Locational Effects of Variability of Injected Poweron Total Cost

Saman CyrusElectrical and Computer Engineering Department

University of Wisconsin-MadisonMadison, Wisconsin, USA

cyrus2@wisc.edu

Bernard LesieutreElectrical and Computer Engineering Department

University of Wisconsin-MadisonMadison, Wisconsin, USA

lesieutre@wisc.edu

AbstractThis paper introduces locational prices associatedwith power injection variability. The sensitivity of uncertainty ininjected power on system cost is calculated using a probabilisticDC optimal power flow (DC-OPF). This sensitivity may be usedas a price for regulation to equitably allocate costs associated withtracking power injections. In this context the price also providesa locational value to energy storage and hence can be used asa tool to investigate best locations for energy storage. As a casestudy, a 24-bus test system with uncertain variable wind powerand uncertain variable loads is analyzed.

I. INTRODUCTION

Electric power injections in the grid are continuouslyvarying as end-use loads turn on/off, shift demand, and certainrenewable resources vary with changes in input (wind,solar).Real-time power balancing is achieved by reserving and ad-justing a portion of controllable generator capacity to followfluctuations in net power injections. These generators aresaid to supply regulation and it is treated as an ancillaryservice in electricity markets. The cost to supply this regulationservice is typically shared among grid users. In this paper weintroduce a method to compute a locational price associatedwith power injection variability with a view that it may beused to equitably allocate regulation costs to those loads andsuppliers with variable injections.

We anticipate further that implementing a regulation pricewill provide an incentive for energy storage as a means tomitigate a regulation charge. Hence, a locational regulationprice provides an indirect valuation for energy storage. Thiscan influence the coupling of energy storage with certainvariable renewable sources such as wind and solar.

This is important as many states and countries have adoptedpolicies to encourage (and even mandate) supply from renew-able resources. Increasing the market penetration of commonrenewable energy technologies will increase the variabilityof power injections and power flows in the grid. Somewhatironically, this necessitates additional controllable resources toprovide more regulation in the current framework. We positthat a locational regulation price will provide an incentive toboth smooth power injections at variable source locations, andencourage non-traditional technologies to supply regulationsuch as demand response.

Other papers have investigated different aspects of energystorage in the grid. In [1] the role of energy storage on anelectricity grid in presence of renewable system as well as

different scenarios to design energy storage systems has beendiscussed. In [2] the properties of storage to allow more useof renewable energy in the network has been discussed. In[3] appropriate size of storage systems has been investigated.In [4] financial analysis has been implemented to computeelectricity storage systems cost. See [5], [6] for calculationof cost of storage and [7] for discussion about differentcategories of energy storage technologies. This paper differsby introducing a locational regulation price that can be usedto value storage.

The price that we compute here is a marginal cost equal tothe sensitivity of total system production cost to a locationalmeasure of power injection variability. This is cast as asensitivity of a probabilistic DC optimal power flow wherethe measure of variability is taken as statistically as a standarddeviation of power injection. Other measures are possible, butnot pursued in this paper.

II. FORMULATION OF PROBABILISTIC DC-OPF

The optimal power flow (OPF) was introduced in the early1960s [8]. The aim of OPF is to find the least-cost operatingpoint considering constraints over transmission lines, voltageconstraints, power generation, etc. The simpler, commonly-used DCOPF is obtained by linearizing the ACOPF assumingall voltage magnitudes are fixed and voltage angle differencesare small [9].

A standard DC optimal power flow formulation is

minCTGPG (1)

subject toEPG B = PL (2)

PminG PG PmaxG (3)

Pmaxij 1

xij(i j) Pmaxij , ij (4)

slack = 0 (5)

where PG is active power injections, is the voltage angles,PmaxG is upper nominal production level, P

minG is lower

nominal production level, PL is vector of loads xij is linereactance for line ij, B is a matrix related to the bus admittancematrix and E is an indicator matrix with all elements are 0 or1 and when is multiplied by PG gives the vector of generated

power on each bus. Matrix E maps the generation of multiplegenerators at a bus to a net bus injected power.

We follow a probabilistic approach to adjust this modelto accommodate uncertainties in power injections. Specifi-cally we use the analytic approach described in detail in[10] in which generation and transmission constraints areenforced with a user-specified probability. That paper focusedon variable wind generation. We augment their example toalso consider variations in load, mathematically treated asnegative generation, and we compute sensitivities to introducea regulation price. Introducing random variations in powerinjection leads to changes on both transmission line andgeneration capacity constraints, equation (3) and equation (4)

Prob

[1

xij(i j) +

NGG=1

GFij,GPG,GPmaxij 0]

1 (6)

Prob

[PG,g +

NGG=1

Dg,GPG,G PmaxG,g] 1 (7)

where GF is a Nl (NG + NG ) matrix and is generalizedgeneration distribution factor. Nl is number of transmissionlines, NG is number of generators which have uncertainty, NGis number of conventional generators, is violation probabilityand D is weighting matrix of changes in generation due todeviation from schedule.

When there are changes in the sysetm due to uncertaintyof G , each generator plays a role to compensate these changes,element Dg,G is the ratio of the change (PG,G ) whichis compensated by generator g. Also here PG,G is thevariation in generated power which is resulted from variationsin generator G . Matrix D is defined as

Dg,l =PmaxG,gNG

j=1,j 6=l PmaxG,j

(8)

where Dg,l describes changes in generator g due to variationsin generator l. Generalized generation distribution factors aredefined by

GF lij =1

xijeijXrl (9)

where eij is a row vector which all of its entries are zero exceptith column which is +1 and jth column which is 1. Alsoxij is the reactance of line which connects buses i and j, andX is the bus reactance matrix. rl describes the changes whichhappen in other buses when power generation at generator lvaries. For example, consider a generator l located at bus k;then for k-th row of rl we can write equation (10) and forother rows which correspond to buses which generator k isnot located on them equation (11) could be written

rl(k) =

gbus#k,g 6=l P

maxG,gNG

m=1,m 6=l PmaxG,m

1 (10)

rl(i) =

gbus#i P

maxG,gNG

j=1,j 6=l PmaxG,j

(11)

If we model the variations by a Gaussian random distribu-tion we can express (6) and (7) as

1

xij(i j) Pmaxij 1(1 )

NGG=1

GF 2ij,G2G (12)

PG,g PmaxG,g 1(1 ) NG

G=1

D2g,G2G (13)

Here 1 is inverse of standard normal distribution functionand G is standard deviation of generator G s power genera-tions distribution function.

In summary, equation (4) in the DC optimal power flowwill be replaced by equation (12), and the right hand sideof equation (3) is replaced by equation (13). We solve thisprobabilistic DCOPF for PG and . Then we perform asensitivity analysis to compute a price.

III. SENSITIVITY ANALYSIS

Using the probabilistic DC-OPF formulas, the KKT con-ditions can be written. Define

=

NGG=1

GF 2ij,G2G , i, j (14)

and write the corresponding Lagrangian as

L = CTGPG + TG(EPG B PL)+ TM (PG PmaxG )+ Tm(P

minG PG)

+ TS(PmaxLine + 1(1 ) + x)

+ TR(PmaxLine + 1(1 ) x) (15)

where is voltage angle vector, is a Nl Nl matrix whereNl is number of transmission lines. At each row of , allelements equal to zero except i-th column which is +1 andjth column which is 1, and x is a Nl 1 column vectorwith 1/xij at each row l.

The KKT conditions [11] are

Stationary:

LPG

= 0 CG + ETG + M m = 0 (16)L

= 0 BTG + T xTS T xTR = 0 (17)Equality Constraints:(Primal Feasibility)

EPG B = PL (18)Inequality Constraints:(Primal Feasibility)

PG,g PmaxG,g 1(1 ) NG

G=1

D2g,G2G , g (19)

PG PminG (20)

1xij(i j) Pmaxij

1(1 ) NG

G=1

GF 2ij,G2G , ij (21)

Complementary Slackness:

Sij

( Pmaxij + 1(1 )

NGG=1

GF 2ij,G2G

+1

xij(i j)

)= 0 ij (22)

Rij

( Pmaxij + 1(1 )

NGG=1

GF 2ij,G2G

1xij

(i j))

= 0 ij (23)

Mi(PGi PmaxGi ) = 0, i (24)

mi(PminGi PGi) = 0, i (25)

Dual Feasibility:

R 0 (26)S 0 (27)M 0 (28)m 0 (29)

IV. CASE STUDY AND LINEARIZATION

Having the formulation above we solve the optimizationproblem and find P G,

, M , m,

S ,

G,

R and linearize all

equations around the solution.

The effect of generation variability on total cost is studiedbased on the 24-bus system which corresponds to the IEEEOne Area RTS-96 system [12] but with the addition of twowind generators installed on buses 8 and 15 (Fig.1) withinstalled capacity of 500 and 700 MW respectively. Thisexample is taken from [10] which considers forecasted windpower in-feed equal to 25% of installed wind power capacityand violation probability (for chance constraint) to be 5%. Theemergency line limit is = 1.1 and nominal capacity availableas regulating power in the case of a contingency is = 1.25.The standard deviation(1 = 2) is considered to be 7.5%of installed capacity. We analyze this system and calculatesensitivities of production cost to uncertain wind generation.

S1. Calculate a regulation price as the sensitivity of pro-duction cost to uncertain wind generation.

Using these data, the sensitivities of total cost to variabilityof wind generators 1 and 2 (500 & 700 MW) are computedand given in Table I. The value are presented in units of $per-unit power per hour.

The sensitivity of total cost due to variability of loads isalso investigated in this paper. Loads are treated as negative

Fig. 1. 24-bus system with wind generators on buses 8 and 15 [10]

TABLE I. TOTAL COST CHANGES DUE TO VARIABILITY OF INJECTEDPOWER

Cost/G1 [$] 2.3252

Cost/G2 [$] 2.9238

generators and two additional scenarios are considered relatedto load variation and wind power variation.

S2. In this scenario, all loads and also two wind generatorslocated on buses 8 and 15 have the same variability(L = 1% for all buses, and G1 = G2 = 1%).

S3. In this scenario the variability on buses with windgenerators are increased. In this case the standarddeviation L remains equals to 1% for all busesexcept buses with wind generators, and for buses withwind generators (buses 8 and 15) standard deviationis substantially increased for loads (L) and windgenerators (G1 and G2 ) to be 71%.

The two different load scenarios are designed to resultin different binding constraints. In first scenario the marginalunits are generators 3 and 4 with all others are at minimumor maximum output. In the second scenario, generators 3 and4 are binding, and the units at bus 7 are marginal. In bothscenarios there is a single binding line constraint. Line 23connecting buses 14 and 16, is binding in both cases.

In all three scenarios, we linearize the KKT conditionsaround the solution, only including binding constraints, andwe obtain

Cost = CG PG (30)ETG + M m = 0 (31)

BTG + T xTS T xTR = 0 (32)EPG B = 0 (33)Mi = 0, i / E (34)

PGi +(1)2

(D2g,G1

11 +D

2g,G2

22) = 0

i E (35)

while

= 1(1 ) NG

G=1

D2g,G2G

E = {7, 8, 15, 21, 23, ..., 33}

diag(PminG P G)m + diag(m)PG = 0 (36)

Rij

(1(1 )GF

2ij,G1

(1)1 +GF2ij,G2

(2)2GF 2ij,G1

21 +GF

2ij,G2

22

1xij

(i j))

= 0, ij = 23

(37)

Rij

( Pmaxij +1(1 )

GF 2ij,G1

21 +GF

2ij,G2

22

1xij

(i j ))

= 0, ij 6= 23 (38)

Sij

( Pmaxij + 1(1 )

GF 2ij,G1

21 +GF

2ij,G2

22

+1

xij(i j )

)= 0 (39)

Rij = 0, ij 6= 23 (40)Sij = 0, ij {1, ..., 24} (41)

The results for the loads are presented in Table II. Thevalue are presented in units of $/per-unit power per hour.

The difference in prices between the two load scenarios arelargely driven by a change in binding constraints. We observethat different cases with the same binding constraints in thisprobabilistic model, the computed regulation prices do notvary, much in the same way that traditional LMP variationsare dominated by congestion costs.

V. CONCLUSION

We have introduced a method for locational pricing ofvariability of power injections and presented an example withwind generators and loads. It is computed as a sensitivity ofproduction cost to a measure of variability for generators andloads. In this paper we use the statistical standard deviation ofpower injection as this measure. The price could be used forpricing regulation and thus would equitably allocate charges tothose generators and loads with variable injections. We suggestthat such a charging scheme may provide incentives for loadenergy storage to reduce variability, and hence the price servesas means to value energy storage by location.

TABLE II. COST CHANGES DUE TO VARIABILITY OF LOADS WITHDIFFERENT VARIABILITIES OF LOADS

Bus Load on CostL [$]

Number bus (MW) Scenario I Scenario II1 108 0.5558 0.12792 97 0.5018 0.11743 180 0.81162 0.10884 74 0.387 0.09365 71 0.3788 0.09696 136 0.7438 0.20287 125 0.677 0.18018 46 0.2491 4.70439 175 0.9233 0.2289

10 195 1.0834 0.306811 0 0 012 0 0 013 265 1.3849 0.334014 194 2.0219 1.201015 142 0.7106 10.587816 100 0.5165 0.120317 0 0 018 333 1.6972 0.378919 181 0.8427 0.130220 128 0.5705 0.067921 0 0 022 0 0 023 0 0 024 0 0 0

More research is needed to consider additional examples tofully assess the potential impact of a location regulation price.Additional technical work can rigorously relate the prices tobinding constraints and should also consider other measuresfor variability.

REFERENCES[1] A. Solomon, D. M. Kammen, and D. Callaway, The role of large-scale

energy storage design and dispatch in the power grid: A study of veryhigh grid penetration of variable renewable resources, Applied Energy,vol. 134, pp. 7589, 2014.

[2] A. Solomon, D. Faiman, and G. Meron, Properties and uses of storagefor enhancing the grid penetration of very large photovoltaic systems,Energy Policy, vol. 38, no. 9, pp. 52085222, 2010.

[3] , Appropriate storage for high-penetration grid-connected photo-voltaic plants, Energy Policy, vol. 40, pp. 335344, 2012.

[4] O. Anuta, A. Crossland, D. Jones, and N. Wade, Regulatory andfinancial hurdles for the installation of energy storage in uk distributionnetworks, in Integration of Renewables into the Distribution Grid,CIRED 2012 Workshop. IET, 2012, pp. 14.

[5] S. Schoenung, Energy storage systems cost update, SAND2011-2730,2011.

[6] I. Pawel, The cost of storagehow to calculate the levelized cost ofstored energy (lcoe) and applications to renewable energy generation,Energy Procedia, vol. 46, pp. 6877, 2014.

[7] H. Ibrahim, A. Ilinca, and J. Perron, Energy storage systemscharacter-istics and comparisons, Renewable and Sustainable Energy Reviews,vol. 12, no. 5, pp. 12211250, 2008.

[8] J. A. Momoh, Electric power system applications of optimization. CRCPress, 2000.

[9] M. B. Cain, R. P. Oneill, and A. Castillo, History of optimal powerflow and formulations, FERC Staff Technical Paper, 2012.

[10] L. Roald, F. Oldewurtel, T. Krause, and G. Andersson, Analytical refor-mulation of security constrained optimal power flow with probabilisticconstraints, in PowerTech (POWERTECH), 2013 IEEE Grenoble, June2013, pp. 16.

[11] J. Nocedal and S. J. Wright, Numerical Optimization 2nd. Springer,2006.

[12] C. Grigg, P. Wong, P. Albrecht, R. Allan, M. Bhavaraju, R. Billinton,Q. Chen, C. Fong, S. Haddad, S. Kuruganty et al., The ieee reliabilitytest system-1996. a report prepared by the reliability test system taskforce of the application of probability methods subcommittee, PowerSystems, IEEE Transactions on, vol. 14, no. 3, pp. 10101020, 1999.