat exchanging processes are physical reactionsTherefrigerant vapour is generated in evaporator of the lithiumbromide absorption refrigerator Then the lithium bromidesolution will become dilute because amount of vapour isabsorbed by the solution in absorber The dilute solution issent back to the heat exchanger by the circulating pumpThesolution is concentrated and further heated by steam or hotwater in the generatorThe temperature of cool dilute solutionrises through heat exchanger between the generator and theabsorber The metal plates are used to transfer heat betweentwo fluids in a plate heat exchanger The lithium bromidesolution and refrigerant water are fed into tubes of the plateheat exchangersThe system also includesmonitoring systemenergy management system and control system The safeand stable running of the distributed generation system isensured

The powers of different subsystems are all subject to thefluctuation of the external environmental state meaning therenewable energy is intermittent The efficient control is aserious challenge in long cycle security and steady runningTherefore it is very necessary for the prediction research ofthe renewable energy system The real information can besupplied to distribution system if applicable The short-termpower forecasting can improve system operating efficiencyand flexibility

3 Phase Space Reconstruction ofChaotic Time Series

Generally only a single scalar time series can be measuredfroma real physical system So the phase space reconstructionfrom time series is the first stage in nonlinear time seriesanalysis of data from chaotic systems [12] All possible statevariables cannot usually be obtained by the observation ofa real process Either not all state variables are measured ornot all of them can be known However due to the couplingsbetween the components of a chaotic system we can recon-struct a phase space from a single observation by a time delayembedding [13]

Supposing that a continuous time dynamical system isdescribed as follows

119889 (119909)

119889119905= 119865 (119909 (119905)) 119909 isin 119877

119899

(1)

where 119909 is the state variable and 119865 is the kinetic equationof the system the continuous system will become a discretesystem after discretization

119909 (119905 + 1) = 119891 (119909 (119905)) 119905 = 0 1 119877119899

997888rarr 119877119898

(2)

where 119909 is discrete state variable and 119891 is discrete kineticequation According to Takenrsquos embedding theorem somestate variants of the original system can be analyzed byobserving the system state 119909(119905)Themost common techniqueis time delay method for phase space reconstruction A scalartime series 119909

119894(119905) where 119894 = 1 2 119873 can be embedded into

an 119898-dimensional space to represent the dynamic system asfollows

119883 (119905) = 119909 (119905) 119909 (119905 minus 120591) 119909 (119905 minus (119898 minus 1) 120591) (3)

where 119905 = 1 2 119898 119898 is embedding dimension and 120591 isthe delay time

The phase space reconstruction system is an orthogonalprojection from 119901-dimensional space to119898-dimensional sub-space The inherent evolution law of the dynamic system canbe described in the form of an119898-dimensional mapping Thelinear smooth mapping119872 is defined as follows

119883 (119905 + ℎ) = 119872 119909 (119905) 119909 (119905 minus 120591) 119909 [119905 minus (119898 minus 1) 120591] (4)

where ℎ is the prediction time domainThe forecasting model is derived from the expression of

the mapping equation (see (4)) But it is difficult to introducethe analytical expression in practice The forecasting modelbased on neural network can be trained by training samplessuch as [119883(119905) 119883(119905 + ℎ)] The neural network model canapproximate the linear smooth mapping 119865 adequately

The delay time and embedding dimension play an impor-tant role in the phase space reconstruction according to ascalar time series However it is often extremely difficult tochoose the parameters such as embedding dimension anddelay time for phase space reconstruction Normally thedelay time and embedding dimension are chosen indepen-dently However in recent years some researchers contendthat the delay time and embedding dimension are interre-lated that is the delay time window should be estimatedfirstly for the choice of delay time and embedding dimension[14] The delay time window can be ascertained using C-C method However the exact selection of delay time canonly be practicable by conventional C-C method This papertries to use the improved C-C method to select embeddingwindow [15]The embedding window is first estimated basedon the mean correlation integral Then the delay time canbe chosen according to the conventional C-C method Theembedding dimension can be obtained according to the rela-tion between embedding window and delay time Owing tothe mean correlation integral with a clear physical meaningthe optimal embedding window can be estimated objectivelyIn accordance with the above method the parameters ofphase space reconstruction can be estimated accurately

4 Echo State Network

Echo state network (ESN) is a new-style recurrent neuralnetwork with a sparsely connected hidden layer [16 17]The weights and connectivity of hidden layer are randomlyassigned and are fixed The weights of output layer canbe learned from training samples The discrete-time neuralnetwork with 119870 input units119873 internal network units and 119871

4 Journal of Control Science and Engineering

x(t)

x(t minus (m minus 1)120591) x(t + 120591)

Figure 3 Architecture of echo state network

output units is governed by the following state updatingequation

119909 (119899 + 1) = 119891 (119882119909 (119899) + 119882in119906 (119899 + 1) + 119882119891119887119910 (119899)) (5)

where 119906(119899) 119909(119899) and 119910(119899) denote the activations of inputunits internal units and output units at time step 119899 respec-tively Furthermore 119906(119899) is a 119870-dimensional input signal119906(119899) = (119906

1(119899) 119906

119870(119899)) 119909(119899) is a 119873-dimensional internal

state 119909(119899) = (1199091(119899) 119909

119873(119899)) and 119910(119899) is a 119871-dimensional

output signal 119910(119899) = (1199101(119899) 119910

119871(119899)) 119891 is a sigmoid func-

tion (usually the logistic sigmoid or the tanh function)119882 isthe reservoir weight matrix 119882in is the input weight matrixand119882

119891119887is the output feedback weight matrix The output of

the neural network can be obtained according to the follow-ing equation

119910 (119899 + 1) = 119891119900(119882119900(119909 (119899 + 1))) (6)

where 119891119900is the activation functions of output units 119909(119899 + 1)

and 119910(119899+1) are the internal activation vectors and the outputactivation vectors at time step 119899 + 1 respectively and119882

119900is

the output weight matrix Figure 3 shows the basic networkarchitecture

The training of echo state network can be done in twostages sampling and weight computation During the sam-pling stage of the training the echo state network is drivenby the input sequence 119906(119899) which yields the internal states119909(119899)The outputs of echo state network can be obtained usingthe system equations (5)-(6) The desired outputs 119889(119899) arewritten into the output units and pumped into the internalstates through the back projection connections The internalstates 119909(119899) are collected into a state collectionmatrixM Like-wise the teacher outputs 119889(119899) are collected into the rows of ateacher output collection matrix T The teacher signal series119889(119899) is approximated as a linear combination of the internalstates 119909

119894(119899) by the following equation

119889 (119899) asymp 119910 (119899) =

119899119900

sum

119894=1

119908119900

119894119909119894(119899) (7)

where 119899119900is the number of the output weights and 119908119900

119894is the

output weight corresponding to internal state 119909119894(119899)

The mean squared training error is defined as

MSE = 1

119873

119873

sum

119899=1

(119889 (119899) minus

119899119900

sum

119894=1

119908119900

119894119909119894(119899))

2

(8)

where 119873 is the time steps for which the teacher signal iswritten into the output unitThe output weightmatrix119882

119900can

be computed such that the mean squared training error MSEis minimized Therefore the optimization of output weightmatrix is to minimize the objective function to make neuralnetwork approximate the mapping from input to output Thecomputation of output weight matrix is a linear regressionproblemThe solution of output weight matrix is ill-posed bylinear regression algorithmThedesired output weightmatrixwhich minimizes MSE can be obtained by multiplying thepseudoinverse ofM with T

119882119900= 119872minus1

119879 (9)

where M is the state collection matrix and T is the teacheroutput collection matrix Then the output weight matrix canbe obtained based on the ready-made functions inMathemat-ica or Matlab The optimal output weight matrix with mini-mized training error is implemented in echo state networkNow the neural network predictor is ready for forecasting ofthe input-output time series

5 Forecasting Algorithm Based onEcho State Network

Nonlinear time series forecasting is an important portion ofcurrent science and technology [18] Generally it is very diffi-cult to obtain the mathematical models of complex dynamicsystems The system parameters such as input output andinternal states are only measured The univariate time seriesof a special parameter of the system can be established Theprediction of time series is very often generated by nonlinearforecastingmodelsThe very common forecastingmodels arebased on the chaos models or neural network models

The forecasting methods of nonlinear time series includeindirect forecasting and direct forecasting The method ofsingle-step forecasting has higher accuracy in practical appli-cationsThe direct multisteps forecasting model is concernedwith the estimation of the system output at some time stepsbased on the mapping of input and output There is noaccumulation error for direct forecasting algorithms withoutthe feedback of errors However with the increase of timesteps the forecastingmodelwill bemore complexThe single-step forecasting model can carry out indirect multistepsforecasting through iterations But the error of forecastingmodel accumulates along with time The method of directmultisteps forecasting is applied to forecast the time series ofdistributed generation system power in this paper A detaileddescription of each implement step of the forecasting algo-rithm is as follows

(1) The chaotic analysis calculate the value of the largestLyapunov exponent to check whether the scalar time

Journal of Control Science and Engineering 5

series is chaotic Positive largest Lyapunov exponentshows the dynamic system is chaotic

(2) The selection of system parameters the embeddingdimension 119898 and the delay time 120591 are chosen basedon the improved C-C method

(3) Phase space reconstruction reconstruct an119898-dimen-sional phase space from a single observation by atime delay embedding based on the delay time andembedding dimension

(4) The training samples the input vectors are 119898(119896)where 119896 is the number of training samplesThe input-output pairs 119898(119896) 119883(119896+ℎ) are the training sampleswhich are obtained according to the prediction timedomain ℎ

(5) The initialization of the forecasting model the fore-casting model based on ESN is initialized accordingto the learning algorithm of ESN The output weightsmatrix is calculated based on (9)

(6) Prediction of the observation the prediction value ofenergy in time domain ℎ can be obtained based on thereal time observation of the system

6 Simulation and Analysis

The equations of the distributed generation system are nor-mally unknown The interaction among the subsystems canbe analyzed based on the dynamics system theoryThe systemlaw in the multidimensional phase space will be investigatedbased on the evolution track of the maximum power Duringthe operation of the distributed generation system the maxi-mum power was recorded every day We gained a rich supplyof data which would normally be accessible In this paperwe collected 1000 data points including 800 training datasetsand 200 test data sequences The data collection systemconsists of two parts data acquisition and normalization pro-cessing The process of data normalization can suppress theelectromagnetic interferences and improve the generalizationability of neural network model The method of normaliza-tion processing is provided The equation of linear transfor-mation can be represented as shown below

119875119894=

119875max minus 119875119894119875max minus 119875min

(119894 = 1 2 ) (10)

where 119875max is the maximum of the power sequence 119875min isthe minimum of the power sequence and 119875

119894is the maximum

power on the 119894th day

61 The Choice of Experimental Parameters The problem ofdetecting and quantifying chaos is a key step for the intensivestudies of nonlinear systems The Lyapunov exponents arethe important indexes to quantify the sensitivity on initialconditions for a dynamical system If the dynamical systemis chaotic at least one Lyapunov exponent must be positiveTherefore we only need to calculate the largest Lyapunovexponent to judge the chaotic characters The largest Lya-punov exponent characterizes the rate exponential diver-gence of the nearby trajectories in the reconstructed phase

0 50 100 150 200 250 3000

50

100

150

0 50 100 150 200 250 300

0

50

100

Slop

ey(i

)

minus50

n

i

Figure 4 The largest Lyapunov exponent

0 0204

06 081

0

05

10

02

04

06

08

1

Maximum power (i)

Maximum power (i minus 10)

Max

imum

pow

er (i

minus20

)

Figure 5 The attractor of chaotic power time series

space There are also lots of algorithms to estimate the largestLyapunov exponent of an experimental time series In thispaper the largest Lyapunov exponent was calculated withsmall amount of data The estimate of the largest Lyapunovexponent is 013 bits (shown in Figure 4)

The embedding dimension 119898 and the delay time 120591 arechosen based on the improved C-C methodThen we obtainembedding dimension 119898 = 10 and delay time 120591 = 8 Thephase space is reconstructed based on the embedding dimen-sion and the delay time The attractor of chaotic power timeseries is shown in Figure 5 In practice there will always besome form of corrupting noise Thus any real time serieseven if mostly deterministic will contain some randomnessThe evolution of the distributed generation system beingchaotic means that every point in the reconstructed spaceis approached arbitrarily but not coincident on the pseudoperiodic orbits

62 The Analysis of Simulation Results The neural networkmodel is trained based on the training samples The teacher

6 Journal of Control Science and Engineering

0 50 100 150 20001

02

03

04

05

06

07

08

09

1

11

Date serial

Max

imum

pow

er (k

W)

Observed powerPredicted power (kW)

Figure 6 The estimated output and observations

signal is written into the output unit for times 119899 = 1 800The internal states 119909(119899) = (119909

1(119899) 119909

20(119899)) for 119899 =

1 800 are collected as a row of a state-collectingmatrixMof size 800times 20 during the sampling period At the same timethe teacher outputs 119889(119899) are collected into a row of a matrixT of size 800 times 1 Then we can compute output weights 119908

119900

for linear output unit 119910(119899) such that the teacher time series119889(119899) is approximated as a linear combination of the internalactivation time series 119909

119894(119899) by (9) The multistep prediction

can be obtained with testing samples The predictor canforecast multistep value directly with different time domain

The result of direct prediction based on echo statenetwork for the maximum power series of the distributedgeneration system is shown in Figure 6The solid line denotesthe predicted value the dotted line is for the observed valueWe observe that the prediction model based on echo statenetwork has high accuracy in direct prediction owing to thecoincidence of prediction and actual curve The max predic-tion error is about 017 (shown in Figure 7) Figure 8 is theerror of the traditional multilayer perceptron The predictorbased on echo state network has higher accuracy

7 Conclusion

Themicroenvironment meteorological data are coupled withthe output power of the distributed generation systemTheoretical proof and experimental results show that theoutput power of the distributed generation system is chaotictime seriesTheprediction of chaotic time series is to establisha nonlinear model of the dynamic system based on real timeseries in the reconstructed phase space Firstly we shoulddetect and quantify chaos for the intensive studies of nonlin-ear systems If the largest Lyapunov exponent is positive thedynamical system must be chaotic Secondly the embeddingdimension and the delay time are chosen based on theimproved C-C method The attractor of chaotic power time

0 50 100 150 200

0

005

01

015

Date serial

Erro

r

minus01

minus015

minus02

minus005

Figure 7 The error of the predictions

0 50 100 150 200

0

005

01

015

02

Date serial

Erro

r

minus01

minus015

minus005

Figure 8 The error of the traditional multilayer perceptron

series can be reconstructed based on the embedding dimen-sion and delay time in the phase space Lastly the neural net-work model is trained based on the training samples whichare obtained from the distributed generation system Theneural network will approximate the curve of output poweradequately The maximum power point of the distributedgeneration system will be tracked based on the meteorologi-cal data

Themeasuring errors are unavoidable among experimen-tal data due to the disturbance and many other factors Theprediction precision of the neural network model is largelydependent on the generalization of neural network There-fore we should optimize the structure and parameters ofthe model for better performance Experimental results showthat the prediction model based on echo state network hashigh accuracyThe chaotic predictionmodel can approximatethe mapping between the maximum power point of thedistributed generation system and the meteorological dataThe echo state network model can make rather remarkably

Journal of Control Science and Engineering 7

accurate predictions about the maximum power of thedistributed generation system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 61473174 61105100 and 51376110)the Specialized Research Fund for the Doctoral Program ofHigher Education (no 20130131130006) Project funded byChina Postdoctoral Science Foundation (no 2014M551907)and Independent Innovation Foundation of Shandong Uni-versity (no 2013ZRQP002)

References

[1] G Angrisani C Roselli and M Sasso ldquoDistributed microtri-generation systemsrdquoProgress in Energy andCombustion Sciencevol 38 no 4 pp 502ndash521 2012

[2] K A Pruitt S Leyffer AMNewman et alOptimal Design andDispatch of Distributed Generation Systems Argonne NationalLaboratory Chicago Ill USA 2012

[3] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[4] Y Song Y Li Q Wang and C Li ldquoMulti-steps prediction ofchaotic time series based on echo state networkrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 669ndash672 IEEE Changsha China September 2010

[5] S Michanos A Tsakoumis P Fessas S Vladov and V Mlade-nov ldquoShort-term load forecasting using a chaotic time seriesrdquo inProceedings of the International Symposium on Signals Circuitsand Systems (SCS rsquo03) pp 437ndash440 Iasi Romania 2003

[6] X An D Jiang M Zhao and C Liu ldquoShort-term prediction ofwind power using EMD and chaotic theoryrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2pp 1036ndash1042 2012

[7] C Qin C-C Chang and L-T Liao ldquoAn adaptive prediction-error expansion oriented reversible information hidingschemerdquo Pattern Recognition Letters vol 33 no 16 pp2166ndash2172 2012

[8] J Zhang K C Lam W J Yan H Gao and Y Li ldquoTime seriesprediction using Lyapunov exponents in embedding phasespacerdquo Computers amp Electrical Engineering vol 30 no 1 pp 1ndash15 2004

[9] J M Guerrero F Blaabjerg T Zhelev et al ldquoDistributedgeneration toward a new energy paradigmrdquo IEEE IndustrialElectronics Magazine vol 4 no 1 pp 52ndash64 2010

[10] G Chicco and P Mancarella ldquoDistributed multi-generationa comprehensive viewrdquo Renewable and Sustainable EnergyReviews vol 13 no 3 pp 535ndash551 2009

[11] G Abdollahi and M Meratizaman ldquoMulti-objective approachin thermoenvironomic optimization of a small-scale distributedCCHP system with risk analysisrdquo Energy and Buildings vol 43no 11 pp 3144ndash3153 2011

[12] D Xia S Song J Wang J Shi H Bi and Z Gao ldquoDetermi-nation of corrosion types from electrochemical noise by phasespace reconstruction theoryrdquo Electrochemistry Communica-tions vol 15 no 1 pp 88ndash92 2012

[13] Z Xie and K Wang ldquoSelection of embedding parameters inphase space reconstructionrdquo in Proceedings of the 2nd Inter-national Conference on Intelligent Computing Technology andAutomation (ICICTA rsquo09) pp 637ndash640 IEEE Hunan ChinaOctober 2009

[14] W-D Cai Y-Q Qin and B-R Yang ldquoSelection of delay timewindow and delay time in phase space reconstructionrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 526ndash530 December 2007

[15] Y-B Li Y Song and C-H Li ldquoSelection of parameters forphase space reconstruction of chaotic time seriesrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 30ndash33IEEE Changsha China September 2010

[16] J HerbertTutorial on Training Recurrent Neural Networks Cov-ering BPPT RTRL EKF and the ldquoEcho State Networkrdquo ApproachGMD-Forschungszentrum Informationstechnik Bremen Ger-many 2002

[17] H Jaeger M Lukosevicius D Popovici and U Siewert ldquoOpti-mization and applications of echo state networks with leaky-integrator neuronsrdquoNeural Networks vol 20 no 3 pp 335ndash3522007

[18] K A Bredahl and T Timo Forecasting with Nonlinear TimeSeries Models School of Economics and Management AarhusUniversity Aarhus Denmark 2011

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2 Journal of Control Science and Engineering

(a) The cooling subsystem (b) The heating subsystem

Figure 1 The combined heating and cooling system

Fieldbus

Plate heatexchanger

Heat dissipating

Exhaustfumes

Electricdevice

Electricitygrid

Controlsystem

Energymanagement

Gas engine

Heatingload

CoolingloadElectric

load

Monitoringsystem

Biogas

Gas

Lithiumbromide

absorptionrefrigeration

V1

V2

V3

V4

Figure 2 The block diagram of distributed generation system

the output power of the distributed generation system basedon neural network and chaos time series forecasting Theneural network is trained based on the training sampleswhich are chaotic time series obtained from the distributedgeneration system The neural network approximates thecurve of output power adequately The functional relation-ships between the output power and the meteorologicaldata are fitted to build the chaos forecasting model of theoutput power The maximum power point is tracked by thechaos forecasting model in the conversion process betweenthe renewable energy sources and the conventional energysourcesThen different kinds of energy system are controlledharmoniously and banded together closely The model of thesystem ismostly established based on accurate dataThe erroris general among experimental data owing to the disturbanceThe precision of forecasting model based on neural networkdepends on the generalization of neural network In orderto improve the neural network predictor the structure andparameters of the model should be optimized

2 The Distributed Generation System

With the increasing demand for energy the shortage ofenergy sources in the world is becoming more and more

serious At the same time the consumption of the tradi-tional energy sources aggravates the environmental pollutionTherefore the utilization and the critical technology ofrenewable energy source have been paid more and moreattentionThe combined cooling heating and power systemshave been considered as the best solution of future energysupply [9 10] This paper deals with the power forecasting ofrenewable distributed energy system as depicted in Figure 1The distributed generation systems operate based on the ideaof the comprehensive step utilization of different grades heatenergy [11]The operation of the system is the conversion pro-cess of different grades energy The different subsystems aremutually interrelated and constrained during the operationof the system So the operation of the distributed generationsystem is a complex process with many factors

The distributed generation system can be conceptuallyseen as being composed of the combination of local subsys-tems producing electricity heat cooling and so on Figure 2shows the basic layout with the relevant energy flows Thecore of the system is represented by one chemical block andtwomain physical blocks the biomass gas generating systemthe lithium bromide absorption refrigeration cycle and theplate heat exchanger

Journal of Control Science and Engineering 3

The chemical reaction is the process in which biomassgas is converted to electrical energy in the biomass gasgenerating systemThe available solid biomass resources withlow calorific value are convertible to gas Small electricitygenerators are powered by biogas or natural gas As onebioenergy technology biomass gas generating technology canprovide clean energyThe biomass gas generating system con-sumes large amounts of solid reject and generates enormousamounts of thermal and electrical energy The waste heatof the generating unit is used to drive the lithium bromideabsorption refrigeration cycle and the plate heat exchanger

The heat exchanging processes are physical reactionsTherefrigerant vapour is generated in evaporator of the lithiumbromide absorption refrigerator Then the lithium bromidesolution will become dilute because amount of vapour isabsorbed by the solution in absorber The dilute solution issent back to the heat exchanger by the circulating pumpThesolution is concentrated and further heated by steam or hotwater in the generatorThe temperature of cool dilute solutionrises through heat exchanger between the generator and theabsorber The metal plates are used to transfer heat betweentwo fluids in a plate heat exchanger The lithium bromidesolution and refrigerant water are fed into tubes of the plateheat exchangersThe system also includesmonitoring systemenergy management system and control system The safeand stable running of the distributed generation system isensured

The powers of different subsystems are all subject to thefluctuation of the external environmental state meaning therenewable energy is intermittent The efficient control is aserious challenge in long cycle security and steady runningTherefore it is very necessary for the prediction research ofthe renewable energy system The real information can besupplied to distribution system if applicable The short-termpower forecasting can improve system operating efficiencyand flexibility

3 Phase Space Reconstruction ofChaotic Time Series

Generally only a single scalar time series can be measuredfroma real physical system So the phase space reconstructionfrom time series is the first stage in nonlinear time seriesanalysis of data from chaotic systems [12] All possible statevariables cannot usually be obtained by the observation ofa real process Either not all state variables are measured ornot all of them can be known However due to the couplingsbetween the components of a chaotic system we can recon-struct a phase space from a single observation by a time delayembedding [13]

Supposing that a continuous time dynamical system isdescribed as follows

119889 (119909)

119889119905= 119865 (119909 (119905)) 119909 isin 119877

119899

(1)

where 119909 is the state variable and 119865 is the kinetic equationof the system the continuous system will become a discretesystem after discretization

119909 (119905 + 1) = 119891 (119909 (119905)) 119905 = 0 1 119877119899

997888rarr 119877119898

(2)

where 119909 is discrete state variable and 119891 is discrete kineticequation According to Takenrsquos embedding theorem somestate variants of the original system can be analyzed byobserving the system state 119909(119905)Themost common techniqueis time delay method for phase space reconstruction A scalartime series 119909

119894(119905) where 119894 = 1 2 119873 can be embedded into

an 119898-dimensional space to represent the dynamic system asfollows

119883 (119905) = 119909 (119905) 119909 (119905 minus 120591) 119909 (119905 minus (119898 minus 1) 120591) (3)

where 119905 = 1 2 119898 119898 is embedding dimension and 120591 isthe delay time

The phase space reconstruction system is an orthogonalprojection from 119901-dimensional space to119898-dimensional sub-space The inherent evolution law of the dynamic system canbe described in the form of an119898-dimensional mapping Thelinear smooth mapping119872 is defined as follows

119883 (119905 + ℎ) = 119872 119909 (119905) 119909 (119905 minus 120591) 119909 [119905 minus (119898 minus 1) 120591] (4)

where ℎ is the prediction time domainThe forecasting model is derived from the expression of

the mapping equation (see (4)) But it is difficult to introducethe analytical expression in practice The forecasting modelbased on neural network can be trained by training samplessuch as [119883(119905) 119883(119905 + ℎ)] The neural network model canapproximate the linear smooth mapping 119865 adequately

The delay time and embedding dimension play an impor-tant role in the phase space reconstruction according to ascalar time series However it is often extremely difficult tochoose the parameters such as embedding dimension anddelay time for phase space reconstruction Normally thedelay time and embedding dimension are chosen indepen-dently However in recent years some researchers contendthat the delay time and embedding dimension are interre-lated that is the delay time window should be estimatedfirstly for the choice of delay time and embedding dimension[14] The delay time window can be ascertained using C-C method However the exact selection of delay time canonly be practicable by conventional C-C method This papertries to use the improved C-C method to select embeddingwindow [15]The embedding window is first estimated basedon the mean correlation integral Then the delay time canbe chosen according to the conventional C-C method Theembedding dimension can be obtained according to the rela-tion between embedding window and delay time Owing tothe mean correlation integral with a clear physical meaningthe optimal embedding window can be estimated objectivelyIn accordance with the above method the parameters ofphase space reconstruction can be estimated accurately

4 Echo State Network

Echo state network (ESN) is a new-style recurrent neuralnetwork with a sparsely connected hidden layer [16 17]The weights and connectivity of hidden layer are randomlyassigned and are fixed The weights of output layer canbe learned from training samples The discrete-time neuralnetwork with 119870 input units119873 internal network units and 119871

4 Journal of Control Science and Engineering

x(t)

x(t minus (m minus 1)120591) x(t + 120591)

Figure 3 Architecture of echo state network

output units is governed by the following state updatingequation

119909 (119899 + 1) = 119891 (119882119909 (119899) + 119882in119906 (119899 + 1) + 119882119891119887119910 (119899)) (5)

where 119906(119899) 119909(119899) and 119910(119899) denote the activations of inputunits internal units and output units at time step 119899 respec-tively Furthermore 119906(119899) is a 119870-dimensional input signal119906(119899) = (119906

1(119899) 119906

119870(119899)) 119909(119899) is a 119873-dimensional internal

state 119909(119899) = (1199091(119899) 119909

119873(119899)) and 119910(119899) is a 119871-dimensional

output signal 119910(119899) = (1199101(119899) 119910

119871(119899)) 119891 is a sigmoid func-

tion (usually the logistic sigmoid or the tanh function)119882 isthe reservoir weight matrix 119882in is the input weight matrixand119882

119891119887is the output feedback weight matrix The output of

the neural network can be obtained according to the follow-ing equation

119910 (119899 + 1) = 119891119900(119882119900(119909 (119899 + 1))) (6)

where 119891119900is the activation functions of output units 119909(119899 + 1)

and 119910(119899+1) are the internal activation vectors and the outputactivation vectors at time step 119899 + 1 respectively and119882

119900is

the output weight matrix Figure 3 shows the basic networkarchitecture

The training of echo state network can be done in twostages sampling and weight computation During the sam-pling stage of the training the echo state network is drivenby the input sequence 119906(119899) which yields the internal states119909(119899)The outputs of echo state network can be obtained usingthe system equations (5)-(6) The desired outputs 119889(119899) arewritten into the output units and pumped into the internalstates through the back projection connections The internalstates 119909(119899) are collected into a state collectionmatrixM Like-wise the teacher outputs 119889(119899) are collected into the rows of ateacher output collection matrix T The teacher signal series119889(119899) is approximated as a linear combination of the internalstates 119909

119894(119899) by the following equation

119889 (119899) asymp 119910 (119899) =

119899119900

sum

119894=1

119908119900

119894119909119894(119899) (7)

where 119899119900is the number of the output weights and 119908119900

119894is the

output weight corresponding to internal state 119909119894(119899)

The mean squared training error is defined as

MSE = 1

119873

119873

sum

119899=1

(119889 (119899) minus

119899119900

sum

119894=1

119908119900

119894119909119894(119899))

2

(8)

where 119873 is the time steps for which the teacher signal iswritten into the output unitThe output weightmatrix119882

119900can

be computed such that the mean squared training error MSEis minimized Therefore the optimization of output weightmatrix is to minimize the objective function to make neuralnetwork approximate the mapping from input to output Thecomputation of output weight matrix is a linear regressionproblemThe solution of output weight matrix is ill-posed bylinear regression algorithmThedesired output weightmatrixwhich minimizes MSE can be obtained by multiplying thepseudoinverse ofM with T

119882119900= 119872minus1

119879 (9)

where M is the state collection matrix and T is the teacheroutput collection matrix Then the output weight matrix canbe obtained based on the ready-made functions inMathemat-ica or Matlab The optimal output weight matrix with mini-mized training error is implemented in echo state networkNow the neural network predictor is ready for forecasting ofthe input-output time series

5 Forecasting Algorithm Based onEcho State Network

Nonlinear time series forecasting is an important portion ofcurrent science and technology [18] Generally it is very diffi-cult to obtain the mathematical models of complex dynamicsystems The system parameters such as input output andinternal states are only measured The univariate time seriesof a special parameter of the system can be established Theprediction of time series is very often generated by nonlinearforecastingmodelsThe very common forecastingmodels arebased on the chaos models or neural network models

The forecasting methods of nonlinear time series includeindirect forecasting and direct forecasting The method ofsingle-step forecasting has higher accuracy in practical appli-cationsThe direct multisteps forecasting model is concernedwith the estimation of the system output at some time stepsbased on the mapping of input and output There is noaccumulation error for direct forecasting algorithms withoutthe feedback of errors However with the increase of timesteps the forecastingmodelwill bemore complexThe single-step forecasting model can carry out indirect multistepsforecasting through iterations But the error of forecastingmodel accumulates along with time The method of directmultisteps forecasting is applied to forecast the time series ofdistributed generation system power in this paper A detaileddescription of each implement step of the forecasting algo-rithm is as follows

(1) The chaotic analysis calculate the value of the largestLyapunov exponent to check whether the scalar time

Journal of Control Science and Engineering 5

series is chaotic Positive largest Lyapunov exponentshows the dynamic system is chaotic

(2) The selection of system parameters the embeddingdimension 119898 and the delay time 120591 are chosen basedon the improved C-C method

(3) Phase space reconstruction reconstruct an119898-dimen-sional phase space from a single observation by atime delay embedding based on the delay time andembedding dimension

(4) The training samples the input vectors are 119898(119896)where 119896 is the number of training samplesThe input-output pairs 119898(119896) 119883(119896+ℎ) are the training sampleswhich are obtained according to the prediction timedomain ℎ

(5) The initialization of the forecasting model the fore-casting model based on ESN is initialized accordingto the learning algorithm of ESN The output weightsmatrix is calculated based on (9)

(6) Prediction of the observation the prediction value ofenergy in time domain ℎ can be obtained based on thereal time observation of the system

6 Simulation and Analysis

The equations of the distributed generation system are nor-mally unknown The interaction among the subsystems canbe analyzed based on the dynamics system theoryThe systemlaw in the multidimensional phase space will be investigatedbased on the evolution track of the maximum power Duringthe operation of the distributed generation system the maxi-mum power was recorded every day We gained a rich supplyof data which would normally be accessible In this paperwe collected 1000 data points including 800 training datasetsand 200 test data sequences The data collection systemconsists of two parts data acquisition and normalization pro-cessing The process of data normalization can suppress theelectromagnetic interferences and improve the generalizationability of neural network model The method of normaliza-tion processing is provided The equation of linear transfor-mation can be represented as shown below

119875119894=

119875max minus 119875119894119875max minus 119875min

(119894 = 1 2 ) (10)

where 119875max is the maximum of the power sequence 119875min isthe minimum of the power sequence and 119875

119894is the maximum

power on the 119894th day

61 The Choice of Experimental Parameters The problem ofdetecting and quantifying chaos is a key step for the intensivestudies of nonlinear systems The Lyapunov exponents arethe important indexes to quantify the sensitivity on initialconditions for a dynamical system If the dynamical systemis chaotic at least one Lyapunov exponent must be positiveTherefore we only need to calculate the largest Lyapunovexponent to judge the chaotic characters The largest Lya-punov exponent characterizes the rate exponential diver-gence of the nearby trajectories in the reconstructed phase

0 50 100 150 200 250 3000

50

100

150

0 50 100 150 200 250 300

0

50

100

Slop

ey(i

)

minus50

n

i

Figure 4 The largest Lyapunov exponent

0 0204

06 081

0

05

10

02

04

06

08

1

Maximum power (i)

Maximum power (i minus 10)

Max

imum

pow

er (i

minus20

)

Figure 5 The attractor of chaotic power time series

space There are also lots of algorithms to estimate the largestLyapunov exponent of an experimental time series In thispaper the largest Lyapunov exponent was calculated withsmall amount of data The estimate of the largest Lyapunovexponent is 013 bits (shown in Figure 4)

The embedding dimension 119898 and the delay time 120591 arechosen based on the improved C-C methodThen we obtainembedding dimension 119898 = 10 and delay time 120591 = 8 Thephase space is reconstructed based on the embedding dimen-sion and the delay time The attractor of chaotic power timeseries is shown in Figure 5 In practice there will always besome form of corrupting noise Thus any real time serieseven if mostly deterministic will contain some randomnessThe evolution of the distributed generation system beingchaotic means that every point in the reconstructed spaceis approached arbitrarily but not coincident on the pseudoperiodic orbits

62 The Analysis of Simulation Results The neural networkmodel is trained based on the training samples The teacher

6 Journal of Control Science and Engineering

0 50 100 150 20001

02

03

04

05

06

07

08

09

1

11

Date serial

Max

imum

pow

er (k

W)

Observed powerPredicted power (kW)

Figure 6 The estimated output and observations

signal is written into the output unit for times 119899 = 1 800The internal states 119909(119899) = (119909

1(119899) 119909

20(119899)) for 119899 =

1 800 are collected as a row of a state-collectingmatrixMof size 800times 20 during the sampling period At the same timethe teacher outputs 119889(119899) are collected into a row of a matrixT of size 800 times 1 Then we can compute output weights 119908

119900

for linear output unit 119910(119899) such that the teacher time series119889(119899) is approximated as a linear combination of the internalactivation time series 119909

119894(119899) by (9) The multistep prediction

can be obtained with testing samples The predictor canforecast multistep value directly with different time domain

The result of direct prediction based on echo statenetwork for the maximum power series of the distributedgeneration system is shown in Figure 6The solid line denotesthe predicted value the dotted line is for the observed valueWe observe that the prediction model based on echo statenetwork has high accuracy in direct prediction owing to thecoincidence of prediction and actual curve The max predic-tion error is about 017 (shown in Figure 7) Figure 8 is theerror of the traditional multilayer perceptron The predictorbased on echo state network has higher accuracy

7 Conclusion

Themicroenvironment meteorological data are coupled withthe output power of the distributed generation systemTheoretical proof and experimental results show that theoutput power of the distributed generation system is chaotictime seriesTheprediction of chaotic time series is to establisha nonlinear model of the dynamic system based on real timeseries in the reconstructed phase space Firstly we shoulddetect and quantify chaos for the intensive studies of nonlin-ear systems If the largest Lyapunov exponent is positive thedynamical system must be chaotic Secondly the embeddingdimension and the delay time are chosen based on theimproved C-C method The attractor of chaotic power time

0 50 100 150 200

0

005

01

015

Date serial

Erro

r

minus01

minus015

minus02

minus005

Figure 7 The error of the predictions

0 50 100 150 200

0

005

01

015

02

Date serial

Erro

r

minus01

minus015

minus005

Figure 8 The error of the traditional multilayer perceptron

series can be reconstructed based on the embedding dimen-sion and delay time in the phase space Lastly the neural net-work model is trained based on the training samples whichare obtained from the distributed generation system Theneural network will approximate the curve of output poweradequately The maximum power point of the distributedgeneration system will be tracked based on the meteorologi-cal data

Themeasuring errors are unavoidable among experimen-tal data due to the disturbance and many other factors Theprediction precision of the neural network model is largelydependent on the generalization of neural network There-fore we should optimize the structure and parameters ofthe model for better performance Experimental results showthat the prediction model based on echo state network hashigh accuracyThe chaotic predictionmodel can approximatethe mapping between the maximum power point of thedistributed generation system and the meteorological dataThe echo state network model can make rather remarkably

Journal of Control Science and Engineering 7

accurate predictions about the maximum power of thedistributed generation system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 61473174 61105100 and 51376110)the Specialized Research Fund for the Doctoral Program ofHigher Education (no 20130131130006) Project funded byChina Postdoctoral Science Foundation (no 2014M551907)and Independent Innovation Foundation of Shandong Uni-versity (no 2013ZRQP002)

References

[1] G Angrisani C Roselli and M Sasso ldquoDistributed microtri-generation systemsrdquoProgress in Energy andCombustion Sciencevol 38 no 4 pp 502ndash521 2012

[2] K A Pruitt S Leyffer AMNewman et alOptimal Design andDispatch of Distributed Generation Systems Argonne NationalLaboratory Chicago Ill USA 2012

[3] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[4] Y Song Y Li Q Wang and C Li ldquoMulti-steps prediction ofchaotic time series based on echo state networkrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 669ndash672 IEEE Changsha China September 2010

[5] S Michanos A Tsakoumis P Fessas S Vladov and V Mlade-nov ldquoShort-term load forecasting using a chaotic time seriesrdquo inProceedings of the International Symposium on Signals Circuitsand Systems (SCS rsquo03) pp 437ndash440 Iasi Romania 2003

[6] X An D Jiang M Zhao and C Liu ldquoShort-term prediction ofwind power using EMD and chaotic theoryrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2pp 1036ndash1042 2012

[7] C Qin C-C Chang and L-T Liao ldquoAn adaptive prediction-error expansion oriented reversible information hidingschemerdquo Pattern Recognition Letters vol 33 no 16 pp2166ndash2172 2012

[8] J Zhang K C Lam W J Yan H Gao and Y Li ldquoTime seriesprediction using Lyapunov exponents in embedding phasespacerdquo Computers amp Electrical Engineering vol 30 no 1 pp 1ndash15 2004

[9] J M Guerrero F Blaabjerg T Zhelev et al ldquoDistributedgeneration toward a new energy paradigmrdquo IEEE IndustrialElectronics Magazine vol 4 no 1 pp 52ndash64 2010

[10] G Chicco and P Mancarella ldquoDistributed multi-generationa comprehensive viewrdquo Renewable and Sustainable EnergyReviews vol 13 no 3 pp 535ndash551 2009

[11] G Abdollahi and M Meratizaman ldquoMulti-objective approachin thermoenvironomic optimization of a small-scale distributedCCHP system with risk analysisrdquo Energy and Buildings vol 43no 11 pp 3144ndash3153 2011

[12] D Xia S Song J Wang J Shi H Bi and Z Gao ldquoDetermi-nation of corrosion types from electrochemical noise by phasespace reconstruction theoryrdquo Electrochemistry Communica-tions vol 15 no 1 pp 88ndash92 2012

[13] Z Xie and K Wang ldquoSelection of embedding parameters inphase space reconstructionrdquo in Proceedings of the 2nd Inter-national Conference on Intelligent Computing Technology andAutomation (ICICTA rsquo09) pp 637ndash640 IEEE Hunan ChinaOctober 2009

[14] W-D Cai Y-Q Qin and B-R Yang ldquoSelection of delay timewindow and delay time in phase space reconstructionrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 526ndash530 December 2007

[15] Y-B Li Y Song and C-H Li ldquoSelection of parameters forphase space reconstruction of chaotic time seriesrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 30ndash33IEEE Changsha China September 2010

[16] J HerbertTutorial on Training Recurrent Neural Networks Cov-ering BPPT RTRL EKF and the ldquoEcho State Networkrdquo ApproachGMD-Forschungszentrum Informationstechnik Bremen Ger-many 2002

[17] H Jaeger M Lukosevicius D Popovici and U Siewert ldquoOpti-mization and applications of echo state networks with leaky-integrator neuronsrdquoNeural Networks vol 20 no 3 pp 335ndash3522007

[18] K A Bredahl and T Timo Forecasting with Nonlinear TimeSeries Models School of Economics and Management AarhusUniversity Aarhus Denmark 2011

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DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 3

The chemical reaction is the process in which biomassgas is converted to electrical energy in the biomass gasgenerating systemThe available solid biomass resources withlow calorific value are convertible to gas Small electricitygenerators are powered by biogas or natural gas As onebioenergy technology biomass gas generating technology canprovide clean energyThe biomass gas generating system con-sumes large amounts of solid reject and generates enormousamounts of thermal and electrical energy The waste heatof the generating unit is used to drive the lithium bromideabsorption refrigeration cycle and the plate heat exchanger

The heat exchanging processes are physical reactionsTherefrigerant vapour is generated in evaporator of the lithiumbromide absorption refrigerator Then the lithium bromidesolution will become dilute because amount of vapour isabsorbed by the solution in absorber The dilute solution issent back to the heat exchanger by the circulating pumpThesolution is concentrated and further heated by steam or hotwater in the generatorThe temperature of cool dilute solutionrises through heat exchanger between the generator and theabsorber The metal plates are used to transfer heat betweentwo fluids in a plate heat exchanger The lithium bromidesolution and refrigerant water are fed into tubes of the plateheat exchangersThe system also includesmonitoring systemenergy management system and control system The safeand stable running of the distributed generation system isensured

The powers of different subsystems are all subject to thefluctuation of the external environmental state meaning therenewable energy is intermittent The efficient control is aserious challenge in long cycle security and steady runningTherefore it is very necessary for the prediction research ofthe renewable energy system The real information can besupplied to distribution system if applicable The short-termpower forecasting can improve system operating efficiencyand flexibility

3 Phase Space Reconstruction ofChaotic Time Series

Generally only a single scalar time series can be measuredfroma real physical system So the phase space reconstructionfrom time series is the first stage in nonlinear time seriesanalysis of data from chaotic systems [12] All possible statevariables cannot usually be obtained by the observation ofa real process Either not all state variables are measured ornot all of them can be known However due to the couplingsbetween the components of a chaotic system we can recon-struct a phase space from a single observation by a time delayembedding [13]

Supposing that a continuous time dynamical system isdescribed as follows

119889 (119909)

119889119905= 119865 (119909 (119905)) 119909 isin 119877

119899

(1)

where 119909 is the state variable and 119865 is the kinetic equationof the system the continuous system will become a discretesystem after discretization

119909 (119905 + 1) = 119891 (119909 (119905)) 119905 = 0 1 119877119899

997888rarr 119877119898

(2)

where 119909 is discrete state variable and 119891 is discrete kineticequation According to Takenrsquos embedding theorem somestate variants of the original system can be analyzed byobserving the system state 119909(119905)Themost common techniqueis time delay method for phase space reconstruction A scalartime series 119909

119894(119905) where 119894 = 1 2 119873 can be embedded into

an 119898-dimensional space to represent the dynamic system asfollows

119883 (119905) = 119909 (119905) 119909 (119905 minus 120591) 119909 (119905 minus (119898 minus 1) 120591) (3)

where 119905 = 1 2 119898 119898 is embedding dimension and 120591 isthe delay time

The phase space reconstruction system is an orthogonalprojection from 119901-dimensional space to119898-dimensional sub-space The inherent evolution law of the dynamic system canbe described in the form of an119898-dimensional mapping Thelinear smooth mapping119872 is defined as follows

119883 (119905 + ℎ) = 119872 119909 (119905) 119909 (119905 minus 120591) 119909 [119905 minus (119898 minus 1) 120591] (4)

where ℎ is the prediction time domainThe forecasting model is derived from the expression of

the mapping equation (see (4)) But it is difficult to introducethe analytical expression in practice The forecasting modelbased on neural network can be trained by training samplessuch as [119883(119905) 119883(119905 + ℎ)] The neural network model canapproximate the linear smooth mapping 119865 adequately

The delay time and embedding dimension play an impor-tant role in the phase space reconstruction according to ascalar time series However it is often extremely difficult tochoose the parameters such as embedding dimension anddelay time for phase space reconstruction Normally thedelay time and embedding dimension are chosen indepen-dently However in recent years some researchers contendthat the delay time and embedding dimension are interre-lated that is the delay time window should be estimatedfirstly for the choice of delay time and embedding dimension[14] The delay time window can be ascertained using C-C method However the exact selection of delay time canonly be practicable by conventional C-C method This papertries to use the improved C-C method to select embeddingwindow [15]The embedding window is first estimated basedon the mean correlation integral Then the delay time canbe chosen according to the conventional C-C method Theembedding dimension can be obtained according to the rela-tion between embedding window and delay time Owing tothe mean correlation integral with a clear physical meaningthe optimal embedding window can be estimated objectivelyIn accordance with the above method the parameters ofphase space reconstruction can be estimated accurately

4 Echo State Network

Echo state network (ESN) is a new-style recurrent neuralnetwork with a sparsely connected hidden layer [16 17]The weights and connectivity of hidden layer are randomlyassigned and are fixed The weights of output layer canbe learned from training samples The discrete-time neuralnetwork with 119870 input units119873 internal network units and 119871

4 Journal of Control Science and Engineering

x(t)

x(t minus (m minus 1)120591) x(t + 120591)

Figure 3 Architecture of echo state network

output units is governed by the following state updatingequation

119909 (119899 + 1) = 119891 (119882119909 (119899) + 119882in119906 (119899 + 1) + 119882119891119887119910 (119899)) (5)

where 119906(119899) 119909(119899) and 119910(119899) denote the activations of inputunits internal units and output units at time step 119899 respec-tively Furthermore 119906(119899) is a 119870-dimensional input signal119906(119899) = (119906

1(119899) 119906

119870(119899)) 119909(119899) is a 119873-dimensional internal

state 119909(119899) = (1199091(119899) 119909

119873(119899)) and 119910(119899) is a 119871-dimensional

output signal 119910(119899) = (1199101(119899) 119910

119871(119899)) 119891 is a sigmoid func-

tion (usually the logistic sigmoid or the tanh function)119882 isthe reservoir weight matrix 119882in is the input weight matrixand119882

119891119887is the output feedback weight matrix The output of

the neural network can be obtained according to the follow-ing equation

119910 (119899 + 1) = 119891119900(119882119900(119909 (119899 + 1))) (6)

where 119891119900is the activation functions of output units 119909(119899 + 1)

and 119910(119899+1) are the internal activation vectors and the outputactivation vectors at time step 119899 + 1 respectively and119882

119900is

the output weight matrix Figure 3 shows the basic networkarchitecture

The training of echo state network can be done in twostages sampling and weight computation During the sam-pling stage of the training the echo state network is drivenby the input sequence 119906(119899) which yields the internal states119909(119899)The outputs of echo state network can be obtained usingthe system equations (5)-(6) The desired outputs 119889(119899) arewritten into the output units and pumped into the internalstates through the back projection connections The internalstates 119909(119899) are collected into a state collectionmatrixM Like-wise the teacher outputs 119889(119899) are collected into the rows of ateacher output collection matrix T The teacher signal series119889(119899) is approximated as a linear combination of the internalstates 119909

119894(119899) by the following equation

119889 (119899) asymp 119910 (119899) =

119899119900

sum

119894=1

119908119900

119894119909119894(119899) (7)

where 119899119900is the number of the output weights and 119908119900

119894is the

output weight corresponding to internal state 119909119894(119899)

The mean squared training error is defined as

MSE = 1

119873

119873

sum

119899=1

(119889 (119899) minus

119899119900

sum

119894=1

119908119900

119894119909119894(119899))

2

(8)

where 119873 is the time steps for which the teacher signal iswritten into the output unitThe output weightmatrix119882

119900can

be computed such that the mean squared training error MSEis minimized Therefore the optimization of output weightmatrix is to minimize the objective function to make neuralnetwork approximate the mapping from input to output Thecomputation of output weight matrix is a linear regressionproblemThe solution of output weight matrix is ill-posed bylinear regression algorithmThedesired output weightmatrixwhich minimizes MSE can be obtained by multiplying thepseudoinverse ofM with T

119882119900= 119872minus1

119879 (9)

where M is the state collection matrix and T is the teacheroutput collection matrix Then the output weight matrix canbe obtained based on the ready-made functions inMathemat-ica or Matlab The optimal output weight matrix with mini-mized training error is implemented in echo state networkNow the neural network predictor is ready for forecasting ofthe input-output time series

5 Forecasting Algorithm Based onEcho State Network

Nonlinear time series forecasting is an important portion ofcurrent science and technology [18] Generally it is very diffi-cult to obtain the mathematical models of complex dynamicsystems The system parameters such as input output andinternal states are only measured The univariate time seriesof a special parameter of the system can be established Theprediction of time series is very often generated by nonlinearforecastingmodelsThe very common forecastingmodels arebased on the chaos models or neural network models

The forecasting methods of nonlinear time series includeindirect forecasting and direct forecasting The method ofsingle-step forecasting has higher accuracy in practical appli-cationsThe direct multisteps forecasting model is concernedwith the estimation of the system output at some time stepsbased on the mapping of input and output There is noaccumulation error for direct forecasting algorithms withoutthe feedback of errors However with the increase of timesteps the forecastingmodelwill bemore complexThe single-step forecasting model can carry out indirect multistepsforecasting through iterations But the error of forecastingmodel accumulates along with time The method of directmultisteps forecasting is applied to forecast the time series ofdistributed generation system power in this paper A detaileddescription of each implement step of the forecasting algo-rithm is as follows

(1) The chaotic analysis calculate the value of the largestLyapunov exponent to check whether the scalar time

Journal of Control Science and Engineering 5

series is chaotic Positive largest Lyapunov exponentshows the dynamic system is chaotic

(2) The selection of system parameters the embeddingdimension 119898 and the delay time 120591 are chosen basedon the improved C-C method

(3) Phase space reconstruction reconstruct an119898-dimen-sional phase space from a single observation by atime delay embedding based on the delay time andembedding dimension

(4) The training samples the input vectors are 119898(119896)where 119896 is the number of training samplesThe input-output pairs 119898(119896) 119883(119896+ℎ) are the training sampleswhich are obtained according to the prediction timedomain ℎ

(5) The initialization of the forecasting model the fore-casting model based on ESN is initialized accordingto the learning algorithm of ESN The output weightsmatrix is calculated based on (9)

(6) Prediction of the observation the prediction value ofenergy in time domain ℎ can be obtained based on thereal time observation of the system

6 Simulation and Analysis

The equations of the distributed generation system are nor-mally unknown The interaction among the subsystems canbe analyzed based on the dynamics system theoryThe systemlaw in the multidimensional phase space will be investigatedbased on the evolution track of the maximum power Duringthe operation of the distributed generation system the maxi-mum power was recorded every day We gained a rich supplyof data which would normally be accessible In this paperwe collected 1000 data points including 800 training datasetsand 200 test data sequences The data collection systemconsists of two parts data acquisition and normalization pro-cessing The process of data normalization can suppress theelectromagnetic interferences and improve the generalizationability of neural network model The method of normaliza-tion processing is provided The equation of linear transfor-mation can be represented as shown below

119875119894=

119875max minus 119875119894119875max minus 119875min

(119894 = 1 2 ) (10)

where 119875max is the maximum of the power sequence 119875min isthe minimum of the power sequence and 119875

119894is the maximum

power on the 119894th day

61 The Choice of Experimental Parameters The problem ofdetecting and quantifying chaos is a key step for the intensivestudies of nonlinear systems The Lyapunov exponents arethe important indexes to quantify the sensitivity on initialconditions for a dynamical system If the dynamical systemis chaotic at least one Lyapunov exponent must be positiveTherefore we only need to calculate the largest Lyapunovexponent to judge the chaotic characters The largest Lya-punov exponent characterizes the rate exponential diver-gence of the nearby trajectories in the reconstructed phase

0 50 100 150 200 250 3000

50

100

150

0 50 100 150 200 250 300

0

50

100

Slop

ey(i

)

minus50

n

i

Figure 4 The largest Lyapunov exponent

0 0204

06 081

0

05

10

02

04

06

08

1

Maximum power (i)

Maximum power (i minus 10)

Max

imum

pow

er (i

minus20

)

Figure 5 The attractor of chaotic power time series

space There are also lots of algorithms to estimate the largestLyapunov exponent of an experimental time series In thispaper the largest Lyapunov exponent was calculated withsmall amount of data The estimate of the largest Lyapunovexponent is 013 bits (shown in Figure 4)

The embedding dimension 119898 and the delay time 120591 arechosen based on the improved C-C methodThen we obtainembedding dimension 119898 = 10 and delay time 120591 = 8 Thephase space is reconstructed based on the embedding dimen-sion and the delay time The attractor of chaotic power timeseries is shown in Figure 5 In practice there will always besome form of corrupting noise Thus any real time serieseven if mostly deterministic will contain some randomnessThe evolution of the distributed generation system beingchaotic means that every point in the reconstructed spaceis approached arbitrarily but not coincident on the pseudoperiodic orbits

62 The Analysis of Simulation Results The neural networkmodel is trained based on the training samples The teacher

6 Journal of Control Science and Engineering

0 50 100 150 20001

02

03

04

05

06

07

08

09

1

11

Date serial

Max

imum

pow

er (k

W)

Observed powerPredicted power (kW)

Figure 6 The estimated output and observations

signal is written into the output unit for times 119899 = 1 800The internal states 119909(119899) = (119909

1(119899) 119909

20(119899)) for 119899 =

1 800 are collected as a row of a state-collectingmatrixMof size 800times 20 during the sampling period At the same timethe teacher outputs 119889(119899) are collected into a row of a matrixT of size 800 times 1 Then we can compute output weights 119908

119900

for linear output unit 119910(119899) such that the teacher time series119889(119899) is approximated as a linear combination of the internalactivation time series 119909

119894(119899) by (9) The multistep prediction

can be obtained with testing samples The predictor canforecast multistep value directly with different time domain

The result of direct prediction based on echo statenetwork for the maximum power series of the distributedgeneration system is shown in Figure 6The solid line denotesthe predicted value the dotted line is for the observed valueWe observe that the prediction model based on echo statenetwork has high accuracy in direct prediction owing to thecoincidence of prediction and actual curve The max predic-tion error is about 017 (shown in Figure 7) Figure 8 is theerror of the traditional multilayer perceptron The predictorbased on echo state network has higher accuracy

7 Conclusion

Themicroenvironment meteorological data are coupled withthe output power of the distributed generation systemTheoretical proof and experimental results show that theoutput power of the distributed generation system is chaotictime seriesTheprediction of chaotic time series is to establisha nonlinear model of the dynamic system based on real timeseries in the reconstructed phase space Firstly we shoulddetect and quantify chaos for the intensive studies of nonlin-ear systems If the largest Lyapunov exponent is positive thedynamical system must be chaotic Secondly the embeddingdimension and the delay time are chosen based on theimproved C-C method The attractor of chaotic power time

0 50 100 150 200

0

005

01

015

Date serial

Erro

r

minus01

minus015

minus02

minus005

Figure 7 The error of the predictions

0 50 100 150 200

0

005

01

015

02

Date serial

Erro

r

minus01

minus015

minus005

Figure 8 The error of the traditional multilayer perceptron

series can be reconstructed based on the embedding dimen-sion and delay time in the phase space Lastly the neural net-work model is trained based on the training samples whichare obtained from the distributed generation system Theneural network will approximate the curve of output poweradequately The maximum power point of the distributedgeneration system will be tracked based on the meteorologi-cal data

Themeasuring errors are unavoidable among experimen-tal data due to the disturbance and many other factors Theprediction precision of the neural network model is largelydependent on the generalization of neural network There-fore we should optimize the structure and parameters ofthe model for better performance Experimental results showthat the prediction model based on echo state network hashigh accuracyThe chaotic predictionmodel can approximatethe mapping between the maximum power point of thedistributed generation system and the meteorological dataThe echo state network model can make rather remarkably

Journal of Control Science and Engineering 7

accurate predictions about the maximum power of thedistributed generation system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 61473174 61105100 and 51376110)the Specialized Research Fund for the Doctoral Program ofHigher Education (no 20130131130006) Project funded byChina Postdoctoral Science Foundation (no 2014M551907)and Independent Innovation Foundation of Shandong Uni-versity (no 2013ZRQP002)

References

[1] G Angrisani C Roselli and M Sasso ldquoDistributed microtri-generation systemsrdquoProgress in Energy andCombustion Sciencevol 38 no 4 pp 502ndash521 2012

[2] K A Pruitt S Leyffer AMNewman et alOptimal Design andDispatch of Distributed Generation Systems Argonne NationalLaboratory Chicago Ill USA 2012

[3] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[4] Y Song Y Li Q Wang and C Li ldquoMulti-steps prediction ofchaotic time series based on echo state networkrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 669ndash672 IEEE Changsha China September 2010

[5] S Michanos A Tsakoumis P Fessas S Vladov and V Mlade-nov ldquoShort-term load forecasting using a chaotic time seriesrdquo inProceedings of the International Symposium on Signals Circuitsand Systems (SCS rsquo03) pp 437ndash440 Iasi Romania 2003

[6] X An D Jiang M Zhao and C Liu ldquoShort-term prediction ofwind power using EMD and chaotic theoryrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2pp 1036ndash1042 2012

[7] C Qin C-C Chang and L-T Liao ldquoAn adaptive prediction-error expansion oriented reversible information hidingschemerdquo Pattern Recognition Letters vol 33 no 16 pp2166ndash2172 2012

[8] J Zhang K C Lam W J Yan H Gao and Y Li ldquoTime seriesprediction using Lyapunov exponents in embedding phasespacerdquo Computers amp Electrical Engineering vol 30 no 1 pp 1ndash15 2004

[9] J M Guerrero F Blaabjerg T Zhelev et al ldquoDistributedgeneration toward a new energy paradigmrdquo IEEE IndustrialElectronics Magazine vol 4 no 1 pp 52ndash64 2010

[10] G Chicco and P Mancarella ldquoDistributed multi-generationa comprehensive viewrdquo Renewable and Sustainable EnergyReviews vol 13 no 3 pp 535ndash551 2009

[11] G Abdollahi and M Meratizaman ldquoMulti-objective approachin thermoenvironomic optimization of a small-scale distributedCCHP system with risk analysisrdquo Energy and Buildings vol 43no 11 pp 3144ndash3153 2011

[12] D Xia S Song J Wang J Shi H Bi and Z Gao ldquoDetermi-nation of corrosion types from electrochemical noise by phasespace reconstruction theoryrdquo Electrochemistry Communica-tions vol 15 no 1 pp 88ndash92 2012

[13] Z Xie and K Wang ldquoSelection of embedding parameters inphase space reconstructionrdquo in Proceedings of the 2nd Inter-national Conference on Intelligent Computing Technology andAutomation (ICICTA rsquo09) pp 637ndash640 IEEE Hunan ChinaOctober 2009

[14] W-D Cai Y-Q Qin and B-R Yang ldquoSelection of delay timewindow and delay time in phase space reconstructionrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 526ndash530 December 2007

[15] Y-B Li Y Song and C-H Li ldquoSelection of parameters forphase space reconstruction of chaotic time seriesrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 30ndash33IEEE Changsha China September 2010

[16] J HerbertTutorial on Training Recurrent Neural Networks Cov-ering BPPT RTRL EKF and the ldquoEcho State Networkrdquo ApproachGMD-Forschungszentrum Informationstechnik Bremen Ger-many 2002

[17] H Jaeger M Lukosevicius D Popovici and U Siewert ldquoOpti-mization and applications of echo state networks with leaky-integrator neuronsrdquoNeural Networks vol 20 no 3 pp 335ndash3522007

[18] K A Bredahl and T Timo Forecasting with Nonlinear TimeSeries Models School of Economics and Management AarhusUniversity Aarhus Denmark 2011

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Propagation

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Navigation and Observation

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DistributedSensor Networks

International Journal of

4 Journal of Control Science and Engineering

x(t)

x(t minus (m minus 1)120591) x(t + 120591)

Figure 3 Architecture of echo state network

output units is governed by the following state updatingequation

119909 (119899 + 1) = 119891 (119882119909 (119899) + 119882in119906 (119899 + 1) + 119882119891119887119910 (119899)) (5)

where 119906(119899) 119909(119899) and 119910(119899) denote the activations of inputunits internal units and output units at time step 119899 respec-tively Furthermore 119906(119899) is a 119870-dimensional input signal119906(119899) = (119906

1(119899) 119906

119870(119899)) 119909(119899) is a 119873-dimensional internal

state 119909(119899) = (1199091(119899) 119909

119873(119899)) and 119910(119899) is a 119871-dimensional

output signal 119910(119899) = (1199101(119899) 119910

119871(119899)) 119891 is a sigmoid func-

tion (usually the logistic sigmoid or the tanh function)119882 isthe reservoir weight matrix 119882in is the input weight matrixand119882

119891119887is the output feedback weight matrix The output of

the neural network can be obtained according to the follow-ing equation

119910 (119899 + 1) = 119891119900(119882119900(119909 (119899 + 1))) (6)

where 119891119900is the activation functions of output units 119909(119899 + 1)

and 119910(119899+1) are the internal activation vectors and the outputactivation vectors at time step 119899 + 1 respectively and119882

119900is

the output weight matrix Figure 3 shows the basic networkarchitecture

The training of echo state network can be done in twostages sampling and weight computation During the sam-pling stage of the training the echo state network is drivenby the input sequence 119906(119899) which yields the internal states119909(119899)The outputs of echo state network can be obtained usingthe system equations (5)-(6) The desired outputs 119889(119899) arewritten into the output units and pumped into the internalstates through the back projection connections The internalstates 119909(119899) are collected into a state collectionmatrixM Like-wise the teacher outputs 119889(119899) are collected into the rows of ateacher output collection matrix T The teacher signal series119889(119899) is approximated as a linear combination of the internalstates 119909

119894(119899) by the following equation

119889 (119899) asymp 119910 (119899) =

119899119900

sum

119894=1

119908119900

119894119909119894(119899) (7)

where 119899119900is the number of the output weights and 119908119900

119894is the

output weight corresponding to internal state 119909119894(119899)

The mean squared training error is defined as

MSE = 1

119873

119873

sum

119899=1

(119889 (119899) minus

119899119900

sum

119894=1

119908119900

119894119909119894(119899))

2

(8)

where 119873 is the time steps for which the teacher signal iswritten into the output unitThe output weightmatrix119882

119900can

be computed such that the mean squared training error MSEis minimized Therefore the optimization of output weightmatrix is to minimize the objective function to make neuralnetwork approximate the mapping from input to output Thecomputation of output weight matrix is a linear regressionproblemThe solution of output weight matrix is ill-posed bylinear regression algorithmThedesired output weightmatrixwhich minimizes MSE can be obtained by multiplying thepseudoinverse ofM with T

119882119900= 119872minus1

119879 (9)

where M is the state collection matrix and T is the teacheroutput collection matrix Then the output weight matrix canbe obtained based on the ready-made functions inMathemat-ica or Matlab The optimal output weight matrix with mini-mized training error is implemented in echo state networkNow the neural network predictor is ready for forecasting ofthe input-output time series

5 Forecasting Algorithm Based onEcho State Network

Nonlinear time series forecasting is an important portion ofcurrent science and technology [18] Generally it is very diffi-cult to obtain the mathematical models of complex dynamicsystems The system parameters such as input output andinternal states are only measured The univariate time seriesof a special parameter of the system can be established Theprediction of time series is very often generated by nonlinearforecastingmodelsThe very common forecastingmodels arebased on the chaos models or neural network models

The forecasting methods of nonlinear time series includeindirect forecasting and direct forecasting The method ofsingle-step forecasting has higher accuracy in practical appli-cationsThe direct multisteps forecasting model is concernedwith the estimation of the system output at some time stepsbased on the mapping of input and output There is noaccumulation error for direct forecasting algorithms withoutthe feedback of errors However with the increase of timesteps the forecastingmodelwill bemore complexThe single-step forecasting model can carry out indirect multistepsforecasting through iterations But the error of forecastingmodel accumulates along with time The method of directmultisteps forecasting is applied to forecast the time series ofdistributed generation system power in this paper A detaileddescription of each implement step of the forecasting algo-rithm is as follows

(1) The chaotic analysis calculate the value of the largestLyapunov exponent to check whether the scalar time

Journal of Control Science and Engineering 5

series is chaotic Positive largest Lyapunov exponentshows the dynamic system is chaotic

(2) The selection of system parameters the embeddingdimension 119898 and the delay time 120591 are chosen basedon the improved C-C method

(3) Phase space reconstruction reconstruct an119898-dimen-sional phase space from a single observation by atime delay embedding based on the delay time andembedding dimension

(4) The training samples the input vectors are 119898(119896)where 119896 is the number of training samplesThe input-output pairs 119898(119896) 119883(119896+ℎ) are the training sampleswhich are obtained according to the prediction timedomain ℎ

(5) The initialization of the forecasting model the fore-casting model based on ESN is initialized accordingto the learning algorithm of ESN The output weightsmatrix is calculated based on (9)

(6) Prediction of the observation the prediction value ofenergy in time domain ℎ can be obtained based on thereal time observation of the system

6 Simulation and Analysis

The equations of the distributed generation system are nor-mally unknown The interaction among the subsystems canbe analyzed based on the dynamics system theoryThe systemlaw in the multidimensional phase space will be investigatedbased on the evolution track of the maximum power Duringthe operation of the distributed generation system the maxi-mum power was recorded every day We gained a rich supplyof data which would normally be accessible In this paperwe collected 1000 data points including 800 training datasetsand 200 test data sequences The data collection systemconsists of two parts data acquisition and normalization pro-cessing The process of data normalization can suppress theelectromagnetic interferences and improve the generalizationability of neural network model The method of normaliza-tion processing is provided The equation of linear transfor-mation can be represented as shown below

119875119894=

119875max minus 119875119894119875max minus 119875min

(119894 = 1 2 ) (10)

where 119875max is the maximum of the power sequence 119875min isthe minimum of the power sequence and 119875

119894is the maximum

power on the 119894th day

61 The Choice of Experimental Parameters The problem ofdetecting and quantifying chaos is a key step for the intensivestudies of nonlinear systems The Lyapunov exponents arethe important indexes to quantify the sensitivity on initialconditions for a dynamical system If the dynamical systemis chaotic at least one Lyapunov exponent must be positiveTherefore we only need to calculate the largest Lyapunovexponent to judge the chaotic characters The largest Lya-punov exponent characterizes the rate exponential diver-gence of the nearby trajectories in the reconstructed phase

0 50 100 150 200 250 3000

50

100

150

0 50 100 150 200 250 300

0

50

100

Slop

ey(i

)

minus50

n

i

Figure 4 The largest Lyapunov exponent

0 0204

06 081

0

05

10

02

04

06

08

1

Maximum power (i)

Maximum power (i minus 10)

Max

imum

pow

er (i

minus20

)

Figure 5 The attractor of chaotic power time series

space There are also lots of algorithms to estimate the largestLyapunov exponent of an experimental time series In thispaper the largest Lyapunov exponent was calculated withsmall amount of data The estimate of the largest Lyapunovexponent is 013 bits (shown in Figure 4)

The embedding dimension 119898 and the delay time 120591 arechosen based on the improved C-C methodThen we obtainembedding dimension 119898 = 10 and delay time 120591 = 8 Thephase space is reconstructed based on the embedding dimen-sion and the delay time The attractor of chaotic power timeseries is shown in Figure 5 In practice there will always besome form of corrupting noise Thus any real time serieseven if mostly deterministic will contain some randomnessThe evolution of the distributed generation system beingchaotic means that every point in the reconstructed spaceis approached arbitrarily but not coincident on the pseudoperiodic orbits

62 The Analysis of Simulation Results The neural networkmodel is trained based on the training samples The teacher

6 Journal of Control Science and Engineering

0 50 100 150 20001

02

03

04

05

06

07

08

09

1

11

Date serial

Max

imum

pow

er (k

W)

Observed powerPredicted power (kW)

Figure 6 The estimated output and observations

signal is written into the output unit for times 119899 = 1 800The internal states 119909(119899) = (119909

1(119899) 119909

20(119899)) for 119899 =

1 800 are collected as a row of a state-collectingmatrixMof size 800times 20 during the sampling period At the same timethe teacher outputs 119889(119899) are collected into a row of a matrixT of size 800 times 1 Then we can compute output weights 119908

119900

for linear output unit 119910(119899) such that the teacher time series119889(119899) is approximated as a linear combination of the internalactivation time series 119909

119894(119899) by (9) The multistep prediction

can be obtained with testing samples The predictor canforecast multistep value directly with different time domain

The result of direct prediction based on echo statenetwork for the maximum power series of the distributedgeneration system is shown in Figure 6The solid line denotesthe predicted value the dotted line is for the observed valueWe observe that the prediction model based on echo statenetwork has high accuracy in direct prediction owing to thecoincidence of prediction and actual curve The max predic-tion error is about 017 (shown in Figure 7) Figure 8 is theerror of the traditional multilayer perceptron The predictorbased on echo state network has higher accuracy

7 Conclusion

Themicroenvironment meteorological data are coupled withthe output power of the distributed generation systemTheoretical proof and experimental results show that theoutput power of the distributed generation system is chaotictime seriesTheprediction of chaotic time series is to establisha nonlinear model of the dynamic system based on real timeseries in the reconstructed phase space Firstly we shoulddetect and quantify chaos for the intensive studies of nonlin-ear systems If the largest Lyapunov exponent is positive thedynamical system must be chaotic Secondly the embeddingdimension and the delay time are chosen based on theimproved C-C method The attractor of chaotic power time

0 50 100 150 200

0

005

01

015

Date serial

Erro

r

minus01

minus015

minus02

minus005

Figure 7 The error of the predictions

0 50 100 150 200

0

005

01

015

02

Date serial

Erro

r

minus01

minus015

minus005

Figure 8 The error of the traditional multilayer perceptron

series can be reconstructed based on the embedding dimen-sion and delay time in the phase space Lastly the neural net-work model is trained based on the training samples whichare obtained from the distributed generation system Theneural network will approximate the curve of output poweradequately The maximum power point of the distributedgeneration system will be tracked based on the meteorologi-cal data

Themeasuring errors are unavoidable among experimen-tal data due to the disturbance and many other factors Theprediction precision of the neural network model is largelydependent on the generalization of neural network There-fore we should optimize the structure and parameters ofthe model for better performance Experimental results showthat the prediction model based on echo state network hashigh accuracyThe chaotic predictionmodel can approximatethe mapping between the maximum power point of thedistributed generation system and the meteorological dataThe echo state network model can make rather remarkably

Journal of Control Science and Engineering 7

accurate predictions about the maximum power of thedistributed generation system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 61473174 61105100 and 51376110)the Specialized Research Fund for the Doctoral Program ofHigher Education (no 20130131130006) Project funded byChina Postdoctoral Science Foundation (no 2014M551907)and Independent Innovation Foundation of Shandong Uni-versity (no 2013ZRQP002)

References

[1] G Angrisani C Roselli and M Sasso ldquoDistributed microtri-generation systemsrdquoProgress in Energy andCombustion Sciencevol 38 no 4 pp 502ndash521 2012

[2] K A Pruitt S Leyffer AMNewman et alOptimal Design andDispatch of Distributed Generation Systems Argonne NationalLaboratory Chicago Ill USA 2012

[3] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[4] Y Song Y Li Q Wang and C Li ldquoMulti-steps prediction ofchaotic time series based on echo state networkrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 669ndash672 IEEE Changsha China September 2010

[5] S Michanos A Tsakoumis P Fessas S Vladov and V Mlade-nov ldquoShort-term load forecasting using a chaotic time seriesrdquo inProceedings of the International Symposium on Signals Circuitsand Systems (SCS rsquo03) pp 437ndash440 Iasi Romania 2003

[6] X An D Jiang M Zhao and C Liu ldquoShort-term prediction ofwind power using EMD and chaotic theoryrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2pp 1036ndash1042 2012

[7] C Qin C-C Chang and L-T Liao ldquoAn adaptive prediction-error expansion oriented reversible information hidingschemerdquo Pattern Recognition Letters vol 33 no 16 pp2166ndash2172 2012

[8] J Zhang K C Lam W J Yan H Gao and Y Li ldquoTime seriesprediction using Lyapunov exponents in embedding phasespacerdquo Computers amp Electrical Engineering vol 30 no 1 pp 1ndash15 2004

[9] J M Guerrero F Blaabjerg T Zhelev et al ldquoDistributedgeneration toward a new energy paradigmrdquo IEEE IndustrialElectronics Magazine vol 4 no 1 pp 52ndash64 2010

[10] G Chicco and P Mancarella ldquoDistributed multi-generationa comprehensive viewrdquo Renewable and Sustainable EnergyReviews vol 13 no 3 pp 535ndash551 2009

[11] G Abdollahi and M Meratizaman ldquoMulti-objective approachin thermoenvironomic optimization of a small-scale distributedCCHP system with risk analysisrdquo Energy and Buildings vol 43no 11 pp 3144ndash3153 2011

[12] D Xia S Song J Wang J Shi H Bi and Z Gao ldquoDetermi-nation of corrosion types from electrochemical noise by phasespace reconstruction theoryrdquo Electrochemistry Communica-tions vol 15 no 1 pp 88ndash92 2012

[13] Z Xie and K Wang ldquoSelection of embedding parameters inphase space reconstructionrdquo in Proceedings of the 2nd Inter-national Conference on Intelligent Computing Technology andAutomation (ICICTA rsquo09) pp 637ndash640 IEEE Hunan ChinaOctober 2009

[14] W-D Cai Y-Q Qin and B-R Yang ldquoSelection of delay timewindow and delay time in phase space reconstructionrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 526ndash530 December 2007

[15] Y-B Li Y Song and C-H Li ldquoSelection of parameters forphase space reconstruction of chaotic time seriesrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 30ndash33IEEE Changsha China September 2010

[16] J HerbertTutorial on Training Recurrent Neural Networks Cov-ering BPPT RTRL EKF and the ldquoEcho State Networkrdquo ApproachGMD-Forschungszentrum Informationstechnik Bremen Ger-many 2002

[17] H Jaeger M Lukosevicius D Popovici and U Siewert ldquoOpti-mization and applications of echo state networks with leaky-integrator neuronsrdquoNeural Networks vol 20 no 3 pp 335ndash3522007

[18] K A Bredahl and T Timo Forecasting with Nonlinear TimeSeries Models School of Economics and Management AarhusUniversity Aarhus Denmark 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 5

series is chaotic Positive largest Lyapunov exponentshows the dynamic system is chaotic

(2) The selection of system parameters the embeddingdimension 119898 and the delay time 120591 are chosen basedon the improved C-C method

(3) Phase space reconstruction reconstruct an119898-dimen-sional phase space from a single observation by atime delay embedding based on the delay time andembedding dimension

(4) The training samples the input vectors are 119898(119896)where 119896 is the number of training samplesThe input-output pairs 119898(119896) 119883(119896+ℎ) are the training sampleswhich are obtained according to the prediction timedomain ℎ

(5) The initialization of the forecasting model the fore-casting model based on ESN is initialized accordingto the learning algorithm of ESN The output weightsmatrix is calculated based on (9)

(6) Prediction of the observation the prediction value ofenergy in time domain ℎ can be obtained based on thereal time observation of the system

6 Simulation and Analysis

The equations of the distributed generation system are nor-mally unknown The interaction among the subsystems canbe analyzed based on the dynamics system theoryThe systemlaw in the multidimensional phase space will be investigatedbased on the evolution track of the maximum power Duringthe operation of the distributed generation system the maxi-mum power was recorded every day We gained a rich supplyof data which would normally be accessible In this paperwe collected 1000 data points including 800 training datasetsand 200 test data sequences The data collection systemconsists of two parts data acquisition and normalization pro-cessing The process of data normalization can suppress theelectromagnetic interferences and improve the generalizationability of neural network model The method of normaliza-tion processing is provided The equation of linear transfor-mation can be represented as shown below

119875119894=

119875max minus 119875119894119875max minus 119875min

(119894 = 1 2 ) (10)

where 119875max is the maximum of the power sequence 119875min isthe minimum of the power sequence and 119875

119894is the maximum

power on the 119894th day

61 The Choice of Experimental Parameters The problem ofdetecting and quantifying chaos is a key step for the intensivestudies of nonlinear systems The Lyapunov exponents arethe important indexes to quantify the sensitivity on initialconditions for a dynamical system If the dynamical systemis chaotic at least one Lyapunov exponent must be positiveTherefore we only need to calculate the largest Lyapunovexponent to judge the chaotic characters The largest Lya-punov exponent characterizes the rate exponential diver-gence of the nearby trajectories in the reconstructed phase

0 50 100 150 200 250 3000

50

100

150

0 50 100 150 200 250 300

0

50

100

Slop

ey(i

)

minus50

n

i

Figure 4 The largest Lyapunov exponent

0 0204

06 081

0

05

10

02

04

06

08

1

Maximum power (i)

Maximum power (i minus 10)

Max

imum

pow

er (i

minus20

)

Figure 5 The attractor of chaotic power time series

space There are also lots of algorithms to estimate the largestLyapunov exponent of an experimental time series In thispaper the largest Lyapunov exponent was calculated withsmall amount of data The estimate of the largest Lyapunovexponent is 013 bits (shown in Figure 4)

The embedding dimension 119898 and the delay time 120591 arechosen based on the improved C-C methodThen we obtainembedding dimension 119898 = 10 and delay time 120591 = 8 Thephase space is reconstructed based on the embedding dimen-sion and the delay time The attractor of chaotic power timeseries is shown in Figure 5 In practice there will always besome form of corrupting noise Thus any real time serieseven if mostly deterministic will contain some randomnessThe evolution of the distributed generation system beingchaotic means that every point in the reconstructed spaceis approached arbitrarily but not coincident on the pseudoperiodic orbits

62 The Analysis of Simulation Results The neural networkmodel is trained based on the training samples The teacher

6 Journal of Control Science and Engineering

0 50 100 150 20001

02

03

04

05

06

07

08

09

1

11

Date serial

Max

imum

pow

er (k

W)

Observed powerPredicted power (kW)

Figure 6 The estimated output and observations

signal is written into the output unit for times 119899 = 1 800The internal states 119909(119899) = (119909

1(119899) 119909

20(119899)) for 119899 =

1 800 are collected as a row of a state-collectingmatrixMof size 800times 20 during the sampling period At the same timethe teacher outputs 119889(119899) are collected into a row of a matrixT of size 800 times 1 Then we can compute output weights 119908

119900

for linear output unit 119910(119899) such that the teacher time series119889(119899) is approximated as a linear combination of the internalactivation time series 119909

119894(119899) by (9) The multistep prediction

can be obtained with testing samples The predictor canforecast multistep value directly with different time domain

The result of direct prediction based on echo statenetwork for the maximum power series of the distributedgeneration system is shown in Figure 6The solid line denotesthe predicted value the dotted line is for the observed valueWe observe that the prediction model based on echo statenetwork has high accuracy in direct prediction owing to thecoincidence of prediction and actual curve The max predic-tion error is about 017 (shown in Figure 7) Figure 8 is theerror of the traditional multilayer perceptron The predictorbased on echo state network has higher accuracy

7 Conclusion

Themicroenvironment meteorological data are coupled withthe output power of the distributed generation systemTheoretical proof and experimental results show that theoutput power of the distributed generation system is chaotictime seriesTheprediction of chaotic time series is to establisha nonlinear model of the dynamic system based on real timeseries in the reconstructed phase space Firstly we shoulddetect and quantify chaos for the intensive studies of nonlin-ear systems If the largest Lyapunov exponent is positive thedynamical system must be chaotic Secondly the embeddingdimension and the delay time are chosen based on theimproved C-C method The attractor of chaotic power time

0 50 100 150 200

0

005

01

015

Date serial

Erro

r

minus01

minus015

minus02

minus005

Figure 7 The error of the predictions

0 50 100 150 200

0

005

01

015

02

Date serial

Erro

r

minus01

minus015

minus005

Figure 8 The error of the traditional multilayer perceptron

series can be reconstructed based on the embedding dimen-sion and delay time in the phase space Lastly the neural net-work model is trained based on the training samples whichare obtained from the distributed generation system Theneural network will approximate the curve of output poweradequately The maximum power point of the distributedgeneration system will be tracked based on the meteorologi-cal data

Themeasuring errors are unavoidable among experimen-tal data due to the disturbance and many other factors Theprediction precision of the neural network model is largelydependent on the generalization of neural network There-fore we should optimize the structure and parameters ofthe model for better performance Experimental results showthat the prediction model based on echo state network hashigh accuracyThe chaotic predictionmodel can approximatethe mapping between the maximum power point of thedistributed generation system and the meteorological dataThe echo state network model can make rather remarkably

Journal of Control Science and Engineering 7

accurate predictions about the maximum power of thedistributed generation system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 61473174 61105100 and 51376110)the Specialized Research Fund for the Doctoral Program ofHigher Education (no 20130131130006) Project funded byChina Postdoctoral Science Foundation (no 2014M551907)and Independent Innovation Foundation of Shandong Uni-versity (no 2013ZRQP002)

References

[1] G Angrisani C Roselli and M Sasso ldquoDistributed microtri-generation systemsrdquoProgress in Energy andCombustion Sciencevol 38 no 4 pp 502ndash521 2012

[2] K A Pruitt S Leyffer AMNewman et alOptimal Design andDispatch of Distributed Generation Systems Argonne NationalLaboratory Chicago Ill USA 2012

[3] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[4] Y Song Y Li Q Wang and C Li ldquoMulti-steps prediction ofchaotic time series based on echo state networkrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 669ndash672 IEEE Changsha China September 2010

[5] S Michanos A Tsakoumis P Fessas S Vladov and V Mlade-nov ldquoShort-term load forecasting using a chaotic time seriesrdquo inProceedings of the International Symposium on Signals Circuitsand Systems (SCS rsquo03) pp 437ndash440 Iasi Romania 2003

[6] X An D Jiang M Zhao and C Liu ldquoShort-term prediction ofwind power using EMD and chaotic theoryrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2pp 1036ndash1042 2012

[7] C Qin C-C Chang and L-T Liao ldquoAn adaptive prediction-error expansion oriented reversible information hidingschemerdquo Pattern Recognition Letters vol 33 no 16 pp2166ndash2172 2012

[8] J Zhang K C Lam W J Yan H Gao and Y Li ldquoTime seriesprediction using Lyapunov exponents in embedding phasespacerdquo Computers amp Electrical Engineering vol 30 no 1 pp 1ndash15 2004

[9] J M Guerrero F Blaabjerg T Zhelev et al ldquoDistributedgeneration toward a new energy paradigmrdquo IEEE IndustrialElectronics Magazine vol 4 no 1 pp 52ndash64 2010

[10] G Chicco and P Mancarella ldquoDistributed multi-generationa comprehensive viewrdquo Renewable and Sustainable EnergyReviews vol 13 no 3 pp 535ndash551 2009

[11] G Abdollahi and M Meratizaman ldquoMulti-objective approachin thermoenvironomic optimization of a small-scale distributedCCHP system with risk analysisrdquo Energy and Buildings vol 43no 11 pp 3144ndash3153 2011

[12] D Xia S Song J Wang J Shi H Bi and Z Gao ldquoDetermi-nation of corrosion types from electrochemical noise by phasespace reconstruction theoryrdquo Electrochemistry Communica-tions vol 15 no 1 pp 88ndash92 2012

[13] Z Xie and K Wang ldquoSelection of embedding parameters inphase space reconstructionrdquo in Proceedings of the 2nd Inter-national Conference on Intelligent Computing Technology andAutomation (ICICTA rsquo09) pp 637ndash640 IEEE Hunan ChinaOctober 2009

[14] W-D Cai Y-Q Qin and B-R Yang ldquoSelection of delay timewindow and delay time in phase space reconstructionrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 526ndash530 December 2007

[15] Y-B Li Y Song and C-H Li ldquoSelection of parameters forphase space reconstruction of chaotic time seriesrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 30ndash33IEEE Changsha China September 2010

[16] J HerbertTutorial on Training Recurrent Neural Networks Cov-ering BPPT RTRL EKF and the ldquoEcho State Networkrdquo ApproachGMD-Forschungszentrum Informationstechnik Bremen Ger-many 2002

[17] H Jaeger M Lukosevicius D Popovici and U Siewert ldquoOpti-mization and applications of echo state networks with leaky-integrator neuronsrdquoNeural Networks vol 20 no 3 pp 335ndash3522007

[18] K A Bredahl and T Timo Forecasting with Nonlinear TimeSeries Models School of Economics and Management AarhusUniversity Aarhus Denmark 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Control Science and Engineering

0 50 100 150 20001

02

03

04

05

06

07

08

09

1

11

Date serial

Max

imum

pow

er (k

W)

Observed powerPredicted power (kW)

Figure 6 The estimated output and observations

signal is written into the output unit for times 119899 = 1 800The internal states 119909(119899) = (119909

1(119899) 119909

20(119899)) for 119899 =

1 800 are collected as a row of a state-collectingmatrixMof size 800times 20 during the sampling period At the same timethe teacher outputs 119889(119899) are collected into a row of a matrixT of size 800 times 1 Then we can compute output weights 119908

119900

for linear output unit 119910(119899) such that the teacher time series119889(119899) is approximated as a linear combination of the internalactivation time series 119909

119894(119899) by (9) The multistep prediction

can be obtained with testing samples The predictor canforecast multistep value directly with different time domain

The result of direct prediction based on echo statenetwork for the maximum power series of the distributedgeneration system is shown in Figure 6The solid line denotesthe predicted value the dotted line is for the observed valueWe observe that the prediction model based on echo statenetwork has high accuracy in direct prediction owing to thecoincidence of prediction and actual curve The max predic-tion error is about 017 (shown in Figure 7) Figure 8 is theerror of the traditional multilayer perceptron The predictorbased on echo state network has higher accuracy

7 Conclusion

Themicroenvironment meteorological data are coupled withthe output power of the distributed generation systemTheoretical proof and experimental results show that theoutput power of the distributed generation system is chaotictime seriesTheprediction of chaotic time series is to establisha nonlinear model of the dynamic system based on real timeseries in the reconstructed phase space Firstly we shoulddetect and quantify chaos for the intensive studies of nonlin-ear systems If the largest Lyapunov exponent is positive thedynamical system must be chaotic Secondly the embeddingdimension and the delay time are chosen based on theimproved C-C method The attractor of chaotic power time

0 50 100 150 200

0

005

01

015

Date serial

Erro

r

minus01

minus015

minus02

minus005

Figure 7 The error of the predictions

0 50 100 150 200

0

005

01

015

02

Date serial

Erro

r

minus01

minus015

minus005

Figure 8 The error of the traditional multilayer perceptron

series can be reconstructed based on the embedding dimen-sion and delay time in the phase space Lastly the neural net-work model is trained based on the training samples whichare obtained from the distributed generation system Theneural network will approximate the curve of output poweradequately The maximum power point of the distributedgeneration system will be tracked based on the meteorologi-cal data

Themeasuring errors are unavoidable among experimen-tal data due to the disturbance and many other factors Theprediction precision of the neural network model is largelydependent on the generalization of neural network There-fore we should optimize the structure and parameters ofthe model for better performance Experimental results showthat the prediction model based on echo state network hashigh accuracyThe chaotic predictionmodel can approximatethe mapping between the maximum power point of thedistributed generation system and the meteorological dataThe echo state network model can make rather remarkably

Journal of Control Science and Engineering 7

accurate predictions about the maximum power of thedistributed generation system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 61473174 61105100 and 51376110)the Specialized Research Fund for the Doctoral Program ofHigher Education (no 20130131130006) Project funded byChina Postdoctoral Science Foundation (no 2014M551907)and Independent Innovation Foundation of Shandong Uni-versity (no 2013ZRQP002)

References

[1] G Angrisani C Roselli and M Sasso ldquoDistributed microtri-generation systemsrdquoProgress in Energy andCombustion Sciencevol 38 no 4 pp 502ndash521 2012

[2] K A Pruitt S Leyffer AMNewman et alOptimal Design andDispatch of Distributed Generation Systems Argonne NationalLaboratory Chicago Ill USA 2012

[3] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[4] Y Song Y Li Q Wang and C Li ldquoMulti-steps prediction ofchaotic time series based on echo state networkrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 669ndash672 IEEE Changsha China September 2010

[5] S Michanos A Tsakoumis P Fessas S Vladov and V Mlade-nov ldquoShort-term load forecasting using a chaotic time seriesrdquo inProceedings of the International Symposium on Signals Circuitsand Systems (SCS rsquo03) pp 437ndash440 Iasi Romania 2003

[6] X An D Jiang M Zhao and C Liu ldquoShort-term prediction ofwind power using EMD and chaotic theoryrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2pp 1036ndash1042 2012

[7] C Qin C-C Chang and L-T Liao ldquoAn adaptive prediction-error expansion oriented reversible information hidingschemerdquo Pattern Recognition Letters vol 33 no 16 pp2166ndash2172 2012

[8] J Zhang K C Lam W J Yan H Gao and Y Li ldquoTime seriesprediction using Lyapunov exponents in embedding phasespacerdquo Computers amp Electrical Engineering vol 30 no 1 pp 1ndash15 2004

[9] J M Guerrero F Blaabjerg T Zhelev et al ldquoDistributedgeneration toward a new energy paradigmrdquo IEEE IndustrialElectronics Magazine vol 4 no 1 pp 52ndash64 2010

[10] G Chicco and P Mancarella ldquoDistributed multi-generationa comprehensive viewrdquo Renewable and Sustainable EnergyReviews vol 13 no 3 pp 535ndash551 2009

[11] G Abdollahi and M Meratizaman ldquoMulti-objective approachin thermoenvironomic optimization of a small-scale distributedCCHP system with risk analysisrdquo Energy and Buildings vol 43no 11 pp 3144ndash3153 2011

[12] D Xia S Song J Wang J Shi H Bi and Z Gao ldquoDetermi-nation of corrosion types from electrochemical noise by phasespace reconstruction theoryrdquo Electrochemistry Communica-tions vol 15 no 1 pp 88ndash92 2012

[13] Z Xie and K Wang ldquoSelection of embedding parameters inphase space reconstructionrdquo in Proceedings of the 2nd Inter-national Conference on Intelligent Computing Technology andAutomation (ICICTA rsquo09) pp 637ndash640 IEEE Hunan ChinaOctober 2009

[14] W-D Cai Y-Q Qin and B-R Yang ldquoSelection of delay timewindow and delay time in phase space reconstructionrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 526ndash530 December 2007

[15] Y-B Li Y Song and C-H Li ldquoSelection of parameters forphase space reconstruction of chaotic time seriesrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 30ndash33IEEE Changsha China September 2010

[16] J HerbertTutorial on Training Recurrent Neural Networks Cov-ering BPPT RTRL EKF and the ldquoEcho State Networkrdquo ApproachGMD-Forschungszentrum Informationstechnik Bremen Ger-many 2002

[17] H Jaeger M Lukosevicius D Popovici and U Siewert ldquoOpti-mization and applications of echo state networks with leaky-integrator neuronsrdquoNeural Networks vol 20 no 3 pp 335ndash3522007

[18] K A Bredahl and T Timo Forecasting with Nonlinear TimeSeries Models School of Economics and Management AarhusUniversity Aarhus Denmark 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 7

accurate predictions about the maximum power of thedistributed generation system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 61473174 61105100 and 51376110)the Specialized Research Fund for the Doctoral Program ofHigher Education (no 20130131130006) Project funded byChina Postdoctoral Science Foundation (no 2014M551907)and Independent Innovation Foundation of Shandong Uni-versity (no 2013ZRQP002)

References

[1] G Angrisani C Roselli and M Sasso ldquoDistributed microtri-generation systemsrdquoProgress in Energy andCombustion Sciencevol 38 no 4 pp 502ndash521 2012

[2] K A Pruitt S Leyffer AMNewman et alOptimal Design andDispatch of Distributed Generation Systems Argonne NationalLaboratory Chicago Ill USA 2012

[3] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[4] Y Song Y Li Q Wang and C Li ldquoMulti-steps prediction ofchaotic time series based on echo state networkrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 669ndash672 IEEE Changsha China September 2010

[5] S Michanos A Tsakoumis P Fessas S Vladov and V Mlade-nov ldquoShort-term load forecasting using a chaotic time seriesrdquo inProceedings of the International Symposium on Signals Circuitsand Systems (SCS rsquo03) pp 437ndash440 Iasi Romania 2003

[6] X An D Jiang M Zhao and C Liu ldquoShort-term prediction ofwind power using EMD and chaotic theoryrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2pp 1036ndash1042 2012

[7] C Qin C-C Chang and L-T Liao ldquoAn adaptive prediction-error expansion oriented reversible information hidingschemerdquo Pattern Recognition Letters vol 33 no 16 pp2166ndash2172 2012

[8] J Zhang K C Lam W J Yan H Gao and Y Li ldquoTime seriesprediction using Lyapunov exponents in embedding phasespacerdquo Computers amp Electrical Engineering vol 30 no 1 pp 1ndash15 2004

[9] J M Guerrero F Blaabjerg T Zhelev et al ldquoDistributedgeneration toward a new energy paradigmrdquo IEEE IndustrialElectronics Magazine vol 4 no 1 pp 52ndash64 2010

[10] G Chicco and P Mancarella ldquoDistributed multi-generationa comprehensive viewrdquo Renewable and Sustainable EnergyReviews vol 13 no 3 pp 535ndash551 2009

[11] G Abdollahi and M Meratizaman ldquoMulti-objective approachin thermoenvironomic optimization of a small-scale distributedCCHP system with risk analysisrdquo Energy and Buildings vol 43no 11 pp 3144ndash3153 2011

[12] D Xia S Song J Wang J Shi H Bi and Z Gao ldquoDetermi-nation of corrosion types from electrochemical noise by phasespace reconstruction theoryrdquo Electrochemistry Communica-tions vol 15 no 1 pp 88ndash92 2012

[13] Z Xie and K Wang ldquoSelection of embedding parameters inphase space reconstructionrdquo in Proceedings of the 2nd Inter-national Conference on Intelligent Computing Technology andAutomation (ICICTA rsquo09) pp 637ndash640 IEEE Hunan ChinaOctober 2009

[14] W-D Cai Y-Q Qin and B-R Yang ldquoSelection of delay timewindow and delay time in phase space reconstructionrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 526ndash530 December 2007

[15] Y-B Li Y Song and C-H Li ldquoSelection of parameters forphase space reconstruction of chaotic time seriesrdquo in Proceed-ings of the IEEE 5th International Conference on Bio-InspiredComputing Theories and Applications (BIC-TA rsquo10) pp 30ndash33IEEE Changsha China September 2010

[16] J HerbertTutorial on Training Recurrent Neural Networks Cov-ering BPPT RTRL EKF and the ldquoEcho State Networkrdquo ApproachGMD-Forschungszentrum Informationstechnik Bremen Ger-many 2002

[17] H Jaeger M Lukosevicius D Popovici and U Siewert ldquoOpti-mization and applications of echo state networks with leaky-integrator neuronsrdquoNeural Networks vol 20 no 3 pp 335ndash3522007

[18] K A Bredahl and T Timo Forecasting with Nonlinear TimeSeries Models School of Economics and Management AarhusUniversity Aarhus Denmark 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of