Power System Model Reduction withGrid-Connected Photovoltaic Systems Based on

Hankel Norm ApproximationImran Maqbool1, Gustav Lammert2, Anton Ishchenko3, Martin Braun1,2

1Fraunhofer IWES, Kassel, Germany2University of Kassel, Kassel, Germany

3Phase to Phase BV, Arnhem, Netherlandsimran maqbool@yahoo.com

Abstract—In order to ensure stability of the system, trans-mission and distribution studies need to be performed. Thesesimulation studies can be time consuming due to high complexityof the system. This paper investigates model reduction on a powersystem with photovoltaic generation. This is done by decreasingthe complexity of the detailed model, which leads to a smallersimulation time, but still maintaining the accuracy of the model.Therefore, the Hankel norm approximation is chosen, whichapplies controllability and observability properties of the systemto compute the singular values. These singular values are helpfulfor determining the order and accuracy of the reduced system.For the verification of the Hankel norm approximation, a smallfour-bus test system is studied in MATLAB/Simulink. The resultsof the reduced linearized model are compared with the originalnonlinear model and show good accuracy.

Index Terms—Dynamic equivalents, dynamic equivalencing,model order reduction, modal model reduction, reduced-ordermodel, Singular Value Decomposition (SVD) methods, powersystem stability.

I. INTRODUCTION

In order to operate power systems reliably and to controlthem properly, power system stability studies must be carriedout. In these stability studies a dynamic response of the systemis accurately studied on computer software simulation toolsthrough specific models of the power system. Furthermore, inrecent decades, power systems have changed rapidly. Powersystems are expanding in both, size and complexity due to everincreasing demand of electricity. Moreover, the integration ofDistributed Generators (DGs) further increases the complexityof the power system. Due to this increase, modelling andsimulation of power systems in detail has become moredifficult as it presents a huge computational burden. Therefore,much efforts have been put into developing efficient com-putational techniques for large complex systems. One of thebest technique is to reduce the complexity of a large powersystem to a computationally feasible size of an equivalentsystem, which retains the dynamic characteristics of the powersystem with reasonable accuracy. The process of reducinga large power system network into an equivalent model fordynamic system studies is called dynamic equivalencing ormodel reduction [1].

In general, dynamic power system equivalents can be clas-sified into three major groups: 1) coherency based methods;2) measurement based methods; and 3) strictly mathematicalmethods [2].

In coherency based methods, the group of generators whichtend to oscillate together (similar rotor angle characteristics)in the event of a disturbance are replaced by an equivalentgenerator. Such a group of generators are called coherentgenerators [1]. There are different methods to identify coherentgroups of generators, e.g., the weak link method [3] or theslow coherency method [4]. A brief overview of these methodscan be found in [5]. Coherency based methods give a goodnonlinear equivalent of the system. The main disadvantage ofthese methods is that they are only applicable for synchronousgenerators as they use rotor angle properties of the genera-tors [6].

Measurement based reduction methods use either real-timemeasurements from phasor measurement units in the powersystem or a simulated response of the power system in orderto create a reduced model of the system. These methods canbe further divided into two groups: Artificial Neural Network(ANN) methods and system identification methods. In [7]ANNs are used to create a nonlinear reduced model of thepower system with fuel cells and microturbines as DGs. Ablack box model is used in [8] to get the reduced model ofthe system with small synchronous generators and doubly-fed induction generators used as DGs. In [9], [10] grey boxmodelling techniques are used to obtain the reduced model ofthe system. The main advantage of these methods is that theyare useful when details of the system are not known. Thesemethods can be applied to distribution networks with largeamount of DGs. For small to medium sized systems thesemethods take more computational time then other methodsas they need a large data set from simulations of predefineddisturbances to create a dynamic equivalent model.

Strictly mathematical methods use existing techniques fromcontrol theory, which can be applied for the reduction ofdifferent large-scale physical systems. These methods mostlyhave a strong mathematical basis and aim to approximatethe input-output behavior of the considered system. Most of

This paper was presented at the 6th Solar Integration Workshop and published in the workshop’s proceedings.

these mathematical methods used for model reduction focus onlinear systems, which in many cases accurately represent thephysical system [6]. Depending on the external area propertiesthat have to be retained in the reduced model, differentmathematical model reduction methods can be applied. In [6]Hankel norm approximation is applied to reduce the system,whereas in [11] an extension of balanced truncation is used.Both of these methods belong to the family of Singular ValueDecomposition (SVD). In [6], [12] Krylov methods are used toreduce large power systems. As these methods require a linearmodel of the system, they are not valid when the operatingpoint of the system is shifted far away from the originalsystem operating point. It should be noted that these methodscan not properly represent the system for highly nonlinearphenomena [6].

As Hankel norm approximation is the most efficient methodout of the SVD based methods [6], it is considered for theinvestigation in this paper.

The paper is organized as follows. Section II providesan introduction to Hankel norm approximation. Section IIIdescribes the test system including the PhotoVoltaic (PV)system on which the Hankel norm approximation is applied.In Section IV the results of the model reduction are illustrated.Finally, the conclusions are presented in Section V.

II. HANKEL NORM APPROXIMATION

Hankel norm is the measure of how much energy can betransferred from the past inputs to the future outputs. Thereduced order model based on Hankel norm attempts to reducethe system in such a way that Hankel norm of the reducedsystem is minimized and thus the worst case error [13]. Thealgorithm of Hankel norm approximation is shown in Fig. 1and can be divided into 5 steps:

1) State space of the system2) Balancing the state space3) Partitioning the state space4) Finding all pass of the system5) Extracting the reduced order system

A. State space of the system

Hankel norm approximation is applicable only to linear (orlinearized) systems. Therefore, consider the linear dynamicalsystem in state space form:

d

dtx(t) = Ax(t) +Bu(t) (1)

y(t) = Cx(t) +Du(t) (2)

where x(t) is the state vector, u(t) is the vector of inputs andy(t) is the vector of outputs. Further, assume that there are nstates, m inputs and p outputs.

Hankel norm approximation uses two important propertiesof the dynamic system, namely controllability and observabil-ity. The controllability is the ability of the system to reachfrom the zero state xo to any other state in finite amount of

State space of the system

d

dtx(t) = Ax(t) +Bu(t)

y(t) = Cx(t) +Du(t)

Wc 6= Wo

Balancing the state space

dz

dt= Az + Bu

y = Cz + Du

Wc = Wo = diagσ1, σ2, . . . , σn

Partitioning the state space

dz1

dt= A11z1 + A12z2 + B1u

dz2

dt= A21z1 + A22z2 + B2u

y = C1z1 + C2z2 + Du

Wc = Wo =

[Σ1 00 Σ2

]Σ1 = diagσ1, . . . , σk, σk+r+1, . . . , σnΣ2 = σk+1Ir

Finding all pass of the system

A = Γ−1(σ2k+1A

T11 + Σ1A11Σ1 − σk+1C

T1UB

T1

)B = Γ−1

(Σ1B1 + σk+1C

T1U

)C = C1Σ1 + σk+1UB

T1

D = D − σk+1U

Extracting the reduced order system

Ak = A− Bk = B−Ck = C− Dk = D−∣∣∣∣G(s)−Gk(s)

∣∣∣∣∞ = σk+1

1. Find T2. Perform similarity transformation x = Tz

1. Select k2. Determine r3. Partition balanced system states

1. Calculate Γ = Σ21 − σ2

k+1I

2. If m > p then replace:(A, B, C, D)→ (AT, CT, BT, DT)

3. Solve B2 + CT2U = 0

1. Extract stable part of all pass dilation:A−, B−, C−, D−

2. If m > p then replace:(A−, B−, C−, D−)→ (AT

−, CT−, B

T−, D

T−)

Fig. 1. Methodology of Hankel norm approximation algorithm [6].

time by the action of some input u(t). In order to reach thisfinal state a limited amount of energy εc is required [14]:

εc = xToW−1c xo (3)

where Wc is called the controllability gramian and defined as:

Wc =

∞∫0

eAtBBTeATt dt (4)

In other words, the controllability provides a quantitativemeasure of how closely the inputs and the states are coupled.

The second important property of the system is observabil-ity. States of a system are internal variables and in generalit might be impossible to directly measure them. At thesame time, the outputs can be measured rather easily. Theobservability is the ability of the system to uniquely determinestate xo through the time from the measured outputs y(t). Acertain amount of output energy is generated by the state xofor a natural response (zero input) [14]:

εo = xToW−1o xo (5)

where Wo is called the observability gramian and defined as:

Wo =

∞∫0

eATtCTCeAt dt (6)

In other words, the observability gramian provides a quanti-tative measure of how closely the outputs and the states arecoupled.

Usually, (4) and (6) are not used for computing thegramians as they involve matrix exponentials and integrals.Instead, gramians are computed by solving the Lyapunovequations [14]:

AWc +WcAT = −BBT (7)

ATWo +WoA = −CTC (8)

B. Balancing the state space

In order to apply model reduction to the system, it should bein balanced state space representation. Thus, the states of thesystem are such that the degree of controllability and degreeof observability of each state is the same. Mathematically,balancing methods consist of the simultaneous diagonalizationof controllability and observability gramians. This can beachieved by the use of similarity transformations which changethe state and the system matrices but retain the same input-output behavior [14]. Suppose the following equation:

x = Tz (9)

where T is a non-singular, constant matrix. Substituting thisexpression into (1) and (2) yields:

dz

dt= T−1ATz + T−1Bu (10)

y = CTz +Du (11)

with A = T−1AT , B = T−1B, C = CT and D = D leadsto:

dz

dt= Az + Bu (12)

y = Cz + Du (13)

It is possible to find such a similarity transformation thatchanges the system in balanced form and diagonalize thegramians:

Wc = Wo = diagσ1, σ2, . . . , σn (14)

where σ1, σ2, . . . , σn are real and positive numbers calledHankel singular values. The amount of Hankel singular valuesis equal to the amount of states of the system. Furthermore,it is assumed that these values are ordered in such way thatσ1 ≥ σ2 ≥ . . . ≥ σn by modifying the rows and columns ofthe matrices of the system.

C. Partitioning the state space

After balancing the system, the order of the reduced systemk is selected using Hankel singular values of the system. Ifthe general situation is considered, it is possible that severalHankel singular values with an index starting from k + 1 areequal, i.e., σk > σk+1 = σk+2 = . . . = σk+r > σk+r+1 ≥. . . ≥ σn > 0, where r is the number of equal Hankel singularvalues with an index starting from k + 1. As seen, if σk+1 6=σk+2 then r = 1. After selecting k and determining r, thesystem can be transformed to a partially balanced form withthe controllability and observability gramians [15]:

Wc = Wo =

[Σ1 00 Σ2

](15)

Σ1 = diagσ1, σ2, . . . , σk, σk+r+1, . . . , σn (16)Σ2 = σk+1Ir (17)

where Ir is the identity matrix with dimension r.The system matrices A, B, C, D can be partitioned in the

same way as the gramians:

dz1

dt= A11z1 + A12z2 + B1u (18)

dz2

dt= A21z1 + A22z2 + B2u (19)

y = C1z1 + C2z2 + Du (20)

where the vector z1 has the dimension n − r and the vectorz2 has the dimension r.

D. Finding all pass of the system

All pass is a key concept used in Hankel norm approxi-mation. Suppose the transfer function of the original systemdefined by (1) and (2) is G(s). Then there exists a systemwith the transfer function G(s) such that the gain of the errortransfer function G(s) − G(s) is constant for all frequenciesand it is said to be all pass. In order to find all pass of thesystem, define a diagonal non-singular matrix [15]:

Γ = Σ21 − σ2

k+1I (21)

If the number of inputs is more than the number of outputs(m > p) then replace A, B, C, D by AT, CT, BT, DT. Thenext step is to find a unitary matrix U by solving:

B2 + CT2U = 0 (22)

Then, the all pass dilation of the system is defined by:

A = Γ−1(σ2k+1A

T11 + Σ1A11Σ1 − σk+1C

T1UB

T1

)(23)

B = Γ−1(Σ1B1 + σk+1C

T1U)

(24)

C = C1Σ1 + σk+1UBT1 (25)

D = D − σk+1U (26)

E. Extracting the reduced order system

The all pass of the system is usually unstable and onlycontains k stable poles [14]. Therefore, the stable part of theall pass dilation A−, B−, C−, D− has to be determined. Thisstable part is the reduced order model of the system.

If the number of inputs is more than the number of outputs(m > p), replace A−, B−, C−, D− by AT

−, CT−, B

T−, D

T−. This

is the last step of the algorithm.One of the important features of the Hankel norm method

is that the error of the approximation is bounded:∣∣∣∣G(s)−Gk(s)∣∣∣∣∞ = σk+1 (27)

where Gk(s) is the transfer function of the k-th order reducedmodel. It means that the largest error for all frequenciesis equal to the k + 1 Hankel singular value. This a prioriknowledge of the achievable errors allows to select a suitablevalue of k.

It is crucial to add that the Hankel norm approximationcan be performed only by means of algebraic operations withthe state space matrices. Therefore, it is fast, reliable andnumerically simple [14].

III. MODELLING OF THE TEST SYSTEM

A. Test system

In power system model reduction, the system is usuallydivided into two subsystems, i.e., study area and external area,as seen Fig. 2.

The study area is the part of the system which is of greatinterest. This area is retained in the power system modelreduction process and all the disturbances and configurationchanges happen in it. Therefore, the modelling of the studyarea is done in detail.

The external area is the rest of the power system containingall the external generators, loads etc. The external area canbe a neighboring area, which does not effect the dynamics

Photovoltaicsystem

External area(reduced linear model)

Study area(detailed nonlinear model)

1 2 3 4

Fig. 2. Test system for performing the Hankel norm approximation.

of the study area very much. Hence, the external area can bereduced to get a simplified equivalent model of the system.Both subsystems are connected to each other by lines andboundary buses.

The test system used in this study is relatively simple inorder to focus on the basic steps of the Hankel norm approxi-mation. A four-bus system is employed in MATLAB/Simulinkas given in Fig. 2. The system consists of a PV system witha coupling branch connected to an infinite bus through atransformer and a feeder branch. The study area contains theinfinite bus, transformer and the feeder branch and it will besubjected to the disturbance. The external area consists of thePV system and the coupling branch, connected to the boundarybus (bus 2). The parameters of the test system are given inTable I.

B. Photovoltaic system

The PV system used in this study is based [16]. In addition,for bulk power system dynamic studies generic PV systemmodels, such as [17], can be used. The control block diagramof the inverter is shown in Fig. 3 and the parameters aregiven in Table I. The power references and the voltages atthe point of connection are used to set references for thecurrent controllers. The inverter is also provided with a DClink voltage from the PV array. These variables constitute theinputs to the model whereas the currents at point of connectionare treated as outputs. The control is performed in a rotatingreference frame (dq-domain) and the reference angle for theabc-dq transformation is provided by the Phased-Locked Loop(PLL). The measured input voltage at the point of connectionis transformed to the dq-domain using the transformationmatrix by the following equation [16]:vOd

vOq

vO0

=

√2

3

cos θ cos

(θ − 2π

3

)cos(θ + 2π

3

)− sin θ − sin

(θ − 2π

3

)− sin

(θ + 2π

3

)1√2

1√2

1√2

·

vOa

vOb

vOc

(28)

The PLL used here is given in Fig. 4. It is based on aligningin closed-loop control the angle of the dq-transformationsuch that the voltage at the connection point has no q-axiscomponent [16]. A PI controller acts on the alignment errorto set the rotational frequency (29) and that frequency isintegrated to give the transformation angle (30):

ω = KPLLP vOq +KPLL

I

∫vOq dt (29)

θ =

∫ω dt (30)

The power controller shown in Fig. 5 is used in open-loopcontrol to calculate the reference currents in dq-form with theaid of the reference active and reactive power and the voltage

TABLE ITEST SYSTEM PARAMETERS

Description Symbol Value Unit

Nominal phase voltage vn 400 [V]Grid frequency f 50 [Hz]Photovoltaic system rated apparent power Sr, PV 20 [kVA]DC bus voltage vDC 1000 [V]Coupling resistance Rc 0.131 [Ω]Coupling inductance Lc 0.96 [mH]Filter resistance R 0.056 [Ω]Filter inductance L 1.35 [mH]Filter capacitance C 50 [µF]Phased-Locked Loop proportional gain KPLL

P 2.1 [−]Phased-Locked Loop integral gain KPLL

I 5000 [1/s]Current controller proportional gain K

dqP 1 [−]

Current controller integral gain KdqI 460 [1/s]

Feeder branch resistance Rf 0.275 [Ω]Feeder branch inductance Lf 0.192 [mH]Transformer rated apparent power Sr 0.1 [MVA]Transformer rated voltage high side Vr, HS 20 [kV]Transformer rated voltage low side Vr, LS 0.4 [kV]Transformer resistance Rtr 0.028 [Ω]Transformer inductance Ltr 0.183 [mH]

Power controller

Current Controller

LC filter and

coupling

PLL

dq

abc

dq

abc

iOabc

iLabc

VOabcVOdq

VOq

iOdq

iLdq

Vldq

VDC

θ ω

P*

Q*

+

−

V*normldq

ierrLdqi*Ldq

Inverter

Fig. 3. Control block diagram of the photovoltaic system [16].

KPPLL

KIPLL∫

∫abc

dq

ω θ +

+

VOabc VOq

Fig. 4. Control block diagram of the Phase-Locked Loop [16].

at the point of connection. The output reference currents arecalculated as:

i*Od =vOdP

* − vOqQ*

v2Od + v2

Oq

(31)

i*Oq =vOqP

* + vOdQ*

v2Od + v2

Oq

(32)

The filter inductor currents are controlled in the current con-troller. Therefore, adjustment must be made to the reference

Second-order low-pass filter

with cut-off frequency ωc

P*

Q*

VOd

VOq

VOd

VOd-VOq

VOq

-1

P*

Q*

i*Od

i*Oq

++ -

++

-

i*Ld

i*Lq

iLd iOd

iOqiLq

iq∑

id∑

Fig. 5. Control block diagram of the power controller [16].

in order to account for the capacitor current of the filter as:

iΣd = i*Od + iCd = i*Od + (iLd − iOd) (33)

iΣq = i*Oq + iCq = i*Oq + (iLq − iOq) (34)

A second-order low-pass filter is used to remove the harmonicsand noises that may result from the distortion of the inputvoltage at the point of connection. The current controllerreferences after filtering are given as:

di*Ld

dt= ω2

c

∫(iΣd − i*Ld) dt−

√2ωci

*Ld (35)

di*Lq

dt= ω2

c

∫(iΣq − i*Lq) dt−

√2ωci

*Lq (36)

where ωc is the cut-off frequency of the filter.The current controller used is given in Fig. 6. It uses two PI

controllers together with cross-axis decoupling terms and feed-forward terms for the connection voltage [16]. The equationsfor this controller are:

v*Id = vOd − ωLiLq +Kd

PierrLd +Kd

I

∫ierrLd dt (37)

v*Iq = vOq − ωLiLd +Kq

PierrLq +Kq

I

∫ierrLq dt (38)

It is assumed that the switching frequency of the inverter issufficiently high. Therefore, the inverter can be simplified toa saturated voltage gain, as shown in Fig. 7.

The last block of the model contains a passive low-pass filterand a coupling impedance. The filter has important dynamic

KPd

KId∫

+

+

KPq

KIq∫

+

+

ωL

ωL

+ +

+

+ +

++

−

qLqerr

qLderr

iLderr

iLqerr

iLq

iLd

VOd

VOq

VDC

V*normld

V*normlq

V*ld

V*lq

Fig. 6. Control block diagram of the current controller [16].

V*normld

V*normlq

Vdsat

Vqsat

VDC

Vld

Vlq

Saturation

Fig. 7. Control block diagram of the inverter model [16].

effects down to quite low frequencies [16]. The equationsdescribing phase a of the filter are:

vIa = iLaR+ LdiLa

dt+ vCa (39)

vOa = vCa − LcdiOa

dt− iOaRc (40)

CdvCa

dt= iLa − iOa (41)

IV. MODEL REDUCTION OF THE SYSTEM

A. Linearization

As previously explained, Hankel norm approximation isonly applicable to linear (or linearized) systems. In orderto reduce the order of the model, the first step involveslinearization. This has been performed for the PV system inthe dq-reference frame. The PLL is considered as part of theconversion system and therefore, it is not linearized. The initialoperating point (states x0, inputs u0 and outputs y0) is definedbased on the power flow results. The Jacobian matrices of thesystem at the operating point are determined as:

A =∂f

∂x

∣∣∣∣x0,u0

B =∂f

∂u

∣∣∣∣x0,u0

C =∂g

∂x

∣∣∣∣x0,u0

D =∂g

∂u

∣∣∣∣x0,u0

(42)

whereas the original nonlinear system is represented by:

d

dtx(t) = f

(x(t), u(t)

)(43)

y(t) = g(x(t), u(t)

)(44)

The linearization is done using numerical perturbation in theControl Design toolbox of MATLAB/Simulink to get the statespace of the external area. In the next step, the linearized statespace equations in incremental form are:

d

dt∆x(t) = A∆x(t) +B∆u(t) (45)

∆y(t) = C∆x(t) +D∆u(t) (46)

with ∆x = x(t)− x0, ∆u = u(t)− u0 and ∆y = y(t)− y0.Accuracy of linearization is checked by comparing the time-domain response of the original system with the linearizedsystem. The results of this comparison are shown in Fig. 8.

−0.05 0.0 0.05 0.1 0.15 0.2 0.25 0.3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Cur

rent

[p.u

.]

Nonlinearlinear

0.02 0.03 0.04 0.05

−0.6

−0.5

Fig. 8. Comparison of the nonlinear and the linearized model. Response ofthe photovoltaic system current of phase a at the boundary bus (bus 2) for a0.8 p.u. voltage dip of 0.1 s.

B. Model reduction

The linearized model has 13 states in total, the same as thenonlinear model. After linearization it is possible to transformthe system into its balanced form and find its Hankel singularvalues. They are plotted in Fig. 9 on a logarithmic scale.From the analysis of the Hankel singular values, it appearsthat the balanced system can be reduced drastically as mostof the states are badly controllable and observable. Furthersimulations have shown that by using Hankel norm approxi-mation, the system might be reduced to the 6th order withoutany significant error. For instance, for the 6th order systemthe Hankel singular value is about 10−1. If the order of thesystem is further decreased, the accuracy will also decrease.The analysis of the Hankel singular values also allows toselect the order of the reduced system without performingsimulations, i.e., to estimate the error and the order a priori.

After checking the Hankel singular values, time-domainsimulation is performed in order to further investigate theresults of the reduced model. Therefore, a distant three-phasefault in the grid is represented by specifying the voltage of the

0 2 4 6 8 10 12 1410−3

10−2

10−1

100

101

Index of singular value

Val

ue

Fig. 9. Hankel singular values of the system.

−0.05 0.0 0.05 0.1 0.15 0.2 0.25 0.3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Cur

rent

[p.u

.]OriginalReduced

0.02 0.03 0.04 0.05

−0.6

−0.5

Fig. 10. Comparison of the original full order (13 states) nonlinear model andthe reduced order (6 states) linearized model. Response of the photovoltaicsystem current of phase a at the boundary bus (bus 2) for a 0.8 p.u. voltagedip of 0.1 s.

−0.05 0.0 0.05 0.1 0.15 0.2 0.25 0.3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Cur

rent

[p.u

.]

OriginalReduced

0.02 0.03 0.04 0.05

−0.6

−0.5

Fig. 11. Comparison of the original full order (13 states) nonlinear model andthe reduced order (5 states) linearized model. Response of the photovoltaicsystem current of phase a at the boundary bus (bus 2) for a 0.8 p.u. voltagedip of 0.1 s.

infinite bus to 0.8 pu for 0.1 s. The fault occurs at t = 0.0 sand is cleared at t = 0.1 s.

In Fig. 10 the comparison of the original full order (13states) nonlinear model and the reduced order (6 states) modelis depicted. By comparing the reduced and the full system, theresults show that the system can be reduced to the 6th orderwith a reasonably good accuracy.

In Fig. 11 the comparison of the original full order (13states) nonlinear model and the reduced order (5 states) modelis shown. By comparing the reduced and the full system, theresults show that the system should not be reduced to the 5thorder due to the relatively poor accuracy. In relation to Fig. 10,it should be noted that the accuracy is considerably decreased.According to the Hankel singular values of the system shownin Fig. 9, this decrease can be also identified in the differenceof the singular values between the 5th and the 6th order. Thismeans further reduction causes the decrease in accuracy andresults in a bad agreement with the original system.

Another comparison is carried out investigating the responseof the reduced order (6 states) linearized model for differentvoltage dips, as shown in Fig. 12. Because Hankel norm ap-proximation is applicable only to linear (or linearized) systems,this study is carried out in order to confirm the accuracy

−0.05 0.0 0.05 0.1 0.15 0.2 0.25 0.3−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

Vol

tage

[p.u

.]

0.8 p.u. voltage dip0.6 p.u. voltage dip

Fig. 12. Different voltage dips with the duration of 0.1 s that are applied forthe reduced order (6 states) linearized system.

−0.05 0.0 0.05 0.1 0.15 0.2 0.25 0.3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Cur

rent

[p.u

.]

0.8 p.u. voltage dip (original)0.8 p.u. voltage dip (reduced)0.6 p.u. voltage dip (original)0.6 p.u. voltage dip (reduced)

Fig. 13. Comparison of the reduced order (6 states) linearized systemapplied to different voltage dips with the duration of 0.1 s. Response of thephotovoltaic system current of phase a at the boundary bus (bus 2).

0.02 0.03 0.04 0.05 0.06 0.07 0.08

−0.95

−0.9

−0.85

−0.8

−0.75

−0.7

−0.65

−0.6

−0.55

−0.5

Time [s]

Cur

rent

[p.u

.]

0.8 p.u. voltage dip (original)0.8 p.u. voltage dip (reduced)0.6 p.u. voltage dip (original)0.6 p.u. voltage dip (reduced)

Fig. 14. Comparison of the reduced order (6 states) linearized systemapplied to different voltage dips with the duration of 0.1 s. Response of thephotovoltaic system current of phase a at the boundary bus (bus 2). Zoom ofFig. 13.

for different deviations from the steady-state operating point.The results of the responses of the PV system currents at theboundary bus (bus 2) are presented in Fig. 13. The zoom ofFig. 13 is shown in Fig. 14. From the results it is obvious thatthe bigger the voltage dip, the worse is the accuracy betweenthe original full order nonlinear model and the reduced orderlinearized model. For voltage dips down to V = 0.8 p.u. theaccuracy is still good that can be seen by the black lines in

Fig. 14. For a voltage dip down to V = 0.6 p.u., indicatedby the grey line, it can be seen that the accuracy betweenthe original full order nonlinear model and the reduced orderlinearized model is very poor. It can be concluded that Hankelnorm approximation can only be used for small disturbancesin the power system.

One of the most important advantages of applying powersystem model reduction methods is the improvement of thecomputational time. Hence, the simulation time of differentsystem orders is also investigated. The simulations are carriedout in MATLAB/Simulink R2013a and executed on a 64-bitWindows 8 operating system running on Intel core i5 1.7 GHzprocessor and 4 GB of RAM. The results of the performancecomparison, considering different system orders, are shown inTable II. For performing a simulation with the original fullorder nonlinear model takes about 11 s, while considering thereduced order linearized model the simulation time decreasesdrastically to about 5 s. Therefore, for the simple test systemused in this study, a 6th order Hankel norm approximationmodel appears to be optimal in the sense of accuracy versusperformance.

V. CONCLUSIONS

The new contribution of this paper is to perform dynamicmodel reduction of a power system using Hankel norm ap-proximation with a new technology, namely grid-connectedPV systems.

The overall methodology of the model reduction algorithmand the test system is presented. Furthermore, Hankel normapproximation is applied for different system orders as wellas various disturbances. The results indicate that for smalldeviations from the operating point, Hankel norm approxima-tion can significantly reduce the order and the simulation timeof the system while retaining reasonably good accuracy withrespect to the original system.

Based on the results, Hankel norm approximation canbe applied in the future to distribution networks with highpenetration levels of grid-connected PV systems for dynamicstudies. The obtained dynamic equivalents can be used on thehigh voltage transmission and the medium voltage distributionlevel to perform different bulk power system studies, suchas transient and small-signal stability as well as voltageand frequency stability studies based on root mean squaresimulations.

TABLE IISIMULATION TIME

Order of the system Simulation time [s]

13 states (original system) 11.4236 states (reduced system) 5.2455 states (reduced system) 5.001

ACKNOWLEDGMENT

The authors gratefully acknowledge the fruitful and insight-ful discussions during this research work with Tina Paschedag(University of Kassel, Kassel, Germany), Salman Zaidi (Uni-versity of Kassel, Kassel, Germany) and Daniel Duckwitz(Fraunhofer IWES, Kassel, Germany).

Furthermore, the authors thank Thomas Stetz (University ofApplied Sciences Mittelhessen, Gießen, Germany) for provid-ing the PV system model in MATLAB/Simulink.

This work was supported by the German Federal Ministryfor Economic Affairs and Energy and the Projekttrager JulichGmbH (PTJ) within the framework of the projects DEA-Stabil(FKZ: 0325585A) and Smart Grid Models (FKZ: 0325616).

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