A Scenario-oriented Approach for Energy-Reserve Joint Procurement and Pricing
AA Scenario-oriented Approach for Energy-ReserveJoint Procurement and Pricing
Jiantao Shi, Ye Guo*
Tsinghua-Berkeley Shenzhen InstituteTsinghua UniversityShenzhen, China
Lang Tong
School of Electrical & Computer EngineeringCornell UniversityIthaca, NY, USA
Wenchuan Wu, Hongbin Sun
Dept. of Electrical EngineeringTsinghua UniversityBeijing, China
Abstract —We propose a scenario-oriented approach forenergy-reserve joint procurement and pricing in electricity mar-kets. In this model, without empirical reserve requirements,reserve is procured according to all possible contingencies andload/renewable generation fluctuations in non-base scenarios, andthe deliverability of reserve is ensured through network con-straints in all scenarios considered. Based on the proposed model,a locational marginal pricing approach has been developed forboth energy and reserve. The associated settlement process isalso discussed in detail. Under certain assumptions, the proposedpricing approach is a set of uniform pricing at the same locationand the property of revenue adequacy for the system operatorhas also been established.
Index Terms —reserve, electricity market, locational marginalprices, energy-reserve co-optimization
I. I
NTRODUCTION
With increasing concerns about energy shortages and globalwarming, integrating more renewable generations into thepower system has become a worldwide trend. However, uncer-tain and intermittent renewable generations have brought newchallenges to power systems’ secure and reliable operations.To deal with these challenges, system operators need to prop-erly procure reserve from dispatch-able resources to defendagainst possible contingencies and load/renewable generationfluctuations. Since energy and reserve are tightly coupled, allthe independent system operators (ISOs) in the U.S. run jointoptimizations [1], by and large with the following model:(I) : minimize { g,r U ,r D } C Tg g + C TU r U + C TD r D (1)subject to λ : T g = T d, (2) µ : S ( g − d ) ≤ f, (3) ( γ U , γ D ) : T r U = R U , T r D = R D , (4) g + r U ≤ G, G + r D ≤ g, ≤ r U ≤ r U , ≤ r D ≤ r D . (5)The objective function (1) aims to minimize the bid-in cost ofenergy and reserve with decision variables ( g, r U , r D ) , whichdenote vectors of energy, upward reserve, and downwardreserve procured from generators. Coefficients ( C g , C U , C D ) denote bid-in price vectors of ( g, r U , r D ) . Constraints (2)-(5) represent, respectively, energy balancing constraints, linecapacity limits, reserve requirement constraints, and generator This work was supported in part by the National Science Foundation ofChina under Grant 51977115.Corresponding author: Ye Guo, e-mail: [email protected] capacity and ramping rate limits. Reserve clearing priceswill be obtained from the Lagrangian multipliers ( γ U , γ D ) ,representing the increase of total bid-in cost when there isone additional unit of upward/downward reserve requirement.Regarding the traditional co-optimization model (I), thereare several key issues that remain unclear. The first is aboutreserve requirements R U and R D . These parameters are ar-tificial and empirical, sometimes specified as the capacity ofthe biggest generator as in PJM [2] or a certain proportion ofsystem loads as in CAISO [3]. However, they will significantlyaffect the clearing results and prices of reserve and even theenergy market. The second is on the locations of reserve re-sources. The traditional model only considers network limits inthe base case. When there are contingencies or load/renewablegeneration fluctuations, the deliverability of reserve may stillbe limited by line capacities. A common solution is to partitionthe entire system into different zones and specify zonal reserverequirements as in ISO-NE [4], which is again empirical. Thethird is about the objective function (1), which is to minimizethe base-case bid-in cost but does not consider possible re-dispatch costs under contingencies and fluctuations.In light of these problems, there are also many researchefforts from academia. Some researches utilize the forecastmethods, such as the density forecasts which consider theprobability distributions of future observations [5], or thescenario forecasts which consider several typical future sce-narios [6], to define reserve requirements in systems withhigh renewable penetrations, see [7] for a survey. In [8], astatistical clustering algorithm is proposed to enable dynamicreserve zone partition. In [9] and [10], the energy balancingand network constraints considering events and re-dispatchesin contingency scenarios are analyzed, and the reserve pricingapproach at the bus level is proposed. In [11], the uncertaintymarginal prices are defined to price both uncertainty sourcesand reserve resources at the bus level.In this paper, a scenario-oriented energy-reserve co-optimization model is developed. These scenarios will corre-spond to possible contingencies or load/renewable generationfluctuations, or their combinations. Reserve procured fromeach generator is equal to the maximum range of its generationre-dispatch among all scenarios, thereby eliminating the needto specify the parameters of reserve requirements artificially.Transmission capacity limits in all scenarios are also includedin the proposed model, therefore the artificial partition ofreserve zones is no longer needed either. Thereupon, we a r X i v : . [ ee ss . S Y ] N ov erive marginal prices of generations, loads and reserve. Undercertain assumptions, energy and reserve marginal prices willbe locational uniform prices. We have also established theproperty of revenue adequacy for SOs: revenue from loadpayments, credits to generators including energy, reserve, andre-dispatches, and congestion rent will reach their balance inthe base case as well as in all scenarios considered.II. E NERGY -R ESERVE C O - OPTIMIZATION M ODEL
To properly address the problems of the traditional model(I), a scenario-oriented energy-reserve joint optimizationmodel is proposed. Consider the current market structure, theproposed model can be regarded as an intermediate look-aheadoptimization stage between the day-ahead (DA) schedulingand the real-time security constrained economic dispatch (RTSCED), like the ancillary service optimizer (ASO) in PJM.As Fig. 1 shows, ASO is one of many look-ahead stages toprocure reserve resources with different flexibility levels. Inpractice, ASO is only financial binding for reserve [2]. Inthis paper, however, we consider the proposed co-optimizationmodel to be financial binding for both energy and reserve.
Ancillary Service Optimizer ( ASO ) : inflexible reserve: recommendations for energy and reserve: energy basepoints and flexible reserve: price calculation Fig. 1. PJM real-time operations including ASO, IT SCED, RT SCED andLPC [2]
The proposed model makes the following assumptions:1) A standard DCOPF model with linear cost functions forenergy and reserve is adopted, which is consistent with thecurrent electricity market design.2) A single-period problem is considered for simplicity .3) For simplicity, we don’t consider the DA market. Ifthe DA market clearing results need to be considered, thequantities in the proposed model can be seen as the deviationsfrom the quantities in the DA scheduling, and all the qualitativeanalyses in this paper will still hold.4) Uncertainties from line outages and load fluctuations areconsidered. Generator outages are not considered .5) Renewable generations are modeled as negative loads.6) The SO’s predictions of possible scenarios as well astheir occurrence probabilities are perfect.Based on assumptions (1)-(6), the proposed model is: F ( g, r U , r D , δg Uk , δg Dk , δd k ) = C Tg g + C TU r U + C TD r D + k ∈K (cid:88) (cid:15) k ( C Tk δg Uk − C Tk δg Dk + C TL δd k ) , (6) A multi-period setting will bring in the coupling between ramping andreserve, which is a highly complicated issue. Generator outages will bring in the complicated issue of non-uniform pric-ing. Essentially, energy and reserve procured from generators with differentoutage probabilities are no longer homogeneous goods. We will skip thistricky problem now and leave it to our future work. (II) : minimize { g,r U ,r D ,δg Uk ,δg Dk ,δd k } F ( · ) , subject to ( λ, µ ) : T g = T d, S ( g − d ) ≤ f, (7) g + r U ≤ G, G + r D ≤ g, ≤ r U ≤ r U , ≤ r D ≤ r D , (8)for all k ∈ K : λ k : T ( g + δg Uk − δg Dk ) = T ( d + π k − δd k ) , (9) µ k : S k (cid:0) ( g + δg Uk − δg Dk ) − ( d + π k − δd k ) (cid:1) ≤ f k , (10) ( α k , α k ) : 0 ≤ δg Uk ≤ r U , (11) ( β k , β k ) : 0 ≤ δg Dk ≤ r D , (12) ( τ k , τ k ) : 0 ≤ δd k ≤ d + π k , (13)where K denotes the set of all non-base scenarios, coefficient (cid:15) k denotes the occurrence probability of scenario k , decisionvariables ( δg Uk , δg Dk , δd k ) denote vectors of generation upwardre-dispatches, downward re-dispatches, and load shedding inscenario k . Coefficients ( C k , C k ) represent vectors of genera-tion upward and downward re-dispatch prices in scenario k .If one generator’s downward reserve is deployed, it will payback to the SO, so there is a negative sign before ( C Tk δg Dk ) in (6). Coefficient C L denotes the vector of load sheddingprices. Coefficient π k denotes the vector of load fluctuationsin scenario k . The objective function (6) aims to minimizethe expected system total cost, including the base-case bid-incost and the expectation of re-dispatch costs in all scenarios.Constraints (7)-(8) denote the base-case limits as constraints(2)-(3) and (5) in the traditional model (I). Constraints (9)-(10) denote the energy balancing constraints and the networkconstraints in all non-base scenarios where S k and f k denotethe shift factor matrix and the line capacity vector in scenario k . In non-base scenarios, the power flows on transmission linesare allowed to exceed line capacities for short duration, and theexceeding ratings are reflected in the setting of f k . Constraints(11)-(12) denote that generation re-dispatches in all scenariosshall not surpass the procured reserve, indicating that theprocured reserve will be modeled as the maximum range ofgeneration re-dispatch among all scenarios. Therefore reservewill be optimally procured wherever necessary to minimize theexpected total cost. Constraint (13) denotes that load sheddingin each scenario must be non-negative and cannot exceedactual load power in that scenario.With such formulations, the solution to the proposed modelwill procure reserve at the best locations considering all kindsof costs. On the contrary, the solution to the traditional modelwill be either sub-optimal or infeasible in order to minimizethe expected total cost subject to energy balancing and linecapacity constraints in both the base case and all scenarios,even with reserve requirements R U and R D being the sum ofthe optimal procurement r ∗ U and r ∗ D in model (II).Moreover, marginal prices derived from the proposedscenario-oriented optimization model (II) have many attractiveproperties, as in the next section. In some ISOs in the U.S., generation re-dispatches will be settled at thereal-time LMPs plus some premiums or adders e.g. ERCOT [12].
II. P
RICING AND S ETTLEMENT
Based on the proposed model, in this section we present thepricing approach for generations, loads, and reserve, as wellas the associated market settlement process.
A. LMP calculations for generations, loads and reserve
The proposed pricing approach for energy and reserve isbased on their marginal contributions to the expected systemtotal cost presented in (6). Namely, consider any generator j , we first fix g ( j ) , r U ( j ) , r D ( j ) at their optimal values g ( j ) ∗ , r U ( j ) ∗ , r D ( j ) ∗ and consider them as parameters insteadof decision variables. Such a modified optimization modelis referred to as model (III). Compared with the originalmodel (II), in model (III) the terms that are only related togenerator j , specifically the bid-in cost of generator j in theobjective function (6) and the j th row of all constraints in(8), will be removed. Next, we evaluate the sensitivity of theoptimal objective function of model (III), which representsthe expected cost of all other market participants except forgenerator j , with respect to the deviations in parameters g ( j ) , r U ( j ) and r D ( j ) . According to the envelop theorem, wehave: ∂C ∗ III ∂g ( j ) = ∂ L III ∂g ( j ) = S (: , m j ) T µ − λ + k ∈K (cid:88) ( S k (: , m j ) T µ k − λ k ) , (14)where C ∗ III denotes the optimal objective function of model(III) and L III denotes the Lagrangian function of model (III).Accordingly the marginal energy price of any generator is: η g ( j ) = − ∂ L III ∂g ( j ) = λ − S (: , m j ) T µ + k ∈K (cid:88) ( λ k − S k (: , m j ) T µ k )= ω ( j ) + k ∈K (cid:88) ω k ( j ) , (15)where η g ( j ) denotes the marginal energy price of generator j ,with the term m j denoting bus m where generator j is located.The term ( λ − S T µ ) , denoted by ω for brevity hereafter,denotes the base-case component of the energy prices ofgenerators. The term ( λ k − S Tk µ k ) , denoted by ω k for brevityhereafter, denote the non-base component of the energy pricesof generators in scenario k . We can see that each componenthas an energy part and a congestion part, as in the standardLMP formulation. Similarly, the marginal energy prices ofloads are calculated as η d = ∂C ∗ III ∂d = ∂ L III ∂d = λ − S T µ + k ∈K (cid:88) ( λ k − S Tk µ k ) − k ∈K (cid:88) τ k = ω + k ∈K (cid:88) ω k − k ∈K (cid:88) τ k , (16)which is consistent with the marginal energy prices of gener-ators except for the last term ( − (cid:80) τ k ) , which are multipliersassociated with the upper bound of load shedding in (13).According to the envelop theorem, there is η U ( j ) = − ∂C ∗ III ∂r U ( j ) = ∂ L III ∂r U ( j ) = k ∈K (cid:88) α k ( j ) , (17) where η U ( j ) denotes the upward reserve marginal price ofgenerator j . Similarly the downward marginal reserve priceis: η D ( j ) = k ∈K (cid:88) β k ( j ) . (18)With such formulations, the connections between reserve andgeneration re-dispatches can be established: if the upwardgeneration re-dispatch δg Uk ( j ) = r U ( j ) in scenario k, then itscorresponding multiplier α k ( j ) will be positive and contributeto the upward reserve marginal price η U ( j ) .Although the energy prices of generators and loads aredefined separately, and the reserve marginal prices in (17)-(18) are defined at the resource level, next we establish theuniform pricing property with some additional assumptions:7) Assume that ( − (cid:80) τ k ) is zero in (16) .8) Assume that generators at the same bus will have thesame ( C k , C k ) in each scenario . Theorem 1 (Uniform Pricing):
Consider any two genera-tors i, j and any load l at the same bus. Under assumptions(1)-(7), there is η g ( i ) = η g ( j ) = η d ( l ) for energy.Moreover, under assumptions (1)-(8) and assume that r U ( i ) , r D ( i ) , r U ( j ) , r D ( j ) > , there are η U ( i ) = η U ( j ) forupward reserve and η D ( i ) = η D ( j ) for downward reserve. The proof is presented in the Appendix A. Due to the pagelimit, we leave the proofs for this Theorem and the followingTheorem 2 to the online version of our manuscript [13].The settlement process based on the proposed pricing ap-proach has some attractive properties, as in the next subsection.
B. Market settlement process
Next the settlement process will be presented. The processcan be separated into two stages. In the ex-ante stage, we don’tknow which prediction is true; and in the ex-post stage, thesettlement depends on which scenario is actually realized.
1) ex-ante stage:
In this stage, generations and reserve willbe financial binding and settled, also the basic load capacities d and load fluctuations π k in all scenarios will be settled.Therefore, the ex-ante stage includes the following payments: • contribution of base-case prices to generator energycredit: Γ g = ω T g ; (19) • contributions of non-base scenario prices to generatorenergy credit: (cid:88) Γ gk = k ∈K (cid:88) ω Tk g ; (20) It is rare to shed a load to its total capacity. If a load is completely shedand another load at the same bus is not, this indicates that they have differentreliability requirements, which will again bring in the non-uniform pricingissue. The upward and downward re-dispatch prices ( C k , C k ) in scenario k can be set as the forecast RT-LMPs in scenario k plus a fixed adder. Withperfect predictions, the forecast RT-LMPs will be equal to the actual RT-LMPs, therefore the upward/downward re-dispatch prices ( C k , C k ) can beused to settle the generation upward/downward redispatches. contributions of base-case price to load energy payment: Γ d = ( ω ) T d ; (21) • contributions of non-base scenario prices to load energypayment: k ∈K (cid:88) Γ dk = k ∈K (cid:88) ( ω k ) T d ; (22) • non-base load fluctuation payment in all scenarios: k ∈K (cid:88) Π k = k ∈K (cid:88) ( ω k ) T π k ; (23) • upward and downward reserve credit: Γ U = ( η U ) T r U = k ∈K (cid:88) α kT r U = k ∈K (cid:88) Γ Uk , (24) Γ D = ( η D ) T r D = k ∈K (cid:88) β kT r D = k ∈K (cid:88) Γ Dk . (25)For the fluctuation payment, note that one load should payfor its fluctuations in all scenarios because the SO will procurereserve to deal with all possible load fluctuations accordingly,therefore the load fluctuation payments should not only dependon the realized scenario in real-time, but should also rely onother scenarios considered.
2) ex-post stage:
In this stage, one of the non-base scenar-ios is actually realized and denoted as scenario k . With perfectpredictions, for each generator j , the deviation of its real-time generation level from its base-case energy procurement g ( j ) ∗ will be δg Uk ( j ) or δg Dk ( j ) . Under the assumption (7)and the corresponding footnote (5), upward and downwardgeneration re-dispatches will be settled with C k and C k , andload shedding will be settled with the shedding prices C L .Therefore, the ex-post stage includes the following payments: • upward redispatch compensation: Φ Uk = C Tk δg Uk ; (26) • downward redispatch pay-back: Φ Dk = C Tk δg Dk ; (27) • load shedding credit: Φ dk = C TL δd k . (28)Note that the net revenue of the SO should be equal tothe congestion rent. For the settlement process, the followingtheorem regarding SO’s revenue adequacy is established: Theorem 2 (Revenue Adequacy):
Under assumptions (1)-(8), the net revenue of the SO is always non-negative.In particular, for the base case, load energy payment (21)is equal to the sum of generator energy credit (19) andcongestion rent: Γ d = Γ g + ∆ = Γ g + f T µ. (29) For each non-base scenario k, its contribution to loadpayment is equal to the sum of its contribution to generatorcredit, load shedding credit, and congestion rent: Γ dk +Π k = Γ gk +Γ Uk +Γ Dk + (cid:15) k Φ Uk − (cid:15) k Φ Dk + (cid:15) k Φ dk +∆ k , k ∈ K , (30) where the left-hand side is the contribution of scenario kto load payment, including energy payment (22) and load fluctuation payment (23). The first five terms on the right-handside represent its contribution to generator credit, includingenergy credit (20), upward and downward reserve credit (24-25), upward and downward re-dispatch payment (26-27). Thesixth term is the expected load shedding credit (28), and thelast term is the congestion rent in that scenario ∆ k = f Tk µ k . Please refer to the Appendix B in [13] for the proof. Withthis Theorem, payments from loads, payments to generatorsand congestion rent will reach their balance in the basecaseas well as in all scenarios. Also, the reserve costs and there-dispatch costs will be allocated in a scenario-oriented way.IV. C
ASE S TUDY
The case studies are performed both on a 2-bus system andon the modified IEEE 118-bus system.
A. Two-bus System
Fig. 2. the One-Line Diagram for the 2-Bus System
We first consider a 2-bus system with its one-line diagrampresented in Fig. 2. The generator bids are presented in Table I.The outage probability of the two-parallel-line branch is 10%,meaning that the system will lose one of these two lines if theoutage happens. The line capacity exceeding rates are set to befor all scenarios. The basic loads are (6 , , MW, with twopossible fluctuation situations: situation I (+2 , +6 , − MWwith probability 20%, and situation II (+3 , +2 , − MW withprobability 20%. Based on the line outage and load fluctuationinformation, we present all possible scenarios in Table II.
TABLE I. G
ENERATORS ’ O
FFER D ATA FOR THE US S YSTEM
Generator
G/G r U /r D C g /C U /C D G1 / / / / G2 / / / / G3 / / / . / . TABLE II. N ON -B ASE S CENARIO D ATA FOR THE US S YSTEM
NO. Outage Load Situation Probability C and C .
06 [19 .
1; 26 . .
02 [19 .
7; 33 . .
02 [19 .
4; 32 . .
18 [19 .
4; 33 . .
18 [19 .
1; 27 . We present the clearing results in Table III. While G1offers the cheapest upward reserve bid and still owns extracapacity/ramping rate, the SO doesn’t clear its entire upwardreserve bid, instead the more expensive resources G2 andG3 are cleared. The reason is that the extra upward reservefrom G1 can’t be delivered in scenarios when the branchoutage happens. It can also be observed that there are uniformenergy and reserve prices at each bus. In the meantime, theload fluctuation payments can calculated according to equation(23): Π d (1) =$23.3, Π d (2) =$91.7, Π d (3) =$-23.5. Note that d3fluctuation payment is negative because d3’s fluctuations willhedge the fluctuations of d1 and d2. ABLE IV. M
ONEY F LOW FOR THE US S YSTEM ($)Base S1 S2 S3 S4 S5 Total Γ d . . . . . . . d . . . . . (cid:15) Φ d . . . g . . . . . . . U . . . D . . (cid:15) Φ U . . . . . . (cid:15) Φ D . . . . .
7∆ 3 . . . . . base / S1-S5: denote the base case / scenario 1-5;Revenue adequacy: (Γ d = Γ g +∆) holds for column 1, (Γ d +Π d − (cid:15) Φ d =Γ g +Γ U +Γ D + (cid:15) Φ U − (cid:15) Φ D +∆) holds for other columns. Fluctuation Level of d59 % F l u c t u a ti on P a y m e n t fr o m d59 $ -48-47-46-45-44 F l u c t u a ti on P a y m e n t fr o m d119 $ Increasing Fluctuation Level of d59 fluctuation payment from d59 fluctuation payment from d119
Fluctuation Level of d119 % F l u c t u a ti on P a y m e n t fr o m d59 $ -120-100-80-60-40 F l u c t u a ti on P a y m e n t fr o m d119 $ Increasing Fluctuation Level of d119
Fig. 3. Fluctuation Payments from d59 (blue) and d119 (red) with theFluctuation Levels of d59 (left) and d119 (right)TABLE III. C
LEARING R ESULTS FOR THE US S YSTEM
Generator g r U r D η g η U η D G1 . . . . . . G2 . . . . . . G3 . . . . . . In Table IV, the money flow is presented. Explicitly revenueadequacy holds in the base case, in each scenario and in total.
B. IEEE 118-Bus System
Simulations on the modified IEEE 118-bus system are alsoreported. Outage probabilities of lines 21, 55, and 102 areset to be 10%. The line capacity exceeding rates are set tobe 1.2 for all scenarios. The original load 59 will be equallyseparated into two loads: new load 59 and load 119. Therewill be two load fluctuation situations, each with occurrenceprobability 10%, and the fluctuation levels of all loads willbe 3% in both situations: in situation I, d119 will increaseby 3% while other loads will decrease by 3%; in situation II,d119 will decrease by 3% while others will rise by 3%. Thegenerators’ energy and reserve bids are modified to be linear.The total revenue inadequacy in this case is $2 . , which israther small compared to the expected total cost $89476 . .In Fig. 3, the fluctuation payment from d119 is negativebecause its fluctuation will hedge the others’ fluctuations in allscenarios, so d119’s fluctuations will be credited. For the leftfigure, With the rising fluctuation level of d59, the fluctuationpayment from d59 will increase while the fluctuation creditto d119 will increase because d59’s rising fluctuation levelwill bring more uncertainties, therefore enhancing the value ofd119’s fluctuations hedge against others’ fluctuations. For theright figure, while the fluctuation credit to d119 will increasewith its rising fluctuation level, fluctuation payment from d59will decrease because the rising fluctuation level of d119 willreduce the impact of d59’s fluctuations on system balance. Fluctuation Level of d15 % P r i ce $ / M W Upward Reserve
Price of r U (7) Fluctuation Level of d66 % P r i ce $ / M W Downward Reserve
Price of r D (29) Fig. 4. Price of r U (7) with the Increasing Fluctuation Levels of d15 (left)and Price of r D (28) with the Increasing Fluctuation Levels of d66 (right) Fig. 4 depicts that with the increasing fluctuation level ofd15, the upward reserve price of G7 at bus 15 will increase(left), and with the increasing fluctuation level of d66, thedownward reserve price of G28 at bus 66 will increase (right).Underlining in these phenomenons is that the increasing fluc-tuation levels of d15 and d66 will bring in more uncertainties,therefore enhancing the value of reserve at the same bus.V. C
ONCLUSIONS
In this paper, we proposed a scenario-oriented energy-reserve co-optimization model which considers the re-dispatchcosts in all non-base scenarios to minimize the expectedsystem total cost, and includes the network constraints in allscenarios to ensure the deliverability of reserve. We defined en-ergy and reserve marginal prices which are locational uniformprices under certain assumptions, and proposed the associatedsettlement process which can guarantee revenue adequacy ofthe system operator. In future studies, we aim to includegenerator outages into the model, and consider the couplingbetween reserve and ramping in multi-period operations.R
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IEEE Transactions on Power Systems A PPENDIX AP ROOF OF T HEOREM i, j and load l at the same bus m , their marginal prices will be: η g ( i ) = η g ( j ) = λ − S (: , m ) T µ + (cid:88) ( λ k − S k (: , m ) T µ k ) , (31) η d ( l ) = λ − S (: , m ) T µ + (cid:88) λ k − (cid:88) S k (: , m ) T µ k , (32)apparently ( η g ( i ) = η g ( j ) = η d ( l )) holds.To consider the reserve prices, we need to fundamentallyanalyze the relationships between the generation redispatchesand procured reserve, so we can substitute the generationredispatches δg Uk , δg Dk as follows: δg Uk = x k ∗ r U , δg Dk = y k ∗ r D , (33)where x k and y k are both diagonal matrices. For eachindicators ( x k ( j ) , y k ( j ) ∈ [0 , , j = 1 , ..N g ) on theirdiagonals, they represent the ratio of generator j ’s up-ward/downward generation redispatch to generator j ’s pro-cured upward/downward reserve in scenario k . With thesesubstitutions, we can transform the original model (II) intoa new model (IV). In the new model, the optimal energy andreserve procurement and redispatches in all non-base scenarioswon’s change, also the corresponding Lagrangian multipliersin these two models will be the same. The new model is: F (cid:48) ( g, r U , r D , x k , y k , δd k ) = C Tg g + C TU r U + C TD r D + K (cid:88) (cid:15) k ( C Tk ( x k r U ) − C Tk ( y k r D ) + C TL δd k ) , ( IV ) minimize { g,r U ,r D ,x k ,y k ,δd k } F (cid:48) ( · ) , subject to (7) , (8) , (13) , for all k ∈ K : ( λ k ) T ( g + x k r U − y k r D ) = T ( d + π k − δd k ) , (34) ( µ k ) S k (cid:0) ( g + x k r U − y k r D ) − ( d + π k − δd k ) (cid:1) ≤ f k , (35) ( α k , α k , β k , β k )0 ≤ x k r U ≤ r U , ≤ y k r D ≤ r D , (36)We denote the Lagrangian function of model (IV) as L IV .According to the KKT condition and ignoring − τ k , we have: ∂ L IV ∂δd k ( l ) = (cid:15) k C L ( l ) − λ k + S k (: , m l ) T µ k = 0 , (37) ∂ L IV ∂x k ( j ) = (cid:15) k C k ( j ) r U ( j ) − α k ( j ) r U ( j )+ α k ( j ) r U ( j ) − λ k r U ( j ) + S k (: , m j ) T µ k r U ( j ) = 0 , (38) ∂ L IV ∂y k ( j ) = − (cid:15) k C k ( j ) r D ( j ) − β k ( j ) r D ( j )+ β k ( j ) r D ( j ) + λ k r D ( j ) − S k (: , m j ) T µ k r D ( j ) = 0 . (39)The assumption ( r U ( i ) , r U ( j ) > indicates that ( α k ( i ) , α k ( j ) = 0) , then according to equation (38) we have: α k ( i ) = − (cid:15) k C k ( i ) + λ k − S k (: , m ) T µ k ,α k ( j ) = − (cid:15) k C k ( j ) + λ k − S k (: , m ) T µ k . Since we assume C k ( i ) = C k ( j ) , then apparently ( α k ( i ) = α k ( j )) holds, so we have: K (cid:88) α k ( i ) = K (cid:88) α k ( j ) , (40)therefore generator i and generator j will receive the sameupward reserve marginal prices.In the meantime, the assumption ( r D ( i ) , r D ( j ) > indi-cates that ( β k ( i ) , β k ( j ) = 0) holds, then according to equation(39) we have: β k ( i ) = (cid:15) k C k ( i ) − λ k + S k (: , m ) T µ k ,β k ( j ) = (cid:15) k C k ( j ) − λ k + S k (: , m ) T µ k . Since we assume C k ( i ) = C k ( j ) , then apparently α k ( i ) = α k ( j ) holds, so we have: K (cid:88) β k ( i ) = K (cid:88) β k ( j ) , (41)therefore generator i and generator j will receive the samedownward reserve marginal prices. With equation (31), (40)and (41) we can prove Theorem 1.A PPENDIX BP ROOF OF T HEOREM δd k ( l ) , x k ( j ) , y k ( j ) to both the left-hand sideand the right-hand side of equations (37)-(39), respectively,then with the complementary slackness of (36) we have: λ k δd k ( l ) = (cid:15) k C L ( l ) δd k ( l ) + S k (: , m l ) T µ k δd k ( l ) , (42) α k ( j ) r U ( j ) = − (cid:15) k C k ( j ) x k ( j ) r U ( j ) + λ k x k ( j ) r U ( j ) − S k (: , m j ) T µ k x k ( j ) r U ( j ) , (43) β k ( j ) r D ( j ) = (cid:15) k C k ( j ) y k ( j ) r D ( j ) − λ k y k ( j ) r D ( j )+ S k (: , m j ) T µ k y k ( j ) r D ( j ) . (44)In the meantime, to consider the congestion rent, the phaseangled based form of model (IV) will be adopted as follows: ( V ) minimize { g,r U ,r D ,x k ,y k ,δd k ,θ,θ k } F (cid:48) ( · ) , subject to (2) , (3) , (5) , (13) , (36) (45) (Λ) g − d = Bθ, (46) ( µ ) F θ ≤ f, (47)for all k ∈ K : (Λ k )( g + x k r U − y k r D ) − ( d + π k − δd k ) = B k θ k , (48) ( µ k ) F k θ k ≤ f k , (49)ith the equivalence of the shift factor based model and thephase angle based model, we have: Λ = λ − S T µ, Λ k = λ k − S Tk µ k , k ∈ K . (50)In the meantime, We denote the Lagrangian function of model(V) as L V , with the KKT condition we have: θ T ∂ L V ∂θ = ( Bθ ) T Λ + (
F θ ) T µ = 0 , (51) θ Tk ∂ L V ∂θ k = ( B k θ k ) T Λ k + ( F k θ k ) T µ k = 0 . (52)In the basecase, we have (Γ d − Γ g = ( λ − S T µ ) T ( d − g )) and (∆ = f T µ ) . To consider revenue adequacy in the basecase,we have: ( λ − S T µ ) T ( d − g ) = Λ T ( d − g ) = − Λ T ( Bθ ) = ( F θ ) T µ = f T µ. (53)These four equations are based on the equation (50), the KKTconditions for constraint (46), the equation (51), and the KKTconditions for constraint (47), respectively. These equationsindicate that (Γ d = Γ g + ∆ ) , which proves revenue adequacyof the SO in the basecase.In the meantime, the congestion rent contributed from anynon-base scenario k will be: f Tk µ k = ( F k θ k ) T µ k = − ( B k θ k ) T Λ k = Λ Tk (( d + π k d − δd k ) − ( g + x k r U − y k r D ))= ( λ k − S Tk µ k ) T (( d + π k d − δd k ) − ( g + x k r U − y k r D ))= ( − S Tk µ k ) T ( d + π k − δd k ) − ( − S Tk µ k ) T ( g + x k r U + y k r D )= ( − S Tk µ k ) T ( d + π k ) + ( S Tk µ k ) T δd k + ( S Tk µ k ) T g + ( S Tk µ k ) T ( x k r U ) − ( S Tk µ k ) T ( y k r D ) . (54)These six equations are based on the KKT conditions forconstraint (49), the equation (52), the KKT conditions forconstraint (48), the equation (50), the KKT conditions forconstraint (34), and the reorganization of the equation, respec-tively. Also with the KKT conditions of constraint (34) andthe equation (42) we have: λ k (cid:88) j ( g ( j ) + x k ( j ) r U ( j ) − y k ( j ) r D ( j )))= (cid:88) l λ k ( d ( l ) + π k ( l ) − δd k ( l ))= (cid:88) l ( λ k d ( l )+ λ k π k ( l ) − (cid:15) k C L ( j ) δd k ( l ) − S k (: , m l ) T (cid:15) k δd k ( l )) , which can be reorganized according to (43)-(44) as follows: (cid:88) l ( λ k d ( l ) + λ k π k ( l ) − (cid:15) k C L ( l ) δd k ( l ) − S k (: , m l ) T µ k δd k ( l )) − (cid:88) j λ k g ( j )= (cid:88) j ( α k ( j ) + (cid:15) k C k ( i ) x k ( j ) + S k (: , m j ) T µ k x k ( j )) r U ( j ) − (cid:88) j ( − β k ( j )+ (cid:15) k C k ( j ) y k ( j )+ S k (: , m j ) T µ k y k ( j )) r D ( j ) . (55) If we add ( (cid:80) l S k (: , m l ) T µ k ( d ( l )+ π k ( l ))+ (cid:80) j S k (: , m j ) T µ k g ( i )) and its opposite to the right hand side of (55) and reorganizethe equation, we have: (cid:88) l ( λ k − S k (: , m l ) T µ k ))( d ( l ) + π k ( l ))= (cid:88) j ( λ k g ( j ) − S k (: , m j ) T µ k g ( j ))+ (cid:88) j α k ( j ) r U ( j ) + (cid:88) j β k ( j ) r D ( j )+ (cid:88) j (cid:15) k C k ( j ) x k ( j ) r U ( j ) − (cid:88) j (cid:15) k C k ( j ) y k ( j ) r D ( j ) − (cid:88) l (cid:15) k C L ( l ) δd k ( l )+ ( (cid:88) j S k (: , m j ) T µ k g ( j ) + (cid:88) j S k (: , m j ) T µ k x k ( j ) r U ( j ) − (cid:88) j S k (: , m j ) T µ k y k ( j ) r D ( j ) + (cid:88) l S k (: , m l ) T µ k δd k ( l )) − (cid:88) l S k (: , m l ) T µ k ( d ( l ) + π k ( l )) . (56)The term on the left-hand side of equation (56) is the contribu-tion of scenario k to load payment, including energy payment(22) and load fluctuation payment (23). The right-hand sideof equation (56) include the energy credit (20) in the st row,upward and downward reserve credit (24-25) in the nd row,expected upward and downward re-dispatch payment (26-27)in the rd row, expected load shedding credit (28) in the th row, the congestion rent in the th - th rows which can bereorganized as equation (54). Therefore equation (56) can alsobe written as: Γ dk +Π k = Γ gk +Γ Uk +Γ Dk + (cid:15) k Φ Uk − (cid:15) k Φ Dk + (cid:15) k Φ dk +∆ k , (57)which can prove revenue adequacy of the SO in each scenario kk