Evaluating the Contribution of Energy Storages to Support Large ...

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON SUSTAINABLE ENERGY

1

Evaluating the Contribution of Energy Storages to Support Large-Scale Renewable Generation in Joint Energy and Ancillary Service Markets Peng Zou, Student Member, IEEE, Qixin Chen, Senior Member, IEEE, Qing Xia, Senior Member, IEEE, Guannan He, and Chongqing Kang, Senior Member, IEEE

Abstract—Energy storage systems (ESSs) are of great value to realize energy management and to support large-scale renewable generation. The combined operation of ESSs and renewables is one way to achieve output levelling and to improve the integration of sustainable energy. However, in a market-based environment, ESSs would make strategic decisions on self-schedules and arbitrage in energy and ancillary service markets, maximizing the overall profits. Will the strategic operation of ESSs promote renewable generation integration? To explicitly answer this question, this paper proposes a multi-period Nash-Cournot equilibrium model for joint energy and ancillary service markets to evaluate the contribution of the ESSs for supporting renewable generation. Then, a reformulation approach based on the potential function is proposed, which can transform the bi-level equilibrium model into an integrated single-level optimization problem to enhance the computation efficiency. Numerical examples are implemented to validate the effectiveness of the reformulation technique. The results of the case study indicate that the ESSs indirectly but substantially provide improved flexibilities while pursuing individual profit maximization. Index Terms—Electricity market, energy storage systems, NashCournot equilibrium, potential function, renewable generation integration.

λR (t) λF C (t) λF P (t) θSi θT i θHi dis ηSi cha ηSi qSi max qT i max qHi max qT i min qHi min PU qTRM i RM P D qT i ESi max EHi max

N OMENCLATURE Indices and Sets T Time index. Si Energy storage system index. T i Thermal unit index. Hi Hydro unit index. Ri Renewable unit index. Parameters and Constants N Total number of market participants. TP Total number of thermal units. HP Total number of hydro units. RE Total number of renewable units. ES Total number of energy storages. Manuscript received March 30, 2015; revised June 30, 2015 and August 12, 2015; accepted October 30, 2015. This work was supported by the National Natural Science Foundation of China under Grants 51107059, 51325702, and 5136113570402. Paper no. TSTE-00226-2015. The authors are with the State Key Laboratory of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSTE.2015.2497283

qSi max qRi max (t) qRi min (t) qD (t) α(t) β(t) R (t) qD F (t) qD

Reserve market price at time t, $/MW. Regulation capacity price at time t, $/MW. Regulation performance price at time t, $/MW. Regulation mileage ratio of storage Si. Regulation mileage ratio of thermal unit Ti. Regulation mileage ratio of hydro unit Hi. Discharging efficiency of storage Si. Charging efficiency of storage Si. Maximum generation output of storage Si, MW. Maximum generation output of thermal unit T i, MW. Maximum generation output of hydro unit Hi, MW. Minimum generation output of thermal unit T i, MW. Minimum generation output of hydro unit Hi, MW. Ramp up rate limit of thermal unit T i, MW. Ramp down rate limit of thermal unit T i, MW. Maximum energy the storage Si can store, MWh. Maximum energy the hydro unit Hi can provide, MWh. Maximum generation output of storage Si, MW. Maximum generation output of renewable unit Ri at time t, MW. Minimum generation output of renewable unit Ri at time t, MW. Load demand at time t, MW. Intercept of the inverse demand function, $/MWh. Slope of the inverse demand function, $/(MWh × MW). Reserve demand at time t, MW. Regulation demand at time t, MW.

Variables πSi (t) Profit of energy storage Si at time t, $. πT i (t) Profit of thermal unit T i at time t, $. πHi (t) Profit of hydro unit Hi at time t, $. πRi (t) Profit of renewable unit Ri at time t, $. λE (t) Energy market price at time t, $/MWh. E (t) Generation output of storage Si at time t, MW. qSi qTEi (t) Output of thermal unit T i at time t, MW. E qHi (t) Output of hydro unit Hi at time t, MW. E (t) Output of renewable unit Ri at time t, MW. qRi

1949-3029 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON SUSTAINABLE ENERGY

R qSi (t) qTRi (t) R (t) qHi F (t) qSi qTF i (t) F (t) qHi dis (t) qSi cha (t) qSi (t) bdis Si

Reserve capacity of storage Si at time t, MW. Reserve capacity of thermal unit T i at time t, MW. Reserve capacity of hydro unit Hi at time t, MW. Regulation capacity of storage Si at time t, MW. Regulation capacity of thermal unit T i at time t, MW. Regulation capacity of hydro unit Hi at time t, MW. Discharging power of storage Si at time t, MW. Charging power of storage Si at time t, MW. Binary variable indicating the discharging status of storage Si at time t. (t) Binary variable indicating the charging status of storbcha Si age Si at time t. ESi (t) Stored energy of storage Si at time t, MWh.

I. I NTRODUCTION

R

ECENT past years have witnessed a continuously changing generation mix around the world. The proportion of renewables is gradually growing, and the conventional fossil energy is undergoing a declining share in the primary inputs to power generation [1]. The ever-increasing wind farms and solar stations produce more clean and eco-friendly power, but put great pressure on the operation of power systems and the electricity markets due to their intermittent and stochastic nature [2], [3]. Meanwhile, emerging energy storage systems (ESSs) are tentatively used to provide a practical solution for more flexibility and to prevent renewable energy from being curtailed. Therefore, several papers have concentrated on the combined operation of ESSs and renewable generations in power markets, which are always price-takers, to make the systems more controllable and to obtain more profits. Reference [4] analyzes the combined planning and operation of wind power and ESSs in the electricity market. Based on the forecasted prices, a procedure is proposed to determine the hourly production profile of this combined system. Reference [5] proposes a technology that combines the concentrating solar power and the ESS as dispatchable to smooth the production profile and obtain more profit in the spot energy market. Reference [6] proposes a strategy for the integrated operation of a wind farm and pumped-hydro storage to increase the benefits from both day-ahead energy and ancillary service markets. Reference [7] uses a risk measure-based, robust bidding strategy for a wind farm in combination with ESS in the day-ahead energy market. An optimal bidding strategy is considered for independently operated ESSs participating in the energy and reserve markets with high penetration of wind power based on a stochastic programming approach [8]. However, the ESSs are regarded as price-takers, and the effects of ESSs to support renewables integration are not discussed in this paper. In fact, in a practical market-based environment, there will be increasing independently invested ESSs and renewable generators that belong to different companies. In this case, the ESSs tend to make strategic decisions on self-schedules and arbitrage in energy and ancillary service markets, maximizing their own profits. Will the strategic operation and behaviors of ESSs still promote renewable generation integration? Specifically, will the ESSs spontaneously or indirectly facilitate more renewable energy to be accommodated while maximizing

individual profits in joint energy and ancillary service (AS) markets? Why and how do these things happen? What factors will influence the strategic behaviors of ESSs and the effects of renewable generation accommodation? To answer the above questions, this paper establishes a multiperiod Nash-Cournot equilibrium model for joint energy and AS markets to accurately study the impacts of the strategic behaviors of independently operated ESSs on the integration of large-scale renewable generations. Multi-scenario settings of wind generation profile and a deterministic model for each scenario are applied to reflect the stochastic nature of the renewables [9]–[11]. In fact, the Cournot model and the supply function model both have their advantages and disadvantages and which one is more appropriate depends on the topic and the purpose of the research. The Cournot competition model may seem less attractive than the supply function equilibrium model in the aspect of reflecting the participants’ real strategic bidding behaviors. However, according to [12], when facing certain demand distributions, a market participant would exactly know its equilibrium residual demand as a result and it has only one choice to maximize its profit, that is, it has no difference to choose either a strategic price or a strategic quantity. In other words, there is no difference to use either the Cournot equilibrium model or the supply function equilibrium model when the uncertainties of demand or renewables are ignored in the model. Although the bidding prices cannot be obtained by using the Cournot model, considering the advantages in computation efficiency, it is still widely used in numerous applications [13], [14], including the market power analysis [15], hydrothermal coordination [16], transmission congestion [17] and co-optimization of energy and ancillary service markets [18]. At present, two traditional way to obtain the Nash-Cournot equilibrium is to simultaneously solve the multi-individual profit-maximization problems with either a nonlinear complementarity problem (NCP) approach or diagonalization methods [19]. Specifically, the diagonalization methods consist of the Jacobi and Guass-Seidel (GS) algorithms, which iteratively solve each generator’s profit-maximization model until a stationary point is obtained. The NCP method collects all of the optimal KKT conditions of each generator’s individual profitmaximization models and then solves them together. However, these two conventional methods are computation intractable and lack robustness [20]. These issues are studied in this paper. Overall, the main contributions of this paper are as follows: 1) A multi-period optimization model is constructed to study the Nash-Cournot equilibrium and the strategic interactions among various generators, including the ESSs, thermal units, hydro units, large-scale wind farms and solar stations in joint energy and AS markets. The payfor-performance mechanism is also considered in the regulation market. 2) The ESSs are considered as price-makers, considering their remarkable potential in the future, instead of pricetakers to precisely describe their strategic behaviors in the energy market. 3) A novel reformulation technique supported by introducing the potential function is proposed to transform the conventional Nash-Cournot equilibrium model into an

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZOU et al.: EVALUATING THE CONTRIBUTION OF ENERGY STORAGES TO SUPPORT LARGE-SCALE RENEWABLE GENERATION

integrated single-level optimization problem, to improve the computation efficiency and identify more optimal solutions. Specifically, the remainder of this paper is organized as follows. The operation pattern of joint energy and AS markets is briefly introduced in Section III. The Nash-Cournot model is derived, and the multi-individual optimization problems are obtained in Section IV. The reformulated single-level optimization model supported by the potential function is deduced in Section V. Numerical examples are tested to validate the reformulation approach and to evaluate the contribution of ESSs for supporting large-scale renewable generation in Section VI. Finally, the conclusions are summarized in Section VII.

II. C O -O PTIMIZATION OF E NERGY AND A NCILLARY S ERVICE M ARKETS Without loss of generality and rationality, multi-period energy and AS markets with one shot game are considered in this paper. Just like the rules implemented in the electricity markets in North America, renewable generators, including wind farms and solar stations, are only permitted to participate in the energy market because of their stochastic outputs, whereas ESSs have the rights to enter the three markets at the same time, as do the thermal power and conventional hydropower.

A. Energy Market In the pool-based energy market, it is assumed that various generators compete in the Cournot manner with complete information, that is, each generator strategically determines its energy quantity to produce, considering the possible behaviors of rivals, and submits this offer to the market operator (MO) before the gate closure. The MO clears the energy market considering the power supply-demand balance constraints and calculates the time-varying marginal clearing price (MCP) to make settlements based on the linear inverse demand function received from the consumer side.

B. Regulation and Reserve Markets Because the regulation and reserve services are mainly established to guarantee secure operation of power systems, the AS demands are usually not considered as price-sensitive but are dependent on the system conditions. In this paper, the regulation and reserve markets are reasonably assumed to have abundant resources and to be relatively competitive. In this case, participants in the AS markets are regarded as price-takers for the purpose of simplicity and the regulation and reserve prices are assumed to be constants. The reserve market implements capacity payments, whereas the performance-based regulation market includes not only capacity payments, but also performance payments to ensure the fast response resources, such as hydropower and ESSs, more incentives and compensation. This mechanism is referenced from the “pay for performance” scheme in the regulation market of PJM [21].

3

C. Co-optimization of Energy and AS Markets When generators are simultaneously providing energy and AS services, the capacity coupling effects of generators should be precisely considered to achieve the optimization of power resource allocation. Therefore, the co-optimization of energy and AS markets is implemented in several major electricity markets operated by the independent system operators (ISO) or regional transmission organizations (RTO) in North America, such as the markets administrated by PJM (short for Pennsylvania, New Jersey and Maryland), ERCOT (short for Electric Reliability Council of Texas) and CAISO (short for California ISO). Accordingly, the objective of the MO is extended to minimize the total production costs of the energy and AS services. Various generators’ technical constraints, especially their capacity coupling constraints for providing energy and AS services, are included to jointly clear the multi-markets. III. M ODELING OF THE NASH -C OURNOT E QUILIBRIUM IN J OINT E NERGY AND AS M ARKETS The Nash-Cournot equilibrium can be formulated as a bilevel optimization model. The upper-level model represents the individual profit-maximization problems of various participants, whereas the lower-level model indicates the clearing conditions of the energy and AS markets, which are shared by each generator. Then, the multi-individual optimization problems can be obtained by substituting the common lower-level model into each upper-level model. The multi-individual optimization problems need to be simultaneously solved to get the Nash-Cournot equilibrium. In this paper, the potential function is introduced to transform the multi-individual optimization problems into an integrated single-level optimization problem to improve the computation efficiency. The specific flowchart of the Nash-Cournot equilibrium modeling is presented as Fig. 1. It should be noted here that there is no objective function in the lower-level model as the classic Cournot model does [22]. But the equilibrium model presented in this paper is still called bi-level optimization model because it has the same mathematical structure as bi-level models. A. Profit Maximization Models for Various Generators The objective functions of thermal units, hydro units and ESSs are to maximize the payments from both the energy and AS markets, whereas for wind and solar power, the revenues are only from the energy market. The explicit variable costs of all generators, except thermal power, are reasonably assumed as zero [23]. The operation and maintenance costs of ESSs have been implicitly considered in the form of round-trip efficiencies. 1) Energy Storage Systems: The objective function of the individual profit-maximization model of ESSs is: T T E R λE (t)qSi (t) + λR (t)qSi (t) πSi (t) = max max F F F F (t)q (t) + θ λ (t)q +λ Si P C Si Si (t) t=1 t=1 (1a)



Fig. 1. The flowchart of the Nash-Cournot equilibrium modeling.

where λE (t) is the MCP of the energy market in period t, λR (t) is the clearing price of the reserve market in period t, λF C (t) is the capability price of the regulation market in period t, λF P (t) is the performance price of the regulation market in period t, θSi E R is the regulation mileage ratio of ESSs, and qSi (t), qSi (t) and F qSi (t) are, respectively, the generation output, reserve capacity and regulation capacity determined by the ESS Si in period t. The charging/discharging states determination equations are: ∀t ∈ T , ⎧ E dis cha ⎪ qSi (t) = qSi (t) + qSi (t) ⎪ ⎪ ⎪ dis dis ⎪ ⎪ ⎨0 ≤ qSi (t) ≤ bSi (t)qSi max cha (1b) −bcha Si (t)qSi max ≤ qSi (t) ≤ 0 ⎪ ⎪ dis cha ⎪ bSi (t) + bSi (t) = 1 ⎪ ⎪ ⎪ ⎩bdis (t), bcha (t) ∈ {0, 1} Si Si dis cha (t) and qSi (t) are, respectively, the discharging where qSi and charging power of the ESS Si, qSi max is the maximum cha capacity limit, and bdis Si (t) and bSi (t) are the binary variables representing the working status of ESSs. The capacity constraints of ESS Si are: ∀t ∈ T , ⎧ E R F ⎪ ⎨qSi (t) + qSi (t) + qSi (t) ≤ qSi max E F (1c) qSi (t) − qSi (t) ≥ −qSi max ⎪ ⎩ R F qSi (t), qSi (t) ≥ 0

The available energy constraints necessarily kept to guarantee the continuous provision of regulation and reserve services for a certain period are: ∀t ∈ T , E R F (t) · h + qSi (t) · h2 + qSi (t) · h4 ≤ ESi (t) qSi (1d) E F (t) · h + qSi (t) · h4 ≤ ESi max ESi (t) − qSi where h equals one hour here, ESi (t) is the stored energy in period t, and ESi max is the maximum energy that the ESS can store. The first constraint of (1d) means that the ESS in period E (t) level, t can continuously discharge for one hour at the qSi R provide reserve service for 30 minutes at the qSi (t) level and F (t) level. These coefficients regulate for 15 minutes at the qSi are referenced from PJM markets and related papers [24], [25].

Obviously, the longer the duration for providing the AS services means less regulation and reserve capacities that the ESS can bid considering its limited stored energy. The energy conversion relations are as below: ∀t ∈ T , ESi |end − ESi |Init = 0 (1e) dis dis cha cha (t)/ηSi − ηSi qSi (t) ESi (t + 1) = ESi (t) − qSi dis cha and ηSi are the efficiencies of discharging and where ηSi charging. The first equation of (1e) ensures that the final state of charge equals the initial state of charge on a daily cycle. The self-discharge of ESS is ignored in the model because many kinds of storages usually have no daily self-charge, such as pumped hydro storage, hydrogen-based energy storage system, compressed air energy storage, zinc-bromine flow battery, polysulphide-bromide flow battery, sodium-sulphur battery [26], and this assumption does not imposes significant effects on the results. 2) Thermal Unit: The objective function of the individual profit-maximization model of thermal unit is formulated as the following:

max

T

πT i (t)

t=1

T F λE (t)qTEi (t) + λR (t)qTRi (t)+λF C (t)qT i (t) = max F F E 2 +θT i λP (t)qT i (t)−{ai [qT i (t)] +bi qTEi (t)+ci ]} t=1 (1f) where θT i is the regulation mileage ratio of thermal power, qTEi (t), qTRi (t) and qTF i (t) are, respectively, the generation output, reserve capacity and regulation capacity, that is, the decision variables determined by thermal unit T i in period t, and ai , bi , ci ≥ 0 are the coefficients of the total generation cost. The capacity constraints of the thermal units are: ∀t ∈ T , ⎧ E R F ⎪ ⎨qT i (t) + qT i (t) + qT i (t) ≤ qT i max E F (1g) qT i (t) − qT i (t) ≥ qT i min ⎪ ⎩ E R F qT i (t), qT i (t), qT i (t) ≥ 0 where qT i min and qT i max are the minimum and maximum generation output limits.


The ramp constraints of thermal unit are: ∀t ∈ T , PU [qTEi (t+1)+qTRi (t+1)+qTF i (t+1)]−[qTEi (t)−qTF i (t)]≤qTRM i PD [qTEi (t)+qTRi (t)+qTF i (t)]−[qTEi (t+1)−qTF i (t+1)]≤qTRM i (1h) PU PD and qTRM are the ramp up and down rate where qTRM i i limits of thermal unit T i. The response time constraints of the reserve and regulation services are: ∀t ∈ T , R PU · tR qT i (t) ≤ qTRM i (1i) PU qTF i (t) ≤ qTRM · tF i where tR is the spinning reserve response time limit, which is assumed to be 10 minutes, and tF is the regulation response time limit, which is assumed to be 5 minutes in this paper. 3) Hydropower: The objective function of the individual profit-maximization model of hydro unit is: T T E R λE (t)qHi (t) + λR (t)qHi (t) πHi (t) = max max F F F F (t)q (t)+θ λ (t)q +λ Hi P C Hi Hi (t) t=1 t=1 (1j) where θHi is the regulation mileage ratio of hydropower, and E R F (t), qHi (t) and qHi (t) are, respectively, the generation outqHi put, reserve capacity and regulation capacity determined by hydro unit Hi in period t. The capacity constraints of hydropower are: ∀t ∈ T , ⎧ E E F ⎪ ⎨qHi (t)+qHi (t) + qHi (t) ≤ qHi max E F (1k) qHi (t) − qHi (t) ≥ qHi min ⎪ ⎩ E E F qHi (t), qHi (t), qHi (t) ≥ 0 where qHi min and qHi max are the minimum and maximum generation output limits. The available energy constraint considering the water resource limit is: T

E qHi (t) ≤ EHi max

(1l)

t=1

where EHi max is the maximum energy hydropower can provide in an entire day. The constraints that guarantee the continuous provision of regulation and reserve services for a certain duration are ignored here for simplicity. Moreover, the response time constraints of the reserve and regulation services are also ignored considering the high flexibility of hydro units. 4) Renewable Unit: The objective function of the individual profit-maximization model of wind farms or solar stations is to maximize the payments only from the energy market: max

T t=1

πRi (t) =

T

E λE (t)qRi (t)

(1m)

t=1

E where qRi (t) is the generation output decided by the renewables Ri in period t.

5

The time-varying weather-dependent capacity constraints of renewable energy are: ∀t ∈ T , E (t) ≤ qRi max (t) qRi min (t) ≤ qRi

(1n)

where qRi max (t) and qRi min (t) are the minimum and maximum generation output limits. B. Market Clearing Conditions The energy market clearing conditions incorporate the multiperiod power balance constraints and the linear inverse demand function. For t = 1, 2, · · · T , we have: ⎧N T P HP RE ES E E E E ⎪ E ⎪ q (t) = q (t)+ q (t)+ q (t)+ qSi (t) ⎪ Xi T i Hi Ri ⎪ ⎨i i=1 i=1 i=1 i=1 N E qXi (t) = qD (t) ⎪ ⎪ ⎪ i ⎪ ⎩ E λ (t) = α(t) − β(t)qD (t) (2a) where N = T P + HP + RE + ES is the total number of E various generators participating in the energy market, qXi (t) represents a certain kind of generation units, qD (t) is the electricity demand in period t, and α(t) and β(t) are the coefficients of the inverse demand function in period t. Based on the decreasing inverse demand function, the generators collectively and strategically determine the energy prices considering the possible reactions of the rivals to maximize its individual profit. The reserve market clearing conditions are the multi-period reserve supply-demand balance constraints: R (t) = qD

TP

qTRi (t) +

i=1

HP

R qHi (t) +

i=1

ES

R qSi (t)

(2b)

i=1

R where qD (t) is the reserve demand in period t, which is determined by three parts: the load reserve, emergency reserve and the extra reserve caused by the stochastic outputs of renewable power. The regulation market clearing conditions are the multiperiod regulation supply-demand balance constraints:

F (t) = qD

TP i=1

qTF i (t) +

HP

F qHi (t) +

i=1

ES

F qSi (t)

(2c)

i=1

F where qD (t) is the regulation demand in period t, which is often a certain ratio of the peak load, for example, 0.7% in the PJM regulation market [21]. Considering that the contributions of ESSs to support largescale renewables are mainly studied from the system-wide perspective, the transmission constraints are not included for simplicity.

C. Multi-individual Optimization Problems By substituting the energy market clearing conditions (2a) into the objective functions of the individual profitmaximization models (1a), (1f), (1j) and (1m) for each



generator, respectively, the compact objective functions can be expressed as: max

Xi=T i,Hi,Ri,Si

T

πXi (t)

t=1

⎧ N T ⎨ E E R qXi (t)]qXi (t) + λR (t)qXi (t) [α(t) − β(t) = max i=1 ⎩ F F F E t=1 +λC (t)qXi (t)+θXi λF P (t)qXi (t)−CXi [qXi (t)] (2d) E where the generation cost function CXi [qXi (t)] is as below: ∀t ∈ T , E CXi [qXi (t)] ai [qTEi (t)]2 + bi qTEi (t) + ci , = 0,

for Xi = T i for Xi = Hi, Ri or Si (2e)

Moreover, the renewables do not participate in the AS marR F (t) and qXi (t) in (2d) for wind farms and kets; thus, the qXi solar stations are always equal to zero. Thus, the multi-individual profit-maximization model of each generator can be obtained by incorporating its own technical constraints, that is, (1b), (1c), (1d) and (1e) for the ESSs; (1g), (1h) and (1i) for thermal power; (1k) and (1l) for hydropower; and (1n) for the renewables. In addition, the reserve and regulation supply-demand balance conditions are also needed as the extra constraints shared by each generator.

difference of its own objective function and that of the potential function are exactly identical when it changes the bidding strategy in the electricity market. In short, the potential function can precisely capture the payoff deviation of each generator when it alters its capacity allocations considering the possible reactions of the rivals. Therefore, taking the potential function as the new objective function and incorporating the technical constraints of all generators (1b)-(1e), (1g)-(1i), (1k)-(1l) and (1n), a reformulated Nash-Cournot equilibrium model can be obtained. The speE R F (t), qXi (t), qXi (t)] is derived as cific potential function Pˆ [qXi below.

B. Procedures to Derive the Potential Function First, take the derivative of the single-period function (2d) E R F with respect to qXi (t), qXi (t) and qXi (t). We then have: ⎧ ∂πXi (t) E ⎪ ⎪ = α(t) − 2β(t)qXi (t) ⎪ E (t) ⎪ ∂q ⎪ Xi ⎪ ⎪ E N ⎪ (t)] dCXi [qXi ⎪ E ⎪ qXj (t) − −β(t) ⎨ E dqXi (t) j=i (t) ∂π ⎪ Xi R ⎪ ⎪ = λ (t) ⎪ R (t) ⎪ ∂qXi ⎪ ⎪ ⎪ ⎪ ∂π (t) ⎪ F ⎩ FXi = λF C (t) + θXi λP (t) ∂qXi (t)

(3b)

Second, sum all of the objective functions (2d) of each individual profit-maximization problem to obtain:

IV. R EFORMULATION W ITH THE P OTENTIAL F UNCTION By introducing the potential function, the conventional NashCournot equilibrium model with multi-individual optimization problems can be equivalently transformed into an integrated single-level optimization model. The optimal results of this new model are strictly equal to the original equilibrium point.

E R F Π[qXi (t), qXi (t), qXi (t)]

= α(t)

N

E qXi (t)

i=1

− β(t)

E [qXi (t)

i=1

E R F Considering the decision variables qXi (t), qXi (t), qXi (t) of each individual profit-maximization model, it can be seen from (3a) that when a participant adjusts its generation output, reserve capacity or regulation capacity, the change of its own profit can be reflected by either the original individual objective function (2d) or the potential funcE R F (t), qXi (t), qXi (t)], that is, the generator’s profit tion Pˆ [qXi

+ λR (t)

E [qXi (t)]

2

i=1

N

A. Principle of the Potential Function If ∀i ∈ N , the objective functions (2d) are continuously E R F differentiable. Then, Pˆ [qXi (t), qXi (t), qXi (t)] is the potential function, if and only if [27]–[29]: ∀t ∈ T , ⎧ E R F ∂ Pˆ [qXi ∂πXi (t) (t), qXi (t), qXi (t)] ⎪ ⎪ = ⎪ ⎪ E (t) E (t) ⎪ ∂q ∂qXi ⎪ Xi ⎪ ⎪ ⎨ ˆ E R F ∂ P [qXi (t), qXi ∂πXi (t) (t), qXi (t)] (3a) = R R (t) ⎪ ∂qXi (t) ∂qXi ⎪ ⎪ ⎪ ⎪ E R F ⎪ ∂ Pˆ [qXi ∂πXi (t) (t), qXi (t), qXi (t)] ⎪ ⎪ ⎩ = F (t) F (t) ∂qXi ∂qXi

− β(t)

N

N i=1

N

E qXj (t)] −

j=i R qXi (t)+λF C (t)

N

E CXi [qXi (t)]

i=1 N i=1

F qXi (t)+λF P (t)

N

F θXi qXi (t)

i=1

(3c) Third, consider (3b) and take the derivative of the above E R F (t), qXi (t) and qXi (t): function (3c) with respect to qXi ⎧ E R F ∂Π[qXi (t), qXi (t), qXi (t)] ⎪ ⎪ ⎪ E ⎪ ∂qXi (t) ⎪ ⎪ ⎪ E N ⎪ (t)] dCXi [qXi ⎪ E E ⎪ = α(t) − 2β(t)qXi (t) − 2β(t) qXj (t) − ⎪ ⎪ E ⎪ dq (t) ⎪ j = i Xi ⎪ ⎨ ∂π (t) N Xi E − β(t) qXj (t) = E (t) ⎪ ∂qXi j=i ⎪ ⎪ ⎪ ∂Π[q E (t), q R (t), q F (t)] ⎪ ∂πXi (t) ⎪ Xi Xi Xi ⎪ = λR (t) = ⎪ R (t) R (t) ⎪ ⎪ ∂q ∂qXi ⎪ Xi ⎪ E R F ⎪ ∂Π[qXi (t), qXi (t), qXi (t)] ∂πXi (t) ⎪ F ⎪ ⎩ = λF C (t)+λP (t)θXi = F F (t) ∂qXi (t) ∂qXi (3d)


7

Finally, considering (3a) and (3d), the potential function is: T

E R F Pˆ [qXi (t), qXi (t), qXi (t)]

t=1

=

⎧ N N ⎪ 2 E E ⎪ ⎪ α(t) qXi (t) − β(t) [qXi (t)] ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎪ N N ⎪ ⎪ E E qXi (t)qXj (t) T ⎪ ⎨−β(t) i=1 j