Robust Scheduling of Virtual Power Plant under Exogenous and Endogenous Uncertainties
Yunfan Zhang, Feng Liu, Zhaojian Wang, Yifan Su, Weisheng Wang, Shuanglei Feng
JJOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. XX, AUGUST XXXX 1
Robust Scheduling of Virtual Power Plant underExogenous and Endogenous Uncertainties
Yunfan Zhang, Feng Liu,
Senior Member, IEEE,
Zhaojian Wang,
Member, IEEE,
Yifan Su,Weisheng Wang, and Shuanglei Feng
Abstract —Virtual power plant (VPP) provides a flexible so-lution to distributed energy resources integration by aggre-gating renewable generation units, conventional power plants,energy storages, and flexible demands. This paper proposesa novel model for determining the optimal offering strategyin the day-ahead energy-reserve market and the optimal self-scheduling plan. It considers exogenous uncertainties (or calleddecision-independent uncertainties, DIUs) associated with marketclearing prices and available wind power generation, as wellas the endogenous uncertainties (or called decision-dependentuncertainties, DDUs) pertaining to real-time reserve deploymentrequests. A tractable solution method based on strong dualitytheory, McCormick relaxation, and the Benders’ decompositionto solve the proposed stochastic adaptive robust optimization withDDUs formulation is developed. Simulation results demonstratethe applicability of the proposed approach.
Index Terms —Adaptive robust optimization, decision depen-dent uncertainty, endogenous uncertainty, virtual power plant N OTATION
In this paper, R n ( R m × n ) depicts the n -dimensional Euclideanspace. [ n ] : = { , , ..., n } denotes the set of integers from 1 to n . For a column vector x ∈ R n (matrix A ∈ R m × n ), x T ( A T )denotes its transpose. We use 1 and 0 to denote vector of onesand zeros, respectively. For x , y ∈ R n , we denote the innerproduct x T y = (cid:80) ni = x i y i where x i , y i stands for the i -th entryof x and y , respectively. We use flourish capital W : X ⇒ Y todenote a set-valued map if W ( x ) is a nonempty subset of Y for all x ∈ X . Sets and IndexT
Sets of time periods indexes t . I G , I W , I D , I ES Sets of conventional power plants (CPP),wind generation units, flexible demand units,and energy storage units indexes i . I S Sets of market price scenarios indexes s . ParametersC i , C i , C SUi , C SDi
Fixed, variable, start-up and shut-down costcoefficient of CPP i . P R + , P R − Maximal up-/down- reserve capacity. E R + , E R − Maximal up-/down- reserve energy that canbe traded in the reserve markets.
EXCH max
Transaction limit between the VPP and dis-tribution energy market. T on i , T off i Minimal on/off time of CPP i . R + i , R − i , R SUi , R SDi
Up-/down-/ start-up/shut-down ramping lim-its of the CPP i . P Geni , P Geni
Power limit of the CPP i . P Di , t , P Di , t , D Di Maximal/minimal power consumption andminimal daily energy consumption of theflexible demand i . r D + i , r D − i Load pick-up/drop ramping limits for flexi-ble demand. P chi , P dci Charging and discharging power capacitiesof the storage unit i . η chi , η dci Charging and discharging efficiency rates ofthe storage unit i . SOC i , SOC i Lower and upper bounds for the stored en-ergy of the storage unit i . SOC i , SOC i Available state of charge of storage unit i . P AWi , t , P AWi , t , P AW , avi , t Maximal/minimal/average value of availablewind power. P AW . hi , t Fluctuation level of available wind genera-tion. µ Et , µ Es , t Energy market price at time period t , as adeterministic coefficient and the value underscenario s , respectively. µ RE + , µ RE − Up-/down- reserve energy prices as a deter-ministic coefficient. µ RE + s , µ RE − s Up-/down- reserve energy prices under sce-nario s . µ RC + t , µ RC − t Up-/down- reserve capacity price as a deter-ministic coefficient. µ RC + s , t , µ RC − s , t Up-/down- reserve capacity price under sce-nario s . ω s Occurrence probability of scenario s . Variableu i , t , v SUi , t , v SDi , t Binary variables representing the state/start-up action/shut-down action of CPP i . p Geni , t Power generation of CPP i . p Di , t Power consumption of flexible demand i . p chi , t , p dci , t Charge/discharge power of storage unit i . SOC i , t State of charge (SoC) of storage unit i . p Wi , t Production of the wind power unit i . p AWi , t Available wind generation of unit i . p EXCHt
Energy transaction between the VPP anddistribution market at time period t . SIG + t , SIG − t Up-/down- regulating signals to the VPP. a r X i v : . [ ee ss . S Y ] F e b OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. XX, AUGUST XXXX 2
I. I
NTRODUCTION
In recent years, virtual power plant (VPP) technique isdeveloped to promote the effective utilization of renewableresources and achieve environmental and economical superi-ority [1]. It combines renewable units with conventional gen-eration units, storage facilities and controllable load demands,etc. Such combination enables distributed energy resourceswith complementary advantages participating in power systemoperation and energy-reserve market as an integrated entity.During this process, uncertainties, distinguished as exogenousand endogenous, are inevitably involved. The former, whichis also known as decision-independent uncertainty (DIU), isindependent of decisions. The latter, which is also knownas decision-dependent uncertainty (DDU), can be affected bydecision variables. This paper addresses the robust schedulingof a VPP participating day-ahead (DA) energy and reservemarket, considering both DIUs and DDUs.Several closely relevant works are [2]–[10], where variousoptimization techniques are applied to hedge against the riskraised by uncertainties. In [2], [3], chance-constrained stochas-tic programs are utilized to achieve risk-aversion of VPP. In[4]–[7], robust optimization (RO) approaches are implementedto maximize the economic profit of VPP under the worst-case realization of the uncertainty in a given set. Reference[4] applies an RO-based model to the self-scheduling of VPPin the volatile day-ahead market environment whereas theuncertainties pertaining to renewable generations are left out.In [5], [7], bidding strategies of VPP in both DA and real-time (RT) markets considering uncertainties of DA marketprices, RT market prices and wind production are presented. Tohedge against multi-stage uncertainties, a standard two-stagerobust model is applied in [5]. Moreover, a four-level robustmodel is formulated in [7] with a tractable algorithm based onstrong duality theorem and column-and-constraint generation(C&CG) algorithm. In [6] communication failures and cyber-attacks on the distributed generators in a VPP are consideredand a robust economic dispatch of the VPP is accordinglyproposed. In [8]–[10], the scenario-based stochastic programand the adaptive robust optimization (ARO) are combined,leading to a stochastic ARO.In spite of the relevance of the aforementioned literature,the dependency of uncertainties on decisions is disregarded.Specifically, the volatile market prices are regarded as ex-ogenously uncertain as the VPP is assumed to be a pricetaker in the market. The uncertainties of renewable generationsare also considered exogenous since they are determined byuncontrollable natural factors. As for the uncertain reservedeployment requests to VPP, equivalent binary-variable-basedrepresentation of the uncertainty set with a given budgetparameter indicates that it is a DIU set. However, when takinginto account the reserve energy provided by the VPP, thepolyhedral uncertainty set pertaining to reserve deploymentrequests becomes endogenous, i.e., dependent on VPP’s of-fering in the reserve market, and cannot be reduced to itsextreme-based exogenous equivalent. To the best of the au-thors’ knowledge, no research work has concurrently modeledexogenous uncertainties and endogenous uncertainties for self- scheduling of a VPP in the RO framework, which is specificto this paper.RO under decision-dependent uncertainties (RO-DDU) re-cently has drawn increasing attention in the optimizationcommunity. Literature regards RO-DDU as two categories:static RO-DDU [11]–[15] and adaptive RO-DDU (ARO-DDU)[16]–[18]. In [11]–[15], the linear decision-dependency ofpolyhedral uncertainty sets on decision variables is consid-ered, rendering a static RO-DDU model. Then, the robustcounterpart, which is a mixed integer linear program (MILP),is derived by applying the strong duality theory and Mc-Cormick Envelopes convex relaxation. In [16]–[18], ARO-DDU models that concurrently incorporate wait-and-see de-cisions and endogenous uncertainties are studied. Due to thecomputational intractability raised by the complex couplingrelationship between uncertainties and decisions in two stages,the current works make considerable simplifications on themodel. Reference [16], [17] assume affine decision rulesfor the wait-and-see decisions, converting the two-stage ROproblem into a static RO problem. To address a two-stageARO-DDU problem without any assumption on affine policies,the extensively-used C&CG algorithm [19] may fail when theuncertainty set is decision-dependent. In this regard, reference[18] focuses on a high-dimensional rectangle DDU set andaccordingly proposes an improved C&CG algorithm with aworst-case-scenario mapping technique. However, to the bestof our knowledge, the solution method for ARO-DDU withgeneral linear dependency has not been addressed.Regarding the aforementioned issues, this paper considersthe robust offering and scheduling strategies of VPP participat-ing in the DA energy-reserve market, where both exogenousand endogenous uncertainties are involved. Specifically, theuncertainties of market prices and renewable generations areexogenous (or called decision-independent), while the un-certainties of reserve deployment requests are endogenous(or called decision-dependent). The main contributions aretwofold:1)
Modeling:
A novel stochastic ARO model incorporatingboth exogenous and endogenous uncertainties is providedfor the robust scheduling of VPP trading in the DAenergy-reserve market. Compared with existing works[8]–[10], we characterize the dependency of uncertainreserve deployment requests on VPP’s decisions in theDA reserve market.2)
Algorithm:
A novel Benders’ decomposition based al-gorithm is proposed to solve the stochastic ARO-DDUproblem with general linear decision dependency. Theproposed algorithm is guaranteed to converge to theoptimum within finite rounds of iterations. To the best ofour knowledge, the computational intractability of non-reduced ARO-DDU with general linear decision depen-dency has not been addressed in the existing literature.The rest of this paper is organized as follows. Section IIpresents the VPP DA robust scheduling formulation with thecharacterization of both exogenous and endogenous uncertain-ties. Section III derives the robust counterpart and a solutionmethodology based on Benders’ decomposition. A case study
OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. XX, AUGUST XXXX 3 is presented in Section IV. Finally, Section V concludes thepaper. II. M
ODEL D ESCRIPTION
A. DA scheduling of VPP
Revenue of VPP in the DA energy market comprises thecost of purchasing energy or the income of selling energy andis calculated as follows: R NRG = (cid:88) t ∈ T µ Et p Et (1)Reserve market revenue consists of the income of providingreserve service that includes two parts: reserve capacity andreserve energy. R RSV = E R + µ RE + + E R − µ RE − + (cid:88) t ∈ T (cid:0) p R + t µ RC + t + p R − t µ RC − t (cid:1) (2)Regarding the generation cost of VPP, operation cost ofwind generation units is assumed to be zero, leaving theinherent cost to be the operation cost of CPPs. The operationcost of CPP is computed as C Gen = (cid:88) t ∈ T , i ∈ I G (cid:0) C i u i , t + C SUi v SUi , t + C SDi v SDi , t + C i p Geni , t (cid:1) (3)which comprises fixed cost, start-up and shut down cost, andthe variable generation cost.The VPP determines the following things as the DA deci-sions: (i) The power sold to/bought from the day-ahead energymarket; (ii) The reserve capacity at each time slots, as wellas the maximum reserve energy that can be provided in theday-ahead reserve market; and (iii) The unit commitment ofCPP. B. Uncertainty Characterization
In this paper, three kinds of uncertainties are taken intoconsideration as follows.
1) Market Clearing Price:
The market clearing prices areexogenously uncertain since the VPP is assumed to be aprice taker in DA energy-reserve market. Price uncertaintiesappear only in the objective function, affecting the optimalityof decisions but not the feasibility of the VPP system. Thusit is suitable to model price uncertainty into a scenario-basedstochastic programming that aims to minimize the expectednet cost of VPP over a set of representative scenarios: E C net = E (cid:0) C Gen − R NRG − R RSV (cid:1) (4a) = C Gen − (cid:88) s ∈ I S ω s (cid:0) µ RE + s E R + + µ RE − s E R − (cid:1) − (cid:88) s ∈ I S , t ∈ T ω s (cid:0) µ Es , t p Et + µ RC + s , t p R + t + µ RC − s , t p R − t (cid:1) (4b)
2) Available Wind Generation:
Available wind generation P AW is exogenously uncertain since it is determined by naturecondition. It appears in the operating constraints of VPP,imposing a significant effect on not only the optimality butalso the feasibility of the solution. Thus wind uncertainty ischaracterized by the following ambiguity set. W = (cid:110) p AWi ∈ R T : P AWi , t ≤ p AWi , t ≤ P AWi , t , ∀ i ∈ I W , t ∈ T (5a) (cid:88) t ∈ T | p AWi , t − P AW , avi , t | / P AW , hi , t ≤ Γ Ti , ∀ i ∈ I W (5b) (cid:88) i ∈ I W | p AWi , t − P AW , avi , t | / P AW , hi , t ≤ Γ St , ∀ t ∈ T (cid:111) (5c) where P AW , avi , t = P AWi , t + P AWi , t and P AW , hi , t = P AWi , t − P AWi , t , ∀ t ∈ T , i ∈ I W .It is assumed that the available wind generation fluctuateswith the interval between P AWi , t and P AWi , t , under a certainconfidence level. P AW , av is the average level for available windpower generation and is calculated as the mean value of thecorresponding upper and lower confidence bounds P AWi , t and P AWi , t . P AW , h denotes half of the interval width. To alleviateconservativeness of the model, space robustness budget Γ S andtime robustness budget Γ T is added to avoid that p AW alwaysachieve boundary values.
3) Reserve Deployment Request:
Considering the uncer-tainty in reserve deployment requests
SIG + and SIG − , energytransaction between the VPP and the distribution energymarket p EXCH is endogenously uncertain since it dependsupon VPP’s decision in DA energy-reserve market. We modelthe uncertainty of p EXCH by exploring its decision-dependentuncertainty set: P ( p E , p R + , p R − , E R + , E R − ) = (cid:110) p EXCH ∈ R | T | : (6a) p EXCHt = p Et + SIG + t − SIG − t , ∀ t ∈ T (6b) SIG + ∈ R | T | , ≤ SIG + t ≤ p R + t , ∀ t ∈ T (6c) SIG − ∈ R | T | , ≤ SIG − t ≤ p R − t , ∀ t ∈ T (6d) (cid:88) t ∈ T SIG + t ≤ E R + , (cid:88) t ∈ T SIG − t ≤ E R − (cid:111) (6e)Constraint (6e) imposes limits on the total reserve energy thatto be deployed. p R + , p R − , E R + , and E R − together control theconservativeness of the ambiguity set associated with the re-quests for reserve deployment SIG + , SIG − . Note that in (6) thecomplementarity constraint to avoid the situation that up- anddown- regulation signals are given simultaneously is omitted.This is because the ambiguity set of p EXCH remains the samewith the relaxation on the complementarity constraint.
C. Formulation
The proposed adaptive robust optimization model aims atminimizing the expected cost over the representative scenar-ios of market clearing price. Moreover, feasibility of real-time operation of VPP is warranted, even under the worst-case uncertainties of available wind generation and reservedeployment requests.minimize E C net , subject to (7a) (cid:8) u , v SU , v SD , p E , p R + , p R − , E R + , E R − (cid:9) ∈ X ∩ X R (7b) (cid:110) p Gen , , p D , , p ch , , p dc , , SOC , p W , (cid:111) ∈ Y ( u , p E ) (7c)where X : = (cid:8) u , v SU , v SD , p E , p R + , p R − , E R + , E R − : (8a)0 ≤ p R + t ≤ P R + , ∀ t ∈ T (8b)0 ≤ p R − t ≤ P R − , ∀ t ∈ T (8c)0 ≤ E R + ≤ min (cid:110) E R + , (cid:88) t ∈ T p R + t (cid:111) (8d)0 ≤ E R − ≤ min (cid:110) E R − , (cid:88) t ∈ T p R − t (cid:111) (8e) − EXCH max ≤ p Et ≤ EXCH max , ∀ t ∈ T (8f) u i , t , v SUi , t , v SDi , t ∈ { , } , ∀ t ∈ T , ∀ i ∈ I G (8g) OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. XX, AUGUST XXXX 4 v SUi , t + v SDi , t ≤ , ∀ t ∈ T , ∀ i ∈ I G (8h) u i , t + = u i , t + v SUi , t − v SDi , t , ∀ t ∈ T , ∀ i ∈ I G (8i) − u i , t − + u i , t ≤ u i , τ , ∀ t ≤ τ ≤ T on i + t − , i ∈ I G (8j) u i , t − − u i , t + u i , τ ≤ , ∀ t ≤ τ ≤ T off i − , i ∈ I G (cid:111) (8k)The feasible region of the wait-and-see decisions is formulatedin (9) where p AW and p EXCH are uncertainties. Y ( u , p AW , p EXCH ) : = (cid:110) p Gen , p D , p ch , p dc , SOC , p W : (9a) u i , t P Geni ≤ p Geni , t ≤ u i , t P Geni , ∀ t ∈ T , i ∈ I G (9b) p Geni , t + − p Geni , t ≤ u i , t R + i + ( − u i , t ) R SUi , ∀ t ∈ T , i ∈ I G (9c) p Geni , t − − p Geni , t ≤ u i , t R − i + ( − u i , t ) R SDi , ∀ t ∈ T , i ∈ I G (9d) P Di , t ≤ p Di , t ≤ P Di , t , ∀ t ∈ T , ∀ i ∈ I D (9e) − r D − i ≤ p Di , t + − p Di , t ≤ r D + i , ∀ t ∈ T , ∀ i ∈ I D (9f) (cid:88) t ∈ T p Di , t ≥ D Di , ∀ i ∈ I D (9g)0 ≤ p chi , t ≤ P chi , ∀ t ∈ T , ∀ i ∈ I ES (9h)0 ≤ p dci , t ≤ P dci , ∀ t ∈ T , ∀ i ∈ I ES (9i) SOC i , t = SOC i , t − + η chi p chi , t − η dci p dci , t , ∀ t ∈ T , i ∈ I ES (9j) SOC i ≤ SOC i , t ≤ SOC i , ∀ i ∈ I ES (9k)0 ≤ p Wi , t ≤ p AWi , t , ∀ t ∈ T , ∀ i ∈ I W (9l) (cid:88) i ∈ I G p Geni , t + (cid:88) i ∈ I W p Wit + (cid:88) i ∈ I ES p dci , t = p EXCHt + (cid:88) i ∈ I D p Di , t + (cid:88) i ∈ I ES p chi , t , ∀ t ∈ T (cid:111) . (9m)Thus the feasible region of the baseline re-dispatch decisions p Gen , , p D , , p ch , , p dc , , SOC , p W , is Y ( u , p E ) : = Y ( u , p AW , av , p E ) Then the robust feasibility set of x is defined as X R : = (cid:8) u , p E , p R + , p R − , E R + , E R − : (10a) ∀ p AW ∈ W defined in (5) , (10b) ∀ p EXCH ∈ P ( p E , p R + , p R − , E R + , E R − ) defined in (6) , (10c) Y ( u , p AW , P EXCH ) (cid:54) = /0 (cid:9) (10d)The feasible region of VPP’s DA decisions ( u , v SU , v SD , p E , p R + , p R − , E R + , E R − ) is denoted by X , with the specificform of (8a)-(8k), including constraints of the DA market(8a)-(8f) which impose limits on the energy and reserveoffering of the VPP, as well as constraints of CPP (8g)-(8k). The feasible region of VPP’s RT decisions ( p Gen , p D , p ch , p dc , SOC , p W ), also called wait-and-see decisions, isdenoted by Y , where constraints of CPP (9b)-(9d), flexi-ble demand (9e)-(9g), the energy storage unit (9h)-(9k), thewind generation unit (9l), and the power balance of VPP(9m) are included. When no uncertainties exist, i.e., p AW = p AW , av , p EXCH = p E , the feasible region of the baseline re-dispatch decisions p Gen , , p D , , p ch , , p dc , , SOC , p W , is de-noted by Y . The uncertain parameters in the DA schedul-ing problem are p AW and p EXCH . The decision independentuncertainty set W for p AW is given in (5) and the decisiondependent uncertainty set P ( · ) for p EXCH is given in (6). Forthe wait and see decisions p Gen , p D , p ch , p dc , SOC , p W , theirfeasible space Y ( u , p AW , p EXCH ) is actually a set-valued map parameterized by the first stage decision u and the uncertainvariables p AW , p EXCH . Besides the aforementioned operationconstraints of each stage, the first-stage decision u , v SU , v SD , p E , p R + , p R − , E R + , E R − has to satisfies robust feasibility, ascharacterized in (10). X R is called robust feasibility region.Next we give the compact form of two-stage stochasticrobust optimization problem (7)-(10). To simplify the formu-lation, the following terminology is adopted. x : = (cid:8) u , v SU , v SD , p E , p R + , p R − , E R + , E R − (cid:9) (11a) y : = (cid:110) p Gen , , p D , , p ch , , p dc , , SOC , p W , (cid:111) (11b) w : = (cid:8) p AW , p EXCH (cid:9) (11c) y : = (cid:110) p Gen , p D , p ch , p dc , SOC , p W (cid:111) (11d)where x ∈ R n R × Z n Z , w ∈ R n w and y , y ∈ R n y . The dimensionof x is denoted by n x = n R + n Z . We denote the cost itemsin (7a) by a unified form f ( x , y ) : R n R + n y × Z n Z → R . Then(7)-(10) is formulated in a compact form ofmin x , y f ( x , y ) (12a)s.t. x ∈ X ∩ X R , y ∈ Y ( x ) (12b) X R = { x |∀ w ∈ W ( x ) , Y ( x , w ) (cid:54) = /0 } (12c) W ( x ) = { w ∈ R n w | Gw ≤ g + ∆ x } (12d) Y ( x , w ) = { y ∈ R n y | Ax + By + Cw ≤ b , y ≥ } (12e)where G ∈ R r × n w , g ∈ R r , ∆ ∈ R r × n x , A ∈ R m × n x , B ∈ R m × n y , C ∈ R m × n w and b ∈ R m are constants. W ( x ) is a unified form of thedecision-independent uncertainty set W in (5) and the decisiondependent uncertainty set P in (6). Note that (12d) modelsgeneral decision dependence, which encompasses the case ofdecision-independent uncertainties by setting the correspond-ing rows of ∆ to zeros. Y ( x , w ) is the compact form of Y in(9).Problem (12) is a two-stage adaptive robust optimizationproblem with decision dependent uncertainties. Regarding thesolution methodology to this type of problem, the C&CGalgorithm is no longer applicable, for the reason that the worst-case uncertainty w ∗ ∈ W ( x ) with a given x may lie outsidethe uncertainty set when giving another x , i.e., w ∗ / ∈ W ( x ) .Then the feasibility cut of the C&CG algorithm may fail toobtain an optimal solution. Moreover, since the vertices setof polytope W ( x ) changes with x , the C&CG algorithm nolonger guarantees finite iterations to convergence.III. S OLUTION M ETHODOLOGY
A. Equivalent Transformation
Given a first stage decision x , the robust feasibility of x , i.e.,whether x locates within X R , can be examined by solving thefollowing relaxed bi-level problem: R ( x ) = max w ∈ W ( x ) min y , s T s (13a)s.t. Ax + By + Cw ≤ b + s , y ≥ , s ≥ s ∈ R m is the supplementary variable introduced to relaxthe constraint Ax + By + Cw ≤ b in Y ( x , w ) . If R ( x ) ≤ x isrobust feasible, i.e., x ∈ X R . Else if R ( x ) >
0, there exists arealization of the uncertain w lying in the W ( x ) that makesno feasible second-stage decision y is available. Since x ∈ X R OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. XX, AUGUST XXXX 5 if and only if R ( x ) ≤
0, we substitute the constraint x ∈ X R in(12) by R ( x ) ≤ R ( x ) . Then, R ( x ) can be equivalently transformedinto the following single-level bi-linear maximization problem R ( x ) = max w , π π T ( b − Ax − Cw ) (14a)s . t . π ∈ Π , w ∈ W ( x ) (14b)where π ∈ R m is the dual variable on constraint (13b) and Π = (cid:8) π | B T π ≤ , − ≤ π ≤ (cid:9) . Therefore, problem (12) canbe reformulated into the following non-linear static robustoptimization problem with DDU:min x , y f ( x , y ) (15a)s . t . x ∈ X , y ∈ Y ( x ) (15b)0 ≥ π T ( b − Ax − Cw ) , ∀ π ∈ Π , w ∈ W ( x ) (15c)Constraint (15c) is decision-dependent static robust constraint.However, due to the bi-linear relationship between variable π and variable w in term − π T Cw , techniques used to derive arobust counterpart of regular static robust optimization are nomore applicable to problem (15). To address the difficulty insolving ARO-DDU problem (12) and its equivalent formula-tion (15), next we provide a novel two-level iterative solutionalgorithm based on Benders decomposition [20]. B. Master Problem (MP)
The master problem at iteration k is formulated below:min x , y f ( x , y ) (16a)s . t . x ∈ X , y ∈ Y ( x ) (16b)0 ≥ π ∗ j T ( b − Ax − Cw ) , ∀ w ∈ W ( x ) , j ∈ [ k ] (16c)where π ∗ , ..., π ∗ k are solutions from the robust feasibility ex-amination problem. If π ∗ , ..., π ∗ k ∈ Π , then the MP (16) is arelaxation to (15). We solve MP (16) to derive a relaxed opti-mum of (15). Constraints (16c) are feasibility cuts to MP. Theyare designed to have the following salient features: (i) Theworst-case uncertainty w ∗ is not involved, to accommodate thecoupling relation between x and w , which is different from theC&CG algorithm. (ii) Dual information of robust feasibilityexamination problem (i.e., π ∗ ) are included, inspired by theBenders dual decomposition. However, they are designed tobe no longer a hyperplane, but a static robust constraint, tocomprise a cluster of worst-case uncertainties.Next, we illustrate how to deal with the robust constraint(16c) by substituting it with its robust counterpart. For anygiven j in [ k ] , constraint (16c) is equivalent to0 ≥ π ∗ j T ( b − Ax ) + (cid:26) max w j − u ∗ j T Cw j s.t. Gw j ≤ g + ∆ x (cid:27) (17)We deploy the KKT conditions of the inner-level problem in(17) as follows G T λ j = − C T π ∗ j (18a) λ j ≥ ⊥ Gw j ≤ g + ∆ x (18b)where λ j ∈ R r is the corresponding dual variable and (18b)denotes the complementary relaxation conditions. The non-linear complementary conditions (18b) can be exactly lin-earized through big-M method by introducing the binary supplementary variable z j ∈ { , } r and a sufficiently largepositive number M as follows:0 ≤ λ j ≤ M ( − z j ) (19a)0 ≤ g + ∆ x − Gw j ≤ Mz j (19b)Then the MP (16) has the following robust counterpart whichis a MILP problem.min x , y , z , λ , w f ( x , y ) (20a)s . t . x ∈ X , y ∈ Y ( x ) (20b)0 ≥ π ∗ j T ( b − Ax ) − π ∗ j T Cw j (18a), (19a), (19b) z j ∈ { , } r , λ j ∈ R r , w j ∈ R n w j ∈ [ k ] (20c) C. Robust Feasibility Examination Subproblem
The subproblem in this subsection examines the robustfeasibility of given x k by solving R ( x k ) . R ( x ) and its equivalentform are given in (13) and (14), respectively. The bi-linearobjective item − π T Cw imposes difficulties on solving R ( x ) .Next we provide linear surrogate formulations of R ( x ) .The robust feasibility examination problem R ( x ) in (14) canbe equivalently written into R ( x ) = max π ∈ Π (cid:26) π T ( b − Ax ) + max w − π T Cw s . t . Gw ≤ g + ∆ x (cid:27) . (21)Then we deploy the KKT conditions of the inner-level prob-lem, which are − π T Cw = ( g + ∆ x ) T ζ (22a) ζ ≥ ⊥ Gw ≤ g + ∆ x (22b) G T ζ = − C T π (22c)where ζ ∈ R r is the corresponding dual variable. The com-plementary constraint (22b) can be linearlized by introducingbinary supplementary variable v ∈ { , } r like what we doto (18b). Moreover, since strong duality holds, we substitute − π T Cw by ( g + ∆ x ) T ζ . Then, the subproblem R ( x ) can beequivalently transformed into the following MILP R ( x ) = max π , w , ζ , v π T ( b − Ax ) + ( g + ∆ x ) T ζ (23a)s.t. π ∈ Π , (22c) , (23b)0 ≤ ζ ≤ M ( − v ) (23c)0 ≤ g + ∆ x − Gw ≤ Mv (23d) v ∈ { , } r , ζ ∈ R r (23e) D. Modified Benders Decomposition Algorithm
Now we have the overall iterative algorithm, as given inAlgorithm 1. Convergence and optimality of the Algorithm1 are justified by Theorem 1. Theorem 1 indicates that theproposed modified Benders decomposition method can findthe optimal solution of ARO-DDU problem (12) within finitesteps. Proof of Theorem 1 is given in the Appendix.
Theorem 1.
Let p be the number of extreme points of Π .Then the Algorithm 1 generates an optimal solution to (12) in O ( p ) iterations. IV. C ASE S TUDIES
In this section, case studies are conducted on MATLABwith a laptop with Intel i5-8250U 1.60GHz CPU and 4GB ofRAM. GUROBI 9.1.0 is used as the solver.
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Algorithm 1
Modified Benders Decomposition Algorithm
Step 0: Initialization
Set k =
0. Choose an initial solution x k ∈ X , y , k ∈ Y ( x k ) . Step 1: Robust Feasibility Examination
Check robust feasibility of x k by solving R ( x k ) in (23). Let ( w ∗ k , π ∗ k ) be the optimum of R ( x k ) . If R ( x k ) > k = k + R ( x k ) =
0, terminate the algorithmand output the optimal solution ( x k , y , k ) . Step 2: Solve Master Problem (MP)
Solve the master problem (20). Let ( x k , y , k ) be the optimumand then go to Step 1. A. Setup
We consider a VPP that consists of four conventionalgenerators, a wind farm, an energy storage facility, a flexibleload, and three fixed loads. The schematic diagram of the VPPis given in Fig.1. For the DA robust scheduling of the VPP,24 hourly periods are considered, i.e., | T | = Bus 1Bus 5 Bus 2 Bus 3 Bus 4main gridconnection D1 D2 D3 D4wind farmstorage G1 Line 1 Line 4 Line 5Line 2Line 6Line 3 ∞ G4 G3 G2
Fig. 1. 5-bus network.
Parameters of the four conventional generators are providedin Table.I. The 400MW wind farm is located at Bus 5, andthe confidence bounds and average levels for available windpower generation are illustrated in Fig.2. Technical data of thefixed and flexible loads are provided in Table.II, and the dailyprofiles of the total fixed load are shown in Fig.2. The storagefacility is located at Bus 5, with a capacity of 100 MW/200MW.h and conversion efficiency of 90%. The maximum andminimum SoC are 180MW.h and 20MW.h, respectively.The VPP is connected to the main grid at Bus 1. Themaximum power that can be obtained from or sold to themain grid is 400 MW. The maximum participation in DAreserve market is 250 MW at each time slot, for both up-and down- reserve market. The maximum deployed reserveenergy is 6000 MW.h (250 MW ×
24 h), for both up- anddown- reserve deployment requests. Market price scenariosare generated from Nord Pool price data from October 25thto November 25th, 2020 [21], through K-means clustering.Therefore, the uncertain market prices are represented by 8typical equiprobable scenarios.
TABLE IP
ARAMETERS OF CONVENTIONAL GENERATORS .Loc [ P Geni , P Geni ](MW) R + i , R − i , R SUi , R SDi (MW) T oni , T of fi (hour) C SUi , C SDi ($/times) C Gen , i ($/h) C Gen , i ($/MW.h)G1 Bus 5 [200,400] 50 6 100 50 40G2 Bus 4 [150,300] 50 5 100 50 60G3 Bus 1 [150,250] 100 4 100 50 70G4 Bus 1 [250,500] 80 6 100 50 50 TABLE IIP ARAMETERS OF L OAD .Loc Type Ratio D Di (MWh) [ P Di , P Di ](MW) r D − i , r D + i (MW)D1 Bus 2 fixed 0.3 - - -D2 Bus 3 fixed 0.3 - - -D3 Bus 4 fixed 0.4 - - -D4 Bus 4 flexible - 1500 [0,200] 110 Time(hour) O u t pu t o r Load ( M W ) confidence interval of available windexpected value of available windtotal fixed load profile Fig. 2. Total fixed load profile; Confidence bounds and average levels foravailable wind power generation.
B. Baseline Results
In this subsection, wind uncertainty budgets are fixed as Γ T = , Γ S =
1. We solve the stochastic robust schedulingproblem of VPP by the proposed Algorithm.1. The algorithmconverges after 25 iteration rounds, the evolution process ofwhich is depicted in Fig.3-5. The increasing net cost and thediminishing reserve revenue represents VPP’s hedging againstthe worst-case realization of uncertainties concerning availablewind generation and reserve deployment requests. 迭代过程
Fig. 3. Evolution of objective value with the number of iterations.Fig. 4. Evolution of reserve revenue with the number of iterations.
C. Sensitivity Analysis1) Impact of Wind Uncertainty Budget:
In this case, 7 timebudgets of wind uncertainty Γ T from 0 to 12 with a gradientof 2 are introduced. Since there is only one wind farm inthe VPP, impact of space robustness parameter Γ S is omittedsince Γ S is fixed as 1. Participation in DA energy-reservemarket in terms of different Γ T is provided in Table.III. Itis observed that the VPP robust scheduling decisions exhibitdifferent tendencies in DA energy and reserve market whenresponding to different wind uncertainty budget: as the value OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. XX, AUGUST XXXX 7
Fig. 5. Evolution of reserve budget with the number of iterations. of Γ T increases, the amount of reserve offering provided inDA market trends to decrease, while there is no obvious trendfor the energy offering in DA market. This is because theVPP would like to keep more ramping resources inside theVPP to hedge against the increasing uncertainty of availablewind generation power. An increasing net cost of VPP canalso be observed as the value of Γ T increases, indicating thata larger uncertainty set always comes with a higher price ofrobustness. TABLE IIII
MPACT OF ROBUSTNESS PARAMETER Γ T ON VPP’
S COST AND REVENUE . Γ T NetCost($) Energy MarketRevenue($) Reserve MarketRevenue($) CPPCost($)0 226438.18 273204.95 154713.67 654356.802 230191.26 291456.07 152588.30 674235.634 231978.20 270391.64 149658.90 652028.746 233943.76 272301.82 148452.17 654697.748 235871.82 274171.84 146816.15 656859.8110 236637.36 268425.54 144729.88 649792.7712 239033.89 269952.13 144610.50 653596.51
2) Impact of µ RE + and µ RE − : In this case, we present theimpact of upward and downward reserve energy price µ RE + and µ RE − on the reserve offering behavior of VPP. The resultsare displayed in Table.IV-V. It is observed that as µ RE + ( µ RE − )increases, the amount of reserve capacity and reserve energytrend to increase. Certainly, the reserve deployment uncertaintywould rise accordingly, but since the reserve revenue is high,the VPP would like to sacrifice more in DA energy market orpay more for CPP generation cost to hedge against a severerrealization of the worst-case reserve deployment. Conversely,if the value of µ RE + and µ RE − are relatively small, the VPPtrends to slash the reserve offering directly to restrict theuncertainty and ensure robust feasibility. TABLE IVI
MPACT OF µ RE + ON VPP’
S COST AND REVENUE WITH FIXED µ RE − = µ RE + NetCost($) EnergyMarketRevenue($) ReserveCapacityRevenue($) ReserveEnergyRevenue($) CPPCost($)6 246468.82 267631.40 62422.43 72524.55 649047.208 235871.82 274171.84 63090.85 83725.30 656859.8110 225351.63 266644.06 63187.46 93450.33 648633.4812 213793.07 253643.08 65379.32 108422.57 641238.0414 202718.58 247247.32 65694.42 119613.80 635274.11
D. Comparative Performance Study1) Comparison between DIU and DDU:
To present thecompetitiveness of the proposed decision-dependent uncertain
TABLE VI
MPACT OF µ RE − ON VPP’
S COST AND REVENUE WITH FIXED µ RE + = µ RE − NetCost($) EnergyMarketRevenue($) ReserveCapacityRevenue($) ReserveEnergyRevenue($) CPPCost($)6 246081.71 263685.24 62631.36 72210.65 644608.968 235871.82 274171.84 63090.85 83725.30 656859.8110 225589.90 276759.48 63254.95 95242.94 660847.2812 213779.46 281404.67 62394.43 107333.27 664911.8414 203425.38 289312.44 63227.29 119525.46 675490.57 regulating signal formulation (6), a decision-independent for-mulation is introduced in (24) as a reference case. In (24), V isa decision-independent set where v R + t and v R − t are the binaryvariables to model the worst-case upward and downwardreserve deployment request, respectively. Γ R ∈ { , , ..., } isthe reserve uncertainty budget parameter which controls theconservativeness of the model in (24) and is pre-determinedbefore the robust scheduling of VPP. p EXCHt = p Et + v R + t p R + t + v R − t p R − t , ∀ t ∈ T (24a)where v R + t , v R − t ∈ V : = v R + , v R − ∈ { , } | T | : v R + t + v R − t ≤ , ∀ t ∈ T (cid:80) | T | t = ( v R + t + v R − t ) ≤ Γ R (24b)Next, we conduct a comparative performance study on theDIU set (24) and the proposed DDU formulation (6). Robustscheduling with DIU set (24) is solved by C&CG algorithm.The first case is set up with Γ R = E R + = E R − = Γ R , E R + , E R − on the price of robustness for VPP respectivelyand the results are depicted in Fig.6-8. As can be observed,price of robustness rises with an increasing uncertainty budget,but exhibits a different rate of change in DIU and DDUformulations. From the view of price of robustness, DIU setwith Γ R =
16 is approximately a counterpart of the DDUset with decisions E R + = , E R − = .
95. Recall thatthe optimal E R + , E R − are 5289.72 and 5175.95 respectivelyaccording to the results in subsection IV-B, indicating that ahigher level of reserve budget is tolerable for VPP, consideringthe reserve energy revenue it provides. The proposed DDUformulation has the capability and incentive to strike thebalance between robustness and profitability, by optimizingover the reserve budget rather than regarding it as a fixedparameter.
2) Comparison between C&CG algorithm and the proposedalgorithm:
To emphasize the necessity of the proposed algo-rithm for decision-dependent robust optimization problem, weapply the widely used C&CG algorithm to the problem andshow how the C&CG algorithm fails to guarantee solutionoptimality when the uncertainty is decision-dependent. Evo-lutions of objective value with the number of iterations in
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Fig. 6. Impact of robustness parameter Γ R on the price of robustness.Fig. 7. Impact of E R + on the price of robustness when fixing E E − to 5175.95.Fig. 8. Impact of E R − on the price of robustness when fixing E R + to 5289.72. both algorithms are depicted in Fig.9. The C&CG algorithmconverges fast, after 4 iteration rounds. However, the net costof VPP derived by C&CG algorithm is much greater than itsoptimal value. This is because, in the C&CG algorithm, fea-sibility cut is directly generated by the worst-case uncertainty,ignoring that the uncertainty set is varying with decisions. Theworst-case uncertainty realization in previous iterations mayno more lie in the uncertainty set under some other decisions.Thus the feasibility cut of C&CG algorithm may ruin theoptimality of the solution, leading to over-conservative results. Fig. 9. Comparision between the standard C&CG algorithm and the proposedalgorithm.
V. C
ONCLUSION
A novel stochastic adaptive robust optimization approachdealing with decision-dependent uncertainties is proposed inthis paper for the DA scheduling strategies of a VPP participat-ing in energy-reserve market. Consideration of the decision de-pendency of uncertain reserve deployment requests on VPP’soffering in reserve market reduced the robustness of robustscheduling. The VPP determined the optimal level of robust-ness, striking a balance between the price of robustness andits profitability in the market. The proposed modified Bendersdecomposition algorithm obtained the optimum scheduling result under decision-dependent uncertainties, covering theshortage of standard C&CG algorithm. Future works willaddress the consideration of better computational efficiencyand a wider variety of decision dependent uncertainty sets.A
PPENDIX
We start the proof of Theorem 1 with the following lemmas.
Lemma 2.
Let f ∗ denote the optimal objective value ofARO-DDU (12). k denotes the iteration round of Algorithm.1.Define f k : = f ( x k , y , k ) . Then,(a) f k is monotonously non-decreasing with respect to k .(b) For any k ∈ Z + , f k ≤ f ∗ .(c) For any k ∈ Z + , if R ( x k ) = f k ≥ f ∗ .(d) For any k ∈ Z + and any j ∈ [ k ] , π ∗ j ∈ vert ( Π ) where theset vert ( Π ) represents all the vertices of the polytope Π .(e) For any k ∈ Z + , ∀ j , j ∈ [ k ] and j (cid:54) = j , π ∗ j (cid:54) = π ∗ j . Proof. Proof of Lemma 2(a):
Recall that f k is the optimal ob-jective to the minimization master problem at iteration k . Sincemore and more constraints which are called feasibility cutsare appended to the minimization master problem (16) duringiterations, thus f k must be monotonously non-decreasing withrespect to k . Proof of Lemma 2(b):
Recall the equivalent formulation ofproblem (12) in (15), thus the master problem (16) is alwaysa relaxation to the minimization ARO-DDU problem (12) forany k ∈ Z + . Thus f k ≤ f ∗ for any k ∈ Z + . Proof of Lemma 2(c):
Recall the definition of R ( x ) in (14), R ( x k ) = x k satisfies constraint (15c). Moreover,since x k is the solution to master problem (16), constrain (15b)(i.e., constraint (16b)) is met with x k . Thus x k is a feasiblesolution to the minimization problem (15), indicating that f k ≥ f ∗ . Proof of Lemma 2(d):
Lemma 2(d) can be easily verified bynoting that the optimal solution of bi-linear programming withpolyhedron feasible set can be achieved at one of the verticesof the polytopes [22]. Specific illustration is given as follows.For given x k , since ( w ∗ k , π ∗ k ) is the optimal solution to R ( x k ) , ( w ∗ k , π ∗ k ) ∈ arg max w ∈ W ( x k ) (cid:26) max π ∈ Π ( b − Ax k − Cw ) T π (cid:27) (25)Then there must be π ∗ k ∈ arg max π ∈ Π ( b − Ax k − Cw ∗ k ) T π . Bynoting that the unique optimal solution of linear programmingmust be found at one of its vertices, we have π ∗ ∈ vert ( Π ) . Proof of Lemma 2(e):
Suppose for the sake of contradictionthat there exists j , j ∈ [ k ] and j (cid:54) = j such that π ∗ j = π ∗ j .Without loss of generality we assume that j < j , and thus j ≤ j − j , j ∈ Z + . Suppose π ∗ j is the optimalsolution to R ( x j ) , there must be R (cid:0) x j (cid:1) >
0, implying thatmax w ∈ W ( x j ) π ∗ j T ( b − Ax j − Cw ) > . (26)Since π ∗ j = π ∗ j , we havemax w ∈ W ( x j ) π ∗ j T ( b − Ax j − Cw ) > . (27)Recall that x j is the optimal solution to the master problemwith the following feasibility cuts0 ≥ π ∗ j T ( b − Ax − Cw ) , ∀ w ∈ W ( x ) , j ∈ [ j − ] . (28) OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. XX, AUGUST XXXX 9
Since j ≤ j −
1, there must be0 ≥ π ∗ j T (cid:0) b − Ax j − Cw (cid:1) , ∀ w ∈ W ( x j ) (29)which contradicts with (27).Now we give the proof of Theorem 1. Proof.
According to Lemma 2(a)-(b), f k is monotonously non-decreasing with respect to k with an upper bound f ∗ . Com-bining Lemma 2(b) and (c), when the Algorithm.1 terminateswith R ( x k ) =
0, we have f k = f ∗ , verifying the optimality ofthe solution.Next we illustrate that the Algorithm.1 terminates withinfinite rounds of iterations. The number of vertexes of Π ,denoted by p , is finite and no vertex of Π can be appendedtwice to the master problem in Algorithm.1 according toLemma 2(d)-(e). Thus the Algorithm.1 terminates within O ( p ) iterations. R EFERENCES[1] S. M. Nosratabadi, R. A. Hooshmand, and E. Gholipour, “A compre-hensive review on microgrid and virtual power plant concepts employedfor distributed energy resources scheduling in power systems,”
Renew.Sust. Energ. Rev. , vol. 67, pp. 341–363, 2017.[2] H. Fu, Z. Wu, X. Zhang, and J. Brandt, “Contributing to dso’s energy-reserve pool: A chance-constrained two-stage µ vpp bidding strategy,” IEEE Power Energy Technol. Syst. J , vol. 4, no. 4, pp. 94–105, 2017.[3] Z. Liang, Q. Alsafasfeh, T. Jin, H. Pourbabak, and W. Su, “Risk-constrained optimal energy management for virtual power plants con-sidering correlated demand response,”
IEEE Trans. Smart Grid , vol. 10,no. 2, pp. 1577–1587, 2019.[4] M. Shabanzadeh, M. Sheikholeslami, and M.-R. Haghifam, “The designof a risk-hedging tool for virtual power plants via robust optimizationapproach,”
Appl. Energy. , vol. 155, p. 766–777, 2015.[5] M. Rahimiyan and L. Baringo, “Strategic bidding for a virtual powerplant in the day-ahead and real-time markets: A price-taker robustoptimization approach,”
IEEE Trans. Power Syst. , vol. 31, no. 4, pp.2676–2687, 2016.[6] P. Li, Y. Liu, H. Xin, and X. Jiang, “A robust distributed economicdispatch strategy of virtual power plant under cyber-attacks,”
IEEETrans. Ind. Informat. , vol. 14, no. 10, pp. 4343–4352, 2018.[7] Y. Zhou, Z. Wei, G. Sun, K. W. Cheung, H. Zang, and S. Chen, “Four-level robust model for a virtual power plant in energy and reservemarkets,”
IET Gener. Transm. Distrib. , vol. 13, no. 11, pp. 2036–2043,2019.[8] A. Baringo and L. Baringo, “A stochastic adaptive robust optimizationapproach for the offering strategy of a virtual power plant,”
IEEE Trans.Power Syst. , vol. 32, no. 5, pp. 3492–3504, 2017.[9] G. Sun, W. Qian, W. Huang, Z. Xu, Z. Fu, Z. Wei, and S. Chen,“Stochastic adaptive robust dispatch for virtual power plants using thebinding scenario identification approach,”
Energies , vol. 12, p. 1918,2019.[10] A. Baringo, L. Baringo, and J. M. Arroyo, “Day-ahead self-schedulingof a virtual power plant in energy and reserve electricity markets underuncertainty,”
IEEE Trans. Power Syst. , vol. 34, no. 3, pp. 1881–1894,2019.[11] N. H. Lappas and C. E. Gounaris, “Robust optimization for decision-making under endogenous uncertainty,”
Comput Chem Eng , vol. 111,pp. 252–266, 2018.[12] O. Nohadani and K. Sharma, “Optimization under decision-dependentuncertainty,”
SIAM J. Optim. , vol. 28, no. 2, p. 1773–1795, 2018.[13] M. Poss, “Robust combinatorial optimization with variable budgeteduncertainty,” , vol. 11, no. 1, pp. 75–92, 2013.[14] R. Vujanic, P. Goulart, and M. Morari, “Robust optimization of schedulesaffected by uncertain events,”
J Optim Theory Appl. , vol. 171, no. 3, pp.1033–1054, 2016.[15] X. Zhang, M. Kamgarpour, A. Georghiou, P. Goulart, and J. Lygeros,“Robust optimal control with adjustable uncertainty sets,”
Automatica ,vol. 75, pp. 249–259, 2017.[16] Q. Zhang and W. Feng, “A unified framework for adjustable robustoptimization with endogenous uncertainty,”
AIChE Journal , vol. 66,no. 12, 2020. [17] N. H. Lappas and C. E. Gounaris, “Multi-stage adjustable robustoptimization for process scheduling under uncertainty,”
AIChE Journal ,vol. 62, no. 5, pp. 1646–1667, 2016.[18] Y. Su, Y. Zhang, F. Liu, S. Feng, Y. Hou, and W. Wang, “Robust dispatchwith demand response under decision-dependent uncertainty,” iSPEC ,2020.[19] B. Zeng and L. Zhao, “Solving two-stage robust optimization problemsusing a column-and-constraint generation method,”
Oper. Res. Let. ,vol. 41, no. 5, pp. 457–461, 2013.[20] J. Benders, “Partitioning procedures for solving mixed-variables pro-gramming problems,”
Numer Math (Heidelb)