Perturbation Analysis of the Wholesale Energy Market Equilibrium in the Presence of Renewables

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by  Arman Kiani

Perturbation Analysis of the Wholesale Energy Market Equilibrium
in the Presence of Renewables
Arman Kiani and Anuradha Annaswamy
Abstract—One of the main challenges in the emerging smart ρn Locational marginal price corresponding to the gen-
grid is the integration of renewable energy resources (RER). erating unit i or the demand j that is located at node
The latter introduces both intermittency and uncertainty into
n
thegrid,bothofwhichcanaffecttheunderlyingenergymarket.
α Dual variable associated with the maximum capacity
An interesting concept that is being explored for mitigating the i
2 integration cost of RERs is Demand Response. Implemented as constraint of generating unit i
1 a time-varying electricity price in real-time, Demand Response φib Dual variable associated with the maximum capacity
0 has a direct impact on the underlying energy market as well. limit for block b of generating unit i
2 Beginningwithanoverallmodelofthemajormarketparticipants σ Dual variable associated with the minimum demand
j
n taongaeltyhzeertwheithentehregcyonmsatrrakienttsinofthtrisanpsampiesrsioanndanddergievneecroantidoint,iownes constraint of demand j
Ja foofrugnlcoebratalinmtiaexsiminumtheuRsinEgRsotanntdhaerdmKarKkeTt ecqriuteilribiar.iuTmheisetfhfeecnt λλUDBjk MPraicregibniadl ubtyilidtyemaassnodcijatteodbwuyithpobwloecrkbkloocfkdkemand j
Djk
7 quantified, with and without real-time pricing. Perturbation λB Price bid by generating unit i to sell power block b
1 analysis methods are used to compare the equilibria in the Gib
λC Marginal operating cost associated with block b of
] nuosimnginaalgaanmdep-etrhteuorrbeetidcmpaorinktetos.fTvhieewse.mSaurffikceitesnatrecoanlsdoitaionnaslyzaerde Gib generator i
C derived for the existence of a unique Pure Nash Equilibrium in Pmax Maximum power demanded in block k of demand j
Djk
O thenominalmarket.Theperturbedmarketisanalyzedusingthe Pmin Minimum power supplied to the demand j
Djk
concept of closeness of two strategic games and the equilibria
. P Power block k that the demand j is consuming
h of close games. This analysis is used to quantify the effect of Djk
P Power block b that the generating unit i is producing
t uncertainty of RERs and its possible mitigation using Demand Gib
ma Response.Finallynumericalstudies arereportedusinganIEEE PGmiabx Maximum power output in block b of generating unit
30-bus to validate the theoretical results. i
[ Pmax Maximum power output of generating unit i
Gi
1 PGmiin Minimum power output of generating unit i
v NOMENCLATURE Pw Power block b that the renewable generating unit l is
Glb
7 producing
θ Set of indices of generating units at node n
6 n Pw Maximum wind power estimation for block b of
4 ϑn Set of indices of demands at node n Glbmax
renewable generating unit l
3 Ωn Set of indices of nodes connected to node n
1. Dq Set of indices of Consumers {1,2,...,ND} PGwlbmin Minimum wind power estimation for block b of
renewable generating unit l
0 Gf Set of indices of generating units {1,2,...,NG} P¯w Mean wind power for block b of renewable generat-
2 Gw Set of indices of renewable generating units Glbmin
1 {1,2,...,N } ing unit l
W
: Pmax Transmission capacity limit of line n−m
v N Number of all buses nm
∆ρ Change of LMP for demand j
Xi ND Number of demands ∆Pn(j) Change of consumption by demand j
N Number of transmission lines Djk
t
r ∆ Uncertainty in block b of wind generating unit i
a NG Number of generating units Gib
κ Curtailment Factor of block k by demand j
N Number of blocks demanded by demand j Djk
Dj
δ voltage angle of bus n
N Number of blocks bid by generating unit i n
Gi
B susceptance of line n−m
N Number of blocks bid by renewable generating unit l nm
Gl
γ Dual variable associated with the transmission capac-
nm
ity constraint of line n−m I. INTRODUCTION
ψ Dual variable associated with the maximum capacity
jk The goal in a Smart Grid is to integrate large-scale renew-
limit for block k of demand j
able generation and emerging storage technologies, control
signals to loads to match supply, and dramatically improve
ThisworkwassupportedbytheTechnischeUniversita¨tMu¨nchen-Institute
for Advanced Study, funded by the German Excellence Initiative and by energyefficiency.Electricitymarkets,theentitythatcarriesout
Deutsche Forschungsgemeinschaft (DFG) through the TUM International power balance by scheduling power using bids from various
GraduateSchoolofScienceandEngineering(IGSSE).
generating companies, are crucial components that can con-
Arman Kiani is with the Institute of Automatic Control Engi-
neering, Technische Universita¨t Mu¨nchen, D-80290 Munich, Germany. tributetoanefficientsmartgrid.Electricitymarketmodelsthat
[email protected] accurately represent the market dynamics are exceedingly im-
Anuradha Annaswamy is with the Department of Mechanical Engineer-
portant. These models must capture the behavior of the dom-
ing, Massachussets Institute of Technology, Cambridge, MA 02319, USA.
[email protected] inant market players such as generators, consumers, and ISO,
market mechanisms such as gaming behaviors (ex. bidding with a perturbed market due to RER. In these papers, a
strategies), real-time prices via Demand Response, diverse stochastic framework is used to capture both overestimation
dynamic drivers (e.g., weather, load, fuel prices, and wind and underestimation of available wind power on the optimal
supply), and physical constraints (e.g., ramping, transmission expected profit of wind power producer, the overall market
congestion) [1]. These market models and analysis of the efficiency, and the overall operation cost. In this paper, we
marketequilibriumdirectlyhelptoidentifyvarioussourcesof utilize a deterministic version of the the model developed in
pricevolatilityanddependenceandsensitivitytouncertainties [41] and analyze the perturbed market and quantify the effect
andintermittenciesinrenewableenergysourcesandvariations of wind uncertainty on the market equilibrium.
in real-time prices due to Demand Response. Our focus in In recent years, there is a significant interest in Demand
this paper is on an electricity market model that captures the Response, a concept that allows the electricity demand to
dynamicsofmarketplayers,physicalconstraints,perturbations response to fluctuations in the electricity price [37], [43]. One
in renewable energy resources (RER), and variations due to way to reduce wind integration costs is to introduce demand
demand response. responseintheformoftime-variantretailelectricityrate,such
Various methods have been proposed in the literature to as real-time pricing (RTP). RTP can potentially reduce wind
determine market models [10], [11], [15]. Given the primary integration and forecast error costs, since consumer demand
purpose of market balance, these methods are primarily fo- could be made to follow the supply of wind generation by
cused on the analysis of market equilibrium. Methods in [10], usingapricesignal.UnderRTP,ifavailablewindgenerationis
[11] model the electricity market participants subject to spot lessthanforecast,thehighcostofdeployingancillaryservices
market equilibrium and their own constraints, leading to a tocoverthegenerationshortfallcanreduceelectricitydemand
method termed Equilibrium Programming with Equilibrium and the cost of serving the load [30]. Similarly, because wind
Constraints (EPECs). In such models, each agent solves a generation has zero marginal cost, electricity demand may
Mathematical Program with Equilibrium Constraints (MPEC) increase when there is more wind resource available than
[15], and by using stationarity theory for MPECs a standard forecast.
mixed complementarity problem (CP) is shown to lead to In this paper, we discuss the equilibrium of an electric
equilibrium [16]. energy market in the presence of renewable energy sources
Another approach to market modeling employs a varia- and demand response. Starting from the framework proposed
tional and complementarity Problem (CP) formulation (see in [14], we use the approach based on linear complementarity
[17], [18], [52], [19]) which models the equilibrium of the problem to delineate the underlying market model. Repre-
electricity market where forward and spot decisions are made senting the effect of forecast errors in renewable energy as
simultaneously. These models may also be specified by the an uncertain parameter, we evaluate its effect on the market
ISOs objective, maximization of social welfare [10], [11] or equilibrium. In a similar manner, focusing on RTP as the
maximization of wheeling or transmission revenue [17], [19]. meansbywhichdemandresponseisrealized,andrepresenting
A single-settlement framework based on a linear complemen- the effect of RTP as a parametric curtailment on the demand,
tarity problem (LCP) is proposed in [14], which leads to a the effect of this parameter on the overall market equilib-
strategic game in which players have coupled constraints and rium is also studied. A comprehensive market framework
the equilibrium properties are analyzed. Yet another approach is introduced in order to evaluate these uncertainty effects.
used for electricity markets is the use of Cournot models. This market framework consists of three main participants,
Here, the strategies of the generator companies are allowed generating companies including renewable energy sources,
to depend on other market participants, who also act so as consumer companies that are capable of responding to market
to maximize their own profits. This in turn allows a more changes, and independent system operators (ISO) who clear
transparent dependence of costs of generators and consumers the market by maximizing social welfare [14] and [44]. The
profit on the market equilibrium (see [20], [21], [52], [22], overall market equilibrium is first evaluated in the absence of
[23], [24], [1], [4], [3], [5]). uncertainties due to either forecast errors in renewable energy
In the context of a market for smart grid, the question that sources or due to demand response, and sufficient conditions
needs to be addressed is how the effect of RERs can be best fortheproposedmarketequilibriumtobeidenticaltotheNash
capturedinamarketmodel.TheintegrationofRERposesthe equilibriumisestablished.Themarketresponseinthepresence
challenge of intermittency and uncertainty, both of which in of uncertainties, and in particular, the market equilibrium is
turn can affect the economic planning and operation of the analyzed using perturbation analysis. A parametric charac-
overall power system. The uncertainties can impose costs due terization of the equilibrium shift is delineated. In both the
to the limited-dispatchability of intermittent generation [31], nominal and perturbed markets, the equilibriums are derived
thevariabilityinresourceavailability,anderrorsinforecasting using KKT conditions We next define formally the closeness
resource availability and loads [34], [35], [36]. There is a oftwostrategicgameandrelatetheequilibriumofclosegames
considerable literature covering many important aspects of using the notion of α−approximation and (cid:15)−equilibrium.
wind power ranging from comprehensive integration studies, IEEE 30-bus system is used to evaluate the results of both
forecasting methods, and technology issues (see [8], [32], uncertainties in renewable energy and in demand curtailment.
[33], [34], [35]). In the specific context of the incorporation The remainder of this paper is organized as follows: In
of renewable energy in a market, the results of [37], [38], Section II we present the necessary preliminaries, Section III
[39], [40], [41], [42] are more pertinent, all of which deal describesthemodelofthethreemarketparticipants.InSection
IV, the overall market equilibrium under nominal conditions are cocave. With these assumptions the optimization problem
isformulated.AperturbationanalysisisintroducedinSection (5) is termed a convex optimization problem. In addition, the
V to address the effects due to uncertainty, and the resulting constraint set for the optimization problem is convex which
effects are summarized. In Section VI numerical simulation allows us to use the method of Lagrange multipliers and the
resultsarereported.SectionVIIincludesconcludingremarks. Karush Kuhn Tucker (KKT) theorem which we state below
[27], [28].
II. PRELIMINARIES Theorem 1: Consider the optimization formulated in (5),
where f(x) is a convex function, g (x) are affine functions,
Inthissection,weprovidesomepreliminariesrelatedtothe n
andh (x)arecocavefunctions.Letx∗ beafeasiblepoint,i.e.
wholesale energy market structure. These include the funda- n
a point that satisfies all the constraints. Suppose there exists
mental theorems of convex optimization which are presented
constants λ and µ ≥0 such that
in Sections II-A. Given the close relationship between convex n m
optimization and Game theory, we present in Section II-B N
(cid:88)
related definitions and theorems related to game theory and ∇f(x∗)+ λ ∇g (x∗)+
n n
Nash equilibrium. In particular, the link between the convex n=1
optimization problem and uniqueness of the Pure Strategy L (6)
(cid:88)
Nash Equilibrium is presented including sufficient conditions µm(Rmn∇hn(x∗)−cm)=0 ∀n=1...N
forthelatter.Theseinturnaredirectlyusedinestablishingthe m=1
equilibrium of the wholesale market in Section III. We start µm(Rmnhn(x∗)−cm)=0 ,∀m=1,...,L
with a few basic definitions.
then x∗ is a global maximum. If f(x) is strictly concave then
Definition 1: A set K ⊆R is convex if for any two points
x∗ is also the unique global maximum.
x,y ∈K,
Proof: see [28].
αx+(1−α)y ∈K, ∀x,y ∈K andα∈[0,1]. (1)
B. Game Theory and NASH Equilibrium
Definition 2: Given a convex set K ⊆ R and a function
An alternative game-theoretic approach can be used to
f(x) : K → R; f is said to be a convex function on K if,
describethemarketequilibrium[51],[52].Todescribeagame,
∀x,y ∈K and α∈(0,1),
there are four things to consider: 1) the players, 2) the rules
f(αx+(1−α)y)≤αf(x)+(1−α)f(y), (2) of the game, 3) the outcomes and 4) the payoffs and the
preferences (utility functions) of the players. A player plays
Furthermore, a function f(x) is concave over a convex set if a game through actions. An action is a choice or election
and only if the function −f(x) is a convex function over the that a player takes, according to his (or her) own strategy.
set. Since a game sets a framework of strategic interdependence,
Definition 3: Given a scalar-valued function f(x) : Rn → aparticipantshouldbeabletohaveenoughinformationabout
R we use the notation ∇f(x) to denote the gradient vector of its own and other players’ past actions. This is called the
f(x) at point x, i.e., informationset.Astrategyisarulethattellstheplayerwhich
(cid:104) (cid:105)T action(s) it should take, according to its own information set
∇f(x)= ∂f(x),...,∂f(x) . (3)
∂x1 ∂xn at any particular stage of a game. Finally, a payoff function
expresses the utility that a player obtains given a strategy
Definition 4: Given a scalar-valued function f(x) :
(cid:81)Ii=1Rmi → R we use the notation ∇if(x) to denote the pdreofifinleedfoasr afollllpolwayse.rs. More formally a strategic form game is
gradient vector of f(x) with respect to x at point x, i.e.,
i
Definition 5: A strategic forms game is a triplet
∇ f(x)=(cid:104)∂f(x),...,∂f(x)(cid:105)T . (4) (cid:104)I,(Si)i∈I,(ui)i∈I(cid:105) such that I is a finite set of players ,
i ∂x1i ∂xmi i i.e. I ={1,...,I}, Si is the set of available actions for player
i, s ∈S is an action for player i, and finally u :S →R is
i i i
A. Convex Optimization the payoff function of player i where S =(cid:81) S is the set of
i i
Consider a generic optimization problem all action profiles.
In addition, we use the notation s = [s ] as a vector of
Maximize f(x) −i (cid:81) j j(cid:54)=i
actionsforallplayersexcepti,S = Sjasthesetofall
s.t.g (x)=0, ∀n=1,...,N −i j(cid:54)=i
n actionprofilesforallplayersexcepti,andfinally(s ,s )∈S
(5) i −i
N denoted a strategy profile of the game. Informally, a set of
(cid:88)
Rmnhn(x)≥cm, ∀m=1,...L strategies is a Nash equilibrium if no player can not do better
n=1 by unilaterally changing his or her strategy.
where f(x) is called the objective function or cost function, Definition 6: APureStrategyNashEquilibriumofastrate-
R is a matrix of constants and c are constants. We assume gicgame(cid:104)I,(S ) ,(u ) (cid:105)isastrategyprofiles∗ ∈Ssuch
m i i∈I i i∈I
that f(x) : Rn → R is a convex function to be maximized that for all i∈I
overthevariablex,thefunctionsg (x)asequalityconstraints
n
u (s ,s∗ )≤u (s∗,s∗ )∀s ∈S . (7)
are affine, and the functions hn(x) as inequality constraints i i −i i i −i i i
Definition 7: Let us assume that for player i ∈ I, the the optimal dispatch but also the optimal Local Marginal
strategy set S is given by Price which is determined as the corresponding shadow price.
i
In what follows, we model each of the components (i)-(iii)
S ={x ∈Rm|h (x )≥0} (8)
i i i i together with their constraints and the optimization goals.
where h : Rm → R is a concave function and the set of
i i
strategy profiles S = (cid:81)I S ⊂ (cid:81)I Rm is a convex set.
i=1 i i=1 i A. Generating Company
The payoff functions (u ,...,u ) are said to be diagonally
1 I It is assumed that the generating company consists of N
strictly concave for x∈S if for every x∗,xˆ∈S, we have G
generating units, and that the production of each generating
(xˆ−x∗)T∇u(x∗)+(x∗−xˆ)T∇u(xˆ)>0. (9) unit i, i = 1,...,N is divided into b power blocks, where
G
b=1,...,N , and is denoted as P . The associated linear
Gi Gib
operating cost is denoted as λC . The goal of the company
In the current context, the optimization problem in (5) can Gib
is to maximize its overall profit, and is stated as
be viewed as a game played by all player i ∈ I, and so that
tcoognedtihteiornth(e6y)cinanthaerroirveemat1a. Nashequilibriumthatsatisfiesthe maximize (cid:88) N(cid:88)Gi(cid:0)ρn(i)−λCGib(cid:1)PGib (11)
We now present a theorem that links the equilibrium point i∈Gf b=1
oftheconvexoptimizationproblemin(5)toNashequilibrium where ρ denotes the LMP of unit i at node n in the
n(i)
using the above definitions. network. The power production is subject to the following
Theorem 2: Consider a strategic form game
constraints:
(cid:104)I,(S ) ,(u ) (cid:105) with S as in Definition 7. Assume that
the syimim∈Ietricimi∈aItrix (U(xi) + UT(x)) is negative definite N(cid:88)GiP ≤Pmax :α ;∀i∈G (12a)
for all x ∈ S, where the matrix U(x) is the Jacobian of Gib Gi i f
b=1
the gradient vector of u(x), i.e. the jth column of U(x) is P ≤Pmax :φ ;∀i∈G ;b=1,...,N (12b)
∂u(x)/∂x , j = 1,...,I. Then the game has a unique pure Gib Gib ib f Gi
j
P ≥0;∀i∈G ;b=1,...,N (12c)
strategy Nash equilibrium identical to the optimal solution Gib f Gi
x∗ ∈(cid:81)Ii=1Rmi of the following optimization problem where αi and φib are the corresponding shadow prices. The
max u (y ,x∗ ) decision variables of this problem are the amounts of power
yi∈Rmi i i −i PGib to be generated by each unit i in each block b. While
s.t.h (x )≥0. (10) ρ are variables in the overall OPF problem, they can be
i i n(i)
viewedasfixedvalueswithrespecttotheoptimizationof(11).
Proof: From Theorem 6 in [29] we have that if U(x)+
Necessary and sufficient conditions for the optimization of
UT(x) is negative definite for all x ∈ S, then the payoff
(11)subjectto(12a)-(12c)areenumeratedbelow,usingdual
functions (u ,...,u ) are diagonally strictly concave. This in
1 I variablesα andφ ,whichcorrespondtotheKKTconditions
turn, using Theorem 2 in [29], implies that there is a unique i ib
[14]
pure strategy Nash equilibrium that is identical to the optimal
solution x∗ ∈ (cid:81)I Rm of the convex optimization problem P (cid:0)−ρ +λC +α +φ (cid:1)=0∀i∈G ; (13a)
i=1 i Gib n(i) Gib i ib f
in (10).
b=1,...,N
Gi
III. MODELINGOFMARKETAGENTBEHAVIOR α (cid:0)−N(cid:88)GiP +Pmax(cid:1)=0;∀i∈G (13b)
i Gib Gi f
Inthissection,wederivethemodeloftheoverallelectricity
b=1
market.Themaincomponentsofthismarketinclude(i)Gener- φ (cid:0)Pmax−P (cid:1)=0;∀i∈G ;b=1,...,N (13c)
ib Gib Gib f Gi
ating Companies (GenCo) (ii) Consumer Companies (ConCo)
(iii) Independent System Operator (ISO) [14]. We focus our
attention in this paper on the wholesale market, and use Definition 8: GenCoiissaidtobehaverationallyiftheas-
a deterministic framework to represent all components. The sociatedlinearoperatingcostdenotedasλCGib isnondecreasing
underlyingproblemthatistobeaddressedisthedetermination with respect to PGib. That is
ofthepowerproductionofthegeneratingunitsandthepower d(λC )/d(P )>0, ∀i∈G , b=1,...,N . (14)
demanded by the consumers such that power is balanced at Gib Gib f Gi
all network nodes and social welfare is maximized while
B. Consumer Modeling
meeting capacity limits. This is a fairly complex optimization
problem since each of the components (i)-(iii) are subjected A consumer company (ConCo) is assumed to consist of
to constraints that may conflict with each other. For instance, N units, and the demand of each unit j, j = 1,...,N ,
D D
consumers are interested in maximizing their benefit, and are is divided into several blocks N , with a block-index k =
Dj
therefore interested in cheap electricity. Availability of such 1,...,N . The associated linear utility function is denoted
Dj
cheap generation is limited due to capacity constraints on as λU which represents the value of using electricity for
Djk
transmission lines. In this case, the ISO may need to force the consumer. The goal of a ConCo is to maximize the total
more expensive units to keep things in equilibrium. The solu- profit while consuming electricity. This profit, for a unit j in
tionoftheresultingoptimizationproblemnotonlydetermines block k connected to node n, is determined as the difference
between the utility λU and the corresponding LMP ρ (j) implies that if the assumed power is not available, power can
Djk n
[14]. Assuming that the corresponding power consumed is be purchased from an alternate source or that loads can be
denoted as P , the maximization problem can be posed as shed. Underestimation implies that surplus power is either
Djk
sold to adjacent utilities, or consumed through fast redispatch
NDj
Maximize (cid:88) (cid:88)(cid:0)λU −ρ (cid:1)P (15) and automatic gain control, or reduced through reduction of
Djk n(j) Djk conventional generation. This case is however not discussed
j∈Dqk=1 in this paper.
As before, this is subjected to the constraints
Using the above discussion, the objective function defined
P ≤Pmax :σ ;∀j ∈D ;k =1,...,N (16a) in (11) is modified as
Djk Djk j q Dj
N(cid:88)DjP ≥Pmin :ψ ;∀j ∈D ;k =1,...,N (16b) maximize (cid:88) N(cid:88)Gl(cid:0)ρ −λC −Cr (∆ )(cid:1)Pw (19)
Djk Djk jk q Dj n(l) Gwlb wlb wlb Glb
k=1 l∈Gwb=1
P ≥0;∀j ∈D ;k =1,...,N (16c)
Djk q Dj where Pw is the wind power from the lth wind generator,
where σj and ψjk are the corresponding dual variables for and assuGmlebd to lie in the interval (cid:2)PGwlbmin,PGwlbmax(cid:3). ∆wlb
(16a) and (16b). While the constraint (16c) is almost always is due to wind uncertainty, given by
satisfied in current-day markets, this may not be the case in
micro-grid topologies. The decision variables of this problem ∆wlb =P¯Gwlb∆Glb (20)
are P , the amounts of power to be consumed by each
Djk where P¯w is the mean wind power, and 0<∆ <1. We
demand j in each block k. While ρ are variables in the Glb Glb
n(j)
note that if there is underestimation, then ∆ needs to be
overall OPF problem, they can be viewed as fixed values, as Glb
negative.
before, with respect to the optimization of (15). Necessary
λC isthemarginalcostfunctionforthelth windgenerator
and sufficient conditions for the optimization of (15) subject Gw
whichlbis very close to zero. Finally, Cr (∆ ) is a cost
to (16a) and (16b) are enumerated below, using dual variables wlb wlb
incurred by committing specific generators as reserves [42],
σ and ψ , which correspond to the KKT conditions [14]
j jk
due to the wind uncertainty ∆ , and is modeled as a
PDjk(cid:0)ρn(j)−λuDjk −σj +ψjk(cid:1)=0 (17a) quadratic function wlb
∀j ∈Dq;k =1,...,NDj Cr (∆ )=b ∆ + cwlb∆2 . (21)
NDj wlb wlb wlb wlb 2 wlb
ψ (cid:0)(cid:88)P −Pmin(cid:1)=0;∀j ∈D ;k =1,...,N
j Djk Djk q Dj Wedefineanotherquantityxw whichrepresentsthepercentage
k=1
of wind penetration as
(17b)
σjk(cid:0)−PDjk+PDmjakx(cid:1)=0;∀j ∈Dq;k =1,...,ND(j17c) xw = (cid:80)(cid:80)l∈Gw(cid:80)(cid:80)NNbD=Gj1lPPmGwlabx (22)
j∈Dq k=1 Djk
Definition 9: ConCo j is said to behave rationally if the
where Pmax is the maximum power demanded by consumer
associated linear operating cost denoted as λu is non- Djk
Djk j inblockk definedin(16a).Wenotethattheimpactofwind
increasing with respect to P . That is
Djk power on the market equilibrium is much smaller if xw is
d(λu )/d(P )<0, ∀j ∈D ,k =1,...,N . (18) small, i.e., if wind penetration is low, than if xw is large.
Djk Djk q Dj
Yet another component that we introduce into the picture
C. Effect of Uncertainties and Demand Response is Demand Response [53], [9]. As for GenCo, categorizing
all ConCo units into dispatchable and non-dispatchable ones,
Themostdominantimpactoftheintroductionofdistributed
we consider a dispatchable load P and assume that it
energy resources is uncertainties, which can directly alter the Djk
is affected by Demand Response and is sensitive to price
power generated. This in turn can affect the overall market
changes. This effect is modeled using a control parameter
equilibrium as well as the corresponding LMP. We take a
κ , and denotes the response of the consumers to a change
first step in this direction by introducing uncertainties in the Djk
in the Real Time Price (RTP) which occurs due to Demand
decision variables introduced in section A. We first separate
Response.
the family of P into PC , i = 1,...,n , and PR ,
Gib Gib C Glb
l = 1,...,nR, where nC denotes the more conventional P¯Djk =PDjk(cid:0)1−κDjk(cid:1) 0<κDjk <1 (23)
dispatchable generating units, and n denotes distributed
R
energyresourcessuchasthosebasedonwindandsolarenergy, P¯ denotes the consumption incorporated with demand
Djk
which are non-dispatchable. At each node, both conventional responsiveness into RTP. A positive κ , denotes a decrease
Djk
generators and wind generators may be committed for pro- in the ConCo consumption while a negative κ , denotes
Djk
duction. We assume that the wind GenCo are competitive and an increase. In this paper, we restrict our attention to positive
thattheysubmittheirbidstothemarketasotherconventional κ since our focus is on cases where there is a shortfall in
Djk
GenCo, and not modeled as a negative demand. It is also the non-dispatchable GenCo, i.e. ∆ >0. It is assumed that
Glb
assumedthattheavailablewindenergyisoverestimatedwhich κ is suitably calibrated to represent the effect of RTP on
Djk
the consumer behavior. Noting that this control parameter is that a generating unit injecting power at a given node is paid
proportional to the elasticity factor (cid:15) defined as [9] the locational marginal price corresponding to that node; and
conversely,ademandreceivingpowerfromagivennodepays
∆PDjk
(cid:15)= PDjk (24) the locational marginal price corresponding to that node. The
∆ρn(j) most dominant network constraints are due to line capacity
ρn(j) limits and network losses. Technical constraints cause the
andisameasureoftheconsumptionresponsetoRTPchanges,
power flow through any line to be limited. The power flow is
we denote this control parameter, κ as Curtailment Factor.
Djk said to be congested when it approaches its maximum limit.
In contrast to ∆ , which may be unknown, curtailment
Gkb This constraint is explicitly included in our model below. The
factor κ , is assumed to be controllable. Nevertheless, the
Djk secondconstraintisduetolosses,mostofwhichareduetothe
equilibriumaswellastheLMPsarealteredduetothepresence
heatlossinthepowerlines.Foreaseofexposition,suchohmic
oftheperturbationparameters∆ ,andκ .InSectionIII,
Gkb Djk losses are not modeled in this paper. The standard market-
asimilaroptimizationprocedureasaboveandequivalentKKT
clearing procedure is based on Social Welfare [46]. Denoting
conditions to (13a)-(13c) and (17a)-(17c) are derived, and the
this as S , it is defined as
W
dependence of the equilibrium and LMPs on the perturbation
parameters are discussed in detail. (cid:88) N(cid:88)Dj (cid:88) N(cid:88)Gi
S = λB P − λB P (25)
It should be noted that the effect of RTP is modeled in this W Djk Djk Gib Gib
paper in the form of κDjk.The inherent assumption here is j∈Dqk=1 i∈Gf b=1
that the nondispatchable ConCo observes the LMP, which is where the first and second term denote the revenue due
the solution of (17a)-(17c), and suitably adjusts its demand. to surpluses stemming from bids from GenCo and ConCo,
This adjustment is represented through the curtailment factor respectively. The market-clearing procedure is then given by
κ and therefore the elasticity factor (cid:15). Therefore, it may
Djk
MaximizeS (26)
be argued that the RTP is assumed to be the same as LMP in W
thispaper,andthattheeffectofRTPismodeledessentiallyas
subject to
a Demand Response parameter, denoted by κ . Further im-
Djk
provements of such a modeling of RTP and demand response (cid:88)N(cid:88)Gi (cid:88)N(cid:88)Dj
P − P −
are the focus of ongoing research. Gib Djk
i∈θ b=1 j∈ϑk=1
(cid:88) (cid:2) (cid:3)
D. Independent System Operator ISO modeling B δ −δ =0;ρ ∀n∈N (27a)
nm n m n
In addition to GenCo and ConCo, a competitive electricity m∈Ω
market also includes a coordinator which is independent of Bnm(cid:2)δn−δm(cid:3)≤Pnmmax;γnm∀n∈N;∀m∈Ω (27b)
these profit-maximization entities [46], [45], [50]. Such an
The constraints (27a) and (27b) are due to power balance and
independent entity is necessary in order to maintain order in
capacity limits, respectively. It can be seen that the associated
the overall market. Denoted as Independent System Operator
Lagrange multipliers, ρ and γ , are indicated in each
n nm
(ISO), this entity is responsible for guaranteeing a non-
constraint.
discriminating access for all participants to the market, and
The underlying optimization problem of the ISO can be
enforces various guidelines such as maintaining transmission
therefore defined as the optimization of (26) subject to con-
capacity limits. While in practice, distinctions are made be-
straints (27a) and (27b). This problem can be restated as
tween an ISO and what is termed a Market Operator (MO)
the solutions of the generation power blocks levels P , the
Gib
where the latter is responsible for financial management and
demand power blocks levels P , the voltage angle δ , and
Djk n
the latter for overall physical supervision, in this paper we
dual variables ρ and γ such that
n nm
combine the two roles into the same rubric of an ISO. The
responsibilitiesoftheISOaretoenforcetransmissioncapacity P (cid:0)λB −ρ (cid:1)=0∀i∈G ;b=1,...,N (28a)
Gib Gib n(i) f Gi
limits,tomaintainindependencefromthemarketparticipants, P (cid:0)ρ −λB (cid:1)=0∀j ∈D ;k =1,...,N (28b)
Djk n(j) Djk q Dj
to avoid discrimination against the market participants, and to
(cid:0) (cid:88) (cid:2) (cid:3)
δ B ρ −ρ +
promote the efficiency of the market. n nm n m
The electricity market that we consider in this paper is m∈Ω
(cid:88) (cid:2) (cid:3)(cid:1)
wholesale and is assumed to function as follows: First, each B γ −γ =0∀n∈N (28c)
nm nm mn
generating company submits the bidding stacks of each of m∈Ω
its units to the pool. Similarly, each consumer submits the (cid:0) (cid:88)N(cid:88)Gi (cid:88)N(cid:88)Dj
bidding stacks of each of its demands to the pool. Then, the ρ − P + P +
n Gib Djk
ISO clears the market using an appropriate market-clearing i∈θ b=1 j∈ϑk=1
procedure resulting in prices and production and consumption (cid:88) (cid:2) (cid:3)(cid:1)
B δ −δ =0∀n∈N;ρ >0 (28d)
nm n m n
schedules. The market-clearing procedure may introduce net-
m∈Ω
work constraints, which model losses [46] and line capacity γ (cid:0)Pmax−B (cid:2)δ −δ (cid:3)(cid:1)=0∀n∈N;∀m∈Ω(28e)
limits [49]. In this paper we include network constraints in nm nm nm n m
the underlying market model and the resulting prices are The decision variables of this problem are the amounts of
therefore the Locational Marginal Prices [48]. This implies power to be generated by each generating unit i in each
block b, i.e., PGib; the amounts of power to be consumed by (cid:0) (cid:88)N(cid:88)Gi (cid:88)N(cid:88)Dj
ρ − P + P +
each demand j in each block k, i.e., P and the locational n Gib Djk
Djk
marginal prices, ρ . It should be noted that (28d) is arrived i∈θ b=1 j∈ϑk=1
n(j) (cid:88) (cid:2) (cid:3)(cid:1)
at using the balance equation, which implies that the term in Bnm δn−δm =0 (29j)
the corresponding parenthesis is zero, and by multiplying this m∈Ω
term by ρ , which is positive.
In summnary, the market model that we consider in this γnm(cid:0)Pnmmax−Bnm(cid:2)δn−δm(cid:3)(cid:1)=0 (29k)
paper include optimization goals of GenCo discussed in sec-
The above problem can be compactly stated as the solution
tion II-A. and formulated in eqs. (11)-(12c), optimization of the following LCP: Find a vector x∗ ∈Rn that solves the
goals of ConCo, (15)-(16c), and formulated in II-B, and
following constraints:
finally optimization of the ISO discussed in section II-D,
and formulated in eqs (25)-(27b). The combined problem xT(cid:0)Mx+q(cid:1)=0
statement is the determination of an equilibrium point which x≥0
collectively optimizes all the above three market contributors. Mx+q ≥0 (30)
The corresponding equilibrium point is therefore defined as
that which satisfies the following criteria: where x ∈ Rn×1 is the variables vector and square matrix
M ∈Rn×nandq ∈Rn×1containsoperatingcostsandbidsof
1) maximum profit for every individual generating com-
generators,utilitiesandbidsofconsumers,maximumandmin-
pany
imum limit of generators and consumers and also maximum
2) maximum utility for every individual consumer
thermal limit of transmission lines (See Nomenclature for all
3) maximum net social welfare for the ISO.
definitions). The corresponding solution x∗ of the underlying
LCP problem, which determines the market equilibrium and
IV. MARKETEQUILIBRIUM is dependent on M and q, is denoted as
We now discuss the overall market equilibrium. In the x∗ (cid:44)LCP(M,q). (31)
GenCoandConCooptimizationproblems,givenin(11)-(12c)
and (15)-(16c), respectively, the Locational Marginal Prices
Definition 10: A matrix M ∈Rn×n is called a P-matrix if
appearasinputs.Ontheotherhand,theseLMPsappearinthe
its all principal minors are positive.
optimization problem of the ISO in (25)-(27b) as dual prices
WenowstateandprovethetheoremthatdiscussestheLCP
corresponding to the balance equations, causing a tight link
solution.
between the three families of optimization problems, thereby
Theorem 3: InLCPproblemdefinedin(30),matrixM isa
making the overall problem nontrivial.
P-matrix if and only if the LCP(M,q) has a unique solution
We note that each of these three sets of optimization
for any q ∈ Rn. Moreover, if M is a P-matrix then there is
problemsarelinearprogrammingproblems.Thus,theKarush-
a neighborhood M of M, such that all matrices in M are
Kuhn-Tucker (KKT) optimality conditions are both necessary
P-matrix.
and sufficient for describing the optimal solutions. The op-
Proof: see [54].
timality conditions of the three sets of problems result in
An additional result that will prove to be useful for our
a Linear Complementarity Problem (LCP) which correspond
following discussions concerns the dependence of x(M,q) on
to the market equilibrium for all i ∈ G , b = 1,...,N ,
f Gi
the market parameters. This is characterized in the following
j ∈D , k =1,...,N , n∈N, and m∈Ω as follows:
q Dj
theorem.
P (cid:0)−ρ +λC +α +φ (cid:1)=0 (29a) Theorem 4: [55] Matrix A is a P-matrix if and only if
Gib n(i) Gib i ib (I−D+DA) is nonsingular for any diagonal matrix D =
αi(cid:0)−N(cid:88)GiPGib+PGmiax(cid:1)=0 (29b) diag(d) with 0 ≤ di ≤ 1, i = 1,2,...,n. Furthermore, if
x(A,q) and x(B,p) are the solution of the corresponding
b=1
φ (cid:0)Pmax−P (cid:1)=0 (29c) LCP(A,q) and LCP(B,p) respectively, we have:
ib Gib Gib
P (cid:0)ρ −λu −σ +ψ (cid:1)=0 (29d) (cid:107)x(B,P)−x(A,q)(cid:107)
Djk n(j) Djk j jk ≤β(A)(cid:107)(A−B)x(B,p)+q−p(cid:107)
(cid:107)x(B,P)(cid:107)
ψ (cid:0)N(cid:88)DjP −Pmin(cid:1)=0 (29e) (32)
j Djk Djk where
k=1
σ (cid:0)−P +Pmax(cid:1)=0 (29f) β(A)=max ||(I−D+DA)−1D||. (33)
jk Djk Djk d∈[0,1]
PGib(cid:0)λBGib−ρn(i)(cid:1)=0 (29g) In the following theorem, the connection between market
P (cid:0)ρ −λB (cid:1)=0 (29h) equilibrium and Pure Strategy Nash Equilibrium is presented.
Djk n(j) Djk
(cid:0) (cid:88) (cid:2) (cid:3)
δ B ρ −ρ +
n nm n m Theorem 5: Assume that all GenCos and ConCos are ra-
m∈Ω tional players. If M is a P-matrix, then equilibrium of the
(cid:88) (cid:2) (cid:3)(cid:1)
Bnm γnm−γmn =0 (29i) market denoted as x∗ (cid:44)LCP(M,q) exists and is identical to
m∈Ω a unique Pure Strategy Nash Equilibrium.
Proof:Weprovethispropositionbyusingcontradictions. a constant η as
That is, assume that GenCos and ConCos are rational, M is
η =(cid:15) β(M)(cid:107)M (cid:107), (40)
a P-matrix and x∗ (cid:44)LCP(M,q) is not a Pure Strategy Nash M
Equilibrium. Since M is a P-matrix, then Theorem 3 implies and a set M as
that x∗ (cid:44)LCP(M,q) exists and is the unique maximizer of
(cid:8) (cid:12) (cid:9)
the GenCo problem denoted in (11) - (12b), ConCo problem M: ∆M(cid:12)β(M)||∆M||≤η (41)
denoted by (15) - (16b), and ISO problem denoted by (26) - We now quantify the relation between x∗ in (37) and x∗
∆
(27b). That is
in (31), in the following theorem.
u (x ,x∗ )≤u (x∗,x∗ )∀x ∈S . (34) Theorem 6: If the nominal market (30) has unique solution
i i −i i i −i i i and η < 1, then the perturbed market in (36) has a unique
Since GenCos and ConCos are rational, the inequalities in solution x∗ and satisfies the following inequality
∆
(14)and(18)hold.FromTheorem2,itfollowsthatthepayoff
(cid:107)x∗−x∗ (cid:107) 2(cid:15)
functions(u ,...,u )arediagonallystrictlyconcave.Theorem ∆ ≤ β(M) (42)
1 I (cid:107)x∗ (cid:107) 1−η
2, in turn, implies that the game between GenCo, ConCo, and
ISO has a unique Pure Strategy Nash equilibrium denoted as Proof: If the nominal electrical market (30) has a
x ∗. Using the Nash Equilibrium definition in (7), it follows unique solution according to Theorem 1, the matrix M is
N
that P-matrix. This in turn, according to Theorem 2, implies
u (x ,x∗ )≤u (x∗ ,x∗ )∀x ∈S . (35) that (I−D+DM) is nonsingular for any diagonal matrix
i i −i i Ni N−i i i D =diag(d ) with 0≤d ≤1, i=1,2,...,n. We have the
i i
Sinceweassumethatx∗ (cid:44)LCP(M,q)isnotaPureStrategy following equality
NashEquilibrium,itimpliesthatbothx∗andx∗N maximizethe (cid:0)I−D+D(M +∆M)(cid:1)=(I−D+DM)(I+M ∆M)
payoffswhichcontradictstheuniquenessofx∗ (cid:44)LCP(M,q) 0
(43)
and therefore completes the proof.
Remark 1: Theorem5providesconditionsunderwhichthe where
markethasauniqueequilibrium,i.e.nodegeneracycanoccur M =(I−D+DM)−1D. (44)
0
[14] in the market if M is a P-matrix.
Noting the definition of β(M), it follows that
We now proceed to the effect of uncertainties and curtail-
ment factor on the market equilibrium. ||M ∆M||≤β(M)||∆M||≤η ∀∆M ∈M (45)
0
Sinceη <1accordingtotheorem3,itfollowsthatI+M ∆M
V. PERTURBATIONANALYSISOFMARKETEQUILIBRIUM 0
is nonsingular for all ∆M ∈M. We therefore conclude from
(cid:0) (cid:1)
Using the results in the previous section, we now present (43) that I−D+D(M+∆M) is a P-matrix and therefore
a perturbation bound for the equilibrium of the electrcial the market in (36) has a unique solution for all ∆M ∈ M
market. As we discussed above, the market equilibrium is and η <1.
the solution of LCP(M,q) which is denoted as x∗ and has Wenowshowthatx∗ satisfies(42).Using(32)inTheorem
∆
beenexplicitlydefinedinEq.(30).AswediscussedinSection 2, we have that:
II.C, the wind forecast error defined in (20), parametrized by
(cid:107)x∗−x∗ (cid:107)
∆ , and demand response defined in (23), parametrized by ∆ ≤β(M +∆M)||∆Mx∗+∆q|| (46)
Glb (cid:107)x∗ (cid:107)
κ , are considered as two sources of perturbations. These
Djk
two components of uncertainty can affect M and q which We rewrite the argument of β(M +∆M) as
explicitlyinouranalysisaredenotedbyM+∆M andq+∆q, (cid:0) (cid:1)−1
I−D+D(M +∆M) D =
respectively. Therefore the underlying LCP problem under
parametric perturbation is altered as (cid:0)I+M (∆M)(cid:1)−1(I−D+DM)−1D (47)
0
xT ((M +∆M)x+(q+∆q))=0 Using Appendix 1 in [2], we have:
x≥0 (cid:0) (cid:1)−1 1 1
|| I+M (∆M) ||≤ ≤ (48)
(M +∆M)x+(q+∆q)≥0 (36) 0 1−β(M)||∆M|| 1−η
Taking norms on both sides of (47), we obtain that:
This in turn leads to a corresponding equilibrium
1
x∗ (cid:44)LCP(M +∆M,q+∆q). (37) β(M +∆M)≤ β(M) (49)
∆ 1−η
In the rest of the paper we are looking for the maximum shift Using (46) implies that:
of the equilibrium. The following definitions are useful.
(cid:107)x∗−x∗ (cid:107) 1
Definition 11: Define non-dimensional perturbation param- ∆ ≤ β(M)||∆M +∆q|| (50)
(cid:107)x∗ (cid:107) 1−η
eters (cid:15) and (cid:15) as
M q
Consideringthedefinitionof∆M and∆q,weobtain,inturn,
||∆M||≤(cid:15) (cid:107)M (cid:107), (38)
M (cid:107)x∗−x∗ (cid:107) 2(cid:15)
∆ ≤ β(M) (51)
||∆q||≤(cid:15)q (cid:107)q (cid:107), (39) (cid:107)x∗ (cid:107) 1−η
TABLEI
where (cid:15) = max{(cid:15) (cid:107) M (cid:107),(cid:15) (cid:107) q (cid:107)} which is the desired
M q COSTFUNCTIONSDATAOFGENERATORS
bound.
Remark 2: Defining µ(∆ ,κ ) for all l ∈ G , b =
Glb Djk w Name Block1 Block2
1,...,NGi, j ∈Dq, k =1,...,NDj as Size(MW) Price($/MW) Size(MW) Price($/MW)
µ(∆Glb,κDjk)= 12−(cid:15)ηβ(M), (52) PPPggg125 111222...572 232404000 448000 1992050000
Pg11 12.2 240 80 1200
Theorem 6 implies that the uncertainty in market can lead Pg13 10 40 20 80
to a maximum shift by the equilibrium of an amount
µ(∆ ,κ ). As this function is nonlinear, determining
Glb Djk
an analytical relationship between µ(∆ ,κ ), ∆ and Using Definitions 13 and 12, it follows that
Glb Djk Glb
κ is exceedingly difficult. As will be shown in the next
Djk u (s ,s∗ )−u (s∗,s∗ )≤2α (57)
section, simulation studies show that as κ increases, µ i i −i i i −i
Djk
decreases.Thisinturnbringstheperturbedequilibriumcloser which in turn implies that s∗ ∈S is an (2α)−equilibrium of
to the nominal equilibrium. G.
Proposition 1: Assume that the perturbed market denoted
by the strategic game G˜ under uncertainties ∆ and κ
A. Game theoretic Interpretations Glb Djk
is an α−approximation of the nominal market denoted by G.
We now evaluate the perturbed market using tools from If x∗ is an equilibrium of the perturbed market G˜, then x∗
∆ ∆
game theory. Consider two strategic games defined by two is an (2α)−equilibrium of the nominal market G.
profiles of utility functions as G = (cid:104)I,(Si)i∈I,(ui)i∈I(cid:105) and Remark 3: If the wind forecast error ∆ increases, from
G˜ = (cid:104)I,(Si)i∈I,(u˜i)i∈I(cid:105). If s∗i is a Nash equilibrium of G, Definition 13, it follows that α will increGalsbe as the relative
then s∗ need not be a Pure Strategy Nash Equilibrium of G˜.
i payoffsofthecorrespondinggeneratorswillhaveanincreased
TheequilibriaofthestrategicgamesGandG˜maybefarapart,
error.Proposition1impliesthatinsuchacase,theequilibrium
even if (ui)i∈I and (u˜i)i∈I are very close to each other. of the perturbed game G˜ is correspondingly far away from
Definition 12: Given (cid:15)≥0, a Pure Strategy Nash Equilib- the nominal market G. It should be noted that as the forecast
rium s∗ ∈ S is called an (cid:15)−equilibrium if for all i ∈ I and error increases, the corresponding cost of deploying ancillary
si ∈Si, servicesincreasesaswell.IfaDemandResponseprogramisin
u (s ,s∗ )≤u (s∗,s∗ )+(cid:15)∀s ∈S . (53) place, this cost increase is conveyed to the consumer, leading
i i −i i i −i i i toadecreaseintheloadquantifiedbythedemandcurtailfactor
Obviously in Definition 12, when (cid:15) = 0, the (cid:15)−equilibrium κDjk as in (23). Similar to our observation in Remark 2, it
isaPureStrategyNashEquilibriuminthesenseofDefinition should be noted that the exact impact of an increasing κDjk
6. on α is difficult to quantify due to its nonlinearity. However,
The following definition formally defines the closeness of simulation studies show, as discussed in Section VI, that as
two strategic form games. κDjk increases, α decreases. This in turn brings the perturbed
Definition 13: Let G = (cid:104)I,(S ) ,(u ) (cid:105) and G˜ = equilibrium closer to the nominal equilibrium.
i i∈I i i∈I
(cid:104)I,(S ) ,(u˜ ) (cid:105) be two strategic form games, then G˜ is
i i∈I i i∈I
a α−approximation of G if for all i∈I, VI. CASESTUDY
An IEEE 30-bus case is used for simulation studies, whose
|u (s)−u˜ (s)|≤α∀s∈S. (54)
i i
interconnections are shown in Figure 1. The size and price
Thenextpropositionrelatesthe(cid:15)−equilibriumofclosegames of each block of each GenCo are shown in Table I and for
as defined in 13. the sake of simplicity we assume that each GenCo bids in
Theorem 7: If G˜ = (cid:104)I,(S ) ,(u˜ ) (cid:105) is an α − its marginal cost. The size of minimum power requirement of
i i∈I i i∈I
approximation to G = (cid:104)I,(S ) ,(u ) (cid:105) and s∗ ∈ S is each demand and the size and price of each block of each
i i∈I i i∈I
an equilibrium of G˜, then s∗ ∈ S is a (2α)−equilibrium of demand are shown in Table II. We assume that only part of
G. the overall loads are participating in the market, and the fixed
Proof: For all i∈I and all si ∈Si, we can write loadsinotherbusesareshowninTableIII.ThereactanceBnm
of the line connecting bus n and bus m can be found in Table
u (s ,s∗ )−u (s∗,s∗ )=u (s ,s∗ )−u˜ (s ,s∗ )+
i i −i i i −i i i −i i i −i IV. The transmission capacity limit of all lines is chosen to
u˜i(si,s∗−i)−u˜i(s∗i,s∗−i)+ (55) be 100MW. The line parameters are perunit with three-phase
u˜ (s∗,s∗ )−u (s∗,s∗ ). base of 230 kV and 10 MVA.
i i −i i −i
Since s∗ is a Pure Nash Equilibrium of G˜, then u˜ (s ,s∗ )−
i i −i A. Nominal Market Equilibrium
u˜ (s∗,s∗ )≤0. This in turn follows that
i i −i
Table V provides equilibrium results concerning generator
ui(si,s∗−i)−ui(s∗i,s∗−i)≤|ui(si,s∗−i)−u˜i(si,s∗−i)|+ output, revenue, and profit for the electric power market if
|u˜ (s∗,s∗ )−u (s∗,s∗ )|. no wind shortfalls are imposed to the system. These results
i i −i i −i
(56) are obtained by directly solving LCP problem (29a)-(29k).
TABLEII
COSTFUNCTIONSDATAOFCONSUMERS
Name Block1 Block2
Size(MW) Price($/MW) Size(MW) Price($/MW)
Pd7 11 150 19 600
Pd15 10 100 21 600
Pd30 12 170 22 600
Pd9 12 150 24 600
Pd26 15.5 150 21 600
Pd27 9.5 150 22 600
TABLEIII
FIXEDLOADDATA
Name Demand Name Demand
Pd3 22.31 Pd14 7.20
Pd4 8.83 Pd16 4.06
Pd8 13.95 Pd17 10.46
Pd10 6.74 Pd18 3.72
Pd12 13.01 Pd19 11.04
Pd20 2.55 Pd21 3.39
Pd23 22.31 Pd24 10.11
Pd26 4.06 Pd29 2.78
Table VI shows the power consumed and the corresponding Fig.1. IEEE30-buscasestudy
demand payments. Since we ignore losses in this study, the
TABLEV
difference between the total generator power output and the
RESULTSOFMARKETEQUILIBRIUMFORGENCO,NOWINDUNCERTAINTY
total power consumed, and are equal to zero and both are
221MW whereas 151MW of the load is a fixed demand as
Name PoweroutputMW Revenues$/h Cost$/h TotalProfit$/h
can be seen in Table III. The Locational Marginal Price in all
Pg1 30 708.6 660 48.6
of the buses is 23.62$/MWh due to the fact that transmission Pg2 11.64 274.93 274.76 0.18
lines are not congested. Pg5 80 1889.6 1000 889.6
Pg11 80 1889.6 1000 889.6
Pg13 20 472.4 8 464.4
TABLEIV
TRANSMISSIONLINESREACTANCE;LINEREACTANCESAREPERUNITED B. Perturbed Market Equilibrium with Wind Uncertainty
BASEDON100MW
We assumed that GenCo at bus 13 in figure 2, is wind
Connected Connected based and is subjected to an uncertainty ∆G13. For the sake
Bus Reactance Bus Reactance ofsimplicityinsimulationsweassumethatuncertaintyforall
Bus Bus (xbk) Bus Bus (xbk) blocksarethesameandcharacterizedby∆ .Theresulting
b k b k G13
1 2 0.0575 1 3 0.1652 perturbed market, discussed in Section V, was simulated.
2 4 0.1737 3 4 0.0379 Figure 2 shows the corresponding market equilibrium shift
2 5 0.1983 2 6 0.1763
µ(∆ ,κ )asdefinein(52)fordifferentwindpenetration.
4 6 0.0414 5 7 0.116 Glb Djk
6 7 0.082 6 8 0.042 As can be seen in Fig 2, with increasing ∆G13, higher
6 9 0.208 6 10 0.556 equilibrium shift µ results. As wind penetration xw% defined
9 11 0.208 9 10 0.11
in (22) increases, the corresponding market equilibrium shift
4 12 0.256 12 13 0.14
12 14 0.2559 12 15 0.1304
12 16 0.1987 14 15 0.1997
16 17 0.1923 15 18 0.2185 TABLEVI
18 19 0.1292 19 20 0.068 RESULTSOFMARKETEQUILIBRIUMFORCONCO,NOWINDUNCERTAINTY
10 20 0.209 10 17 0.0845
10 21 0.0749 10 22 0.1499
21 22 0.0236 15 23 0.202 Name Powerconsumed TotalPayment$/h
22 24 0.179 23 24 0.27 Pd7 11 259.82
24 25 0.3292 25 26 0.38 Pd15 12 283.44
25 27 0.2087 28 27 0.396 Pd30 10 236.2
27 29 0.4153 27 30 0.6027 Pd9 15.5 366.1
29 30 0.4533 8 28 0.2 Pd26 9.5 224.4
6 28 0.0599 Pd27 12.0 283.44