11 Market Power in Convex Hull Pricing
Jian Sun and
Chenye Wu
Abstract —The start up costs in many kinds of generators leadto complex cost structures, which in turn yield severe marketloopholes in the locational marginal price (LMP) scheme. Convexhull pricing (a.k.a. extended LMP) is proposed to improve themarket efficiency by providing the minimal uplift payment to thegenerators. In this letter, we consider a stylized model where allgenerators share the same generation capacity. We analyze thegenerators’ possible strategic behaviors in such a setting, andthen propose an index for market power quantification in theconvex hull pricing schemes.
Index Terms —Convex Hull Pricing, Market Power, UpliftPayment, Incentive Design
I. I
NTRODUCTION
In general, the electricity market is organized by a sequenceof processes: unit commitment, day-ahead market, real timemarket, balancing market, etc . Since the focuses of differentprocesses are diverse, the incentives they provide do notnecessarily align with each other. To relieve the tensionbetween these processes, convex hull pricing (CHP) is onepromising solution: it provides the minimal uplift payment tothe generators for incentive alignment [1–3].However, although remarkable in improving the systemefficiency by reallocating the market surplus between supplyand demand [4], we find that the incentive issues in the CHPscheme have not been fully solved due to the non-convex coststructures. To the best of our knowledge, we are the first toidentify market power in the CHP schemes.In this letter, we first revisit the CHP scheme in Section II.And then we seek to understand each individual firm’s possiblestrategic behaviors in Section III. Based on the identifiedbehaviors, we propose the market power index and highlightthe existence of market power via numerical studies in SectionIV. Finally, concluding remarks are given in Section V. Weprovide all the necessary proofs in the Appendix.II. C
ONVEX H ULL P RICING : A B
RIEF R EVISIT
We consider the CHP scheme in the electricity pool modelwith n generators in the system. To highlight the existenceof strategic behaviors, we assume all the generators have thesame capacity G . In such a model, the system operator seeksto minimize the total generation costs at each time slot: c ( y ) := min g (cid:88) ni =1 f i ( g i ) s.t. (cid:88) ni =1 g i = y, ≤ g i ≤ G, (1)where g i denotes the output of generator i , f i ( g i ) denotes thegeneration cost for generator i , vector g is [ g , ..., g n ] , and y The authors are with the Institute of Interdisciplinary Information Sciences(IIIS), Tsinghua University, Beijing, China, 100084. C. Wu is the correspon-dence author. Email:[email protected] is the total load in the system. The optimal solution g ∗ ( y ) andthe optimal objective value c ( y ) are both functions of y .If f i ( g i ) is linear, then the operator could conduct theeconomic dispatch according to the merit order of marginalgeneration cost and set the price as the marginal cost.However, in practice, there are start up costs associatedwith generation. Hence, for generator i , f i ( g i ) is often of thefollowing form: f i ( g i ) = s i · I ( g i >
0) + v i · g i , (2)where s i is the sum of fixed cost and start up cost, and v i is the variable cost. To highlight the non-convexity in f i ( g i ) , we employ the indicator function I ( · ) . Such a coststructure challenges the conventional scheme in terms ofdispatch profile, price design, and incentive analysis.The solution proposed by CHP is to use the uplift paymentto incentivize the generators to follow the system operator’sdispatch profile.For each generator i , given price p , its desired generationlevel g di ( p ) is associated with its maximal profits, and can bedifferent from the dispatch profile g ∗ i : g di ( p ) = sup (cid:26) arg max z ∈ [0 ,G ] { p · z − f i ( z ) } (cid:27) . (3)And this generation level leads to the maximal profits given p , denoted by π ∗ i ( p ) , i.e., π ∗ i ( p ) = p · g di ( p ) − f i ( g di ( p )) . (4)The difference in the profits between generating g di ( p ) and g ∗ i needs to be compensated by uplift payment:Uplift i ( p, y ) = π ∗ i ( p ) − [ p · g ∗ i − f i ( g ∗ i )] . (5)And the CHP scheme is proved to guarantee that it leads tothe optimal price p ∗ in terms of minimizing the total upliftpayment [5]. In this letter, we adopt an equivalent form tocharacterize p ∗ : Lemma 1 : The optimal price p ∗ in CHP can be expressed asfollows: p ∗ = inf (cid:110) p (cid:12)(cid:12) (cid:88) ni =1 g di ( p ) ≥ y (cid:111) . (6)Given the structure of f i ( g i ) , we claim for demand of y , thereexists an optimal dispatch profile g ∗ , which is composed ofthe following dispatch profiles:1) m − elements are G , and n − m elements are 0.2) one generator is dispatched at x units.Here, m = (cid:100) yG (cid:101) , and x = y − ( m − · G . This allowsthe system operator to dispatch the generators roughly in theorder of their average generation costs . In our case, since allthe generators share the same capacity, the system operator This order plays the most important role in the optimal dispatch. Thereis minor issue in dispatching the generator which is scheduled to generate x units. We omit the detailed discussion due to the page limits. a r X i v : . [ m a t h . O C ] F e b can directly sort them with respect to f i ( G ) . For notationalsimplicity, we assume the subscript i for generator i alsodenotes its ascending order in the average generation costs.Hence, we can make the following observations. Fact 1 : We can establish the mapping between generator’sorder in average generation cost and its dispatch profile.1) If i < m , then g ∗ i ∈ { x, G } ; if i > m , then g ∗ i ∈ { , x } .2) If g ∗ i = 0 , then i ≥ m ; if g ∗ i = G , then i ≤ m .However, the CHP scheme only guarantees the minimaltotal uplift payment, it does not fully mitigate generators’opportunities in obtaining more profits by strategic bidding.III. S TRATEGIC B EHAVIOR A NALYSIS
We seek to understand generator’s strategic behavior viaanalyzing its profits. For each generator, its total profits aregained from the uplift payment as well as selling electricityaccording to the dispatch profile from the system operator.We want to emphasize that the second component is notnecessarily positive, which further implies the importance ofuplift payment.In practice, the generators are allowed to bid their costfunctions as well as their available capacities to the operator.In this letter, we assume the generator is not allowed towithhold its capacity. Hence, it can only strategically bid itscost function. Since the fixed cost and start up cost are ratherstable, the only remaining opportunity for manipulation is thevariable cost v i , for each generator i .Mathematically, if generator i truthfully bids its v i , given ademand of y , we denote its profits P i as benchmark: P i = π i ( p ∗ ( v i )) = { f m ( G ) − f i ( G ) } + . (7)Recall m = (cid:100) yG (cid:101) , and f m ( · ) denotes the cost function forgenerator m , which ranks the m th in terms of the averagegeneration cost. It is crucial in determining the optimal price p ∗ in CHP. We provide the detailed analysis in the Appendix.On the other hand, if the generator strategically reports itsgeneration cost as ˜ f i ( · ) (more precisely, a different variablecost ˜ v i ), which may lead to a potentially different ˜ p ∗ given byCHP, and a potentially different dispatch profile ˜ g ∗ . This strate-gic bidding will also reshuffle the order of generators in termsof average bid generation cost. We denote the reshuffled bidgeneration cost by ˜ f (1) , · · · , ˜ f ( n ) . Hence, generator i ’s totalprofits ˜ P i (˜ v i ) via strategic bidding by can be straightforwardlycharacterized as follows: ˜ P i (˜ v i ) = (cid:110) ˜ f ( m ) ( G ) − ˜ f i ( G ) (cid:111) + (cid:124) (cid:123)(cid:122) (cid:125) Profits by Strategic Bidding + (cid:16) ˜ f i (˜ g ∗ i ) − f i (˜ g ∗ i ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) Profits in Generation Cost . (8)Since for every generator i , ˜ g ∗ i ∈ { , x, G } , we can defineits maximal profits via strategic bidding as follows: sup { ˜ P i } = sup u ∈{ ,x,G } { ˜ P i | ˜ g ∗ i = u } . (9)This allows us to quantify each generation’s maximal addi-tional profits through strategic manipulation: M ( i ) := sup { ˜ P i } − P i . (10)Surprisingly, M ( i ) has a uniform and rather neat expression. &