Bidding Strategy of a Wind-Thermal GENCO considering Piecewise Linear AC Power Flow and Correlated Uncertainties

In deregulated electricity markets, generation companies (GENCO) try to maximize their economic beneﬁts considering the electricity demand, transmission network condition, and other participants’ behaviors. The increasing penetration of renewable sources such as wind power generation with intermittent nature poses several challenges to the participation of GENCOs in the electricity market. Thus, this paper presents a stochastic bilevel optimization model to determine the coordinated bidding strategy of a wind-thermal GENCO with the aim of maximizing its proﬁt in the day-ahead and real-time balancing market. Herein, the model aims to maximize the proﬁt of GENCO in the day-ahead and the balancing market in the upper-level problem while minimizing the operation cost of the system in the lower-level problem. The uncertainties of wind power generation and electricity demand are modeled by deﬁning a set of scenarios considering their mutual correlation using the copula technique. Additionally, incorporating AC power ﬂow constraints in the proposed optimization model oﬀers a better solution to the coordinated bidding strategy of the wind-thermal GENCO. Further, the nonlinear AC power ﬂow equations are linearized using the piecewise approximation technique to reduce the computational complexity and enhance the accuracy of the optimal solution. In the end, the developed algorithm is implemented on the IEEE 24-bus RTS, and the simulation results are provided to validate the eﬃciency and applicability of the proposed coordinated bidding strategy model. The results advocate that the participation of the thermal unit along with the wind farm might mitigate the risk of uncertainties, but it causes an intense increase in the locational marginal price of the system. Importantly, the simulation results indicate the computational eﬃciency of the model by developing an exact AC power ﬂow model without compromising the results. Notably, it has been found that the proﬁt of the wind-thermal GENCO would be increased by 35.2% employing the copula technique to model the mutual correlation of uncertain parameters.


Aim.
e intense increase in fossil fuel prices and environmental concerns of greenhouse gases (GHG) has made policy-makers and power system planners find alternatives for conventional thermal and gas power plants [1,2]. In recent decades, exploiting renewable energy sources (RESs) in the generation sector of power systems has been increasing rapidly [3,4]. e leading technologies of RESs include wind and solar power, hydropower, biomass, and geothermal [5,6]. A remarkable penetration of wind power, among other technologies, has been realized in the last decade [7]. e total deployed capacity of wind power at the end of 2019 was 651 GW worldwide, which means more than 10% growth compared to 2018 [8]. However, the intermittency and unpredictability nature of wind power poses significant challenges in the power system's operation [9].
In a restructured environment with competitive electricity markets, the generation companies (GENCO) determine the optimum bidding strategy for participating in the wholesale electricity market [10]. e main goal of market participants in the electricity market is maximizing their profit considering the prevailing uncertainties [11]. For a GENCO with thermal power plants, merely the market-clearing price is the uncertain parameter. In contrast, the uncertainties of market-clearing price and output power of wind turbines must be addressed for a GENCO with wind turbines. e uncertainties make a large variability in the profit of market participants. us, the GENCO must properly model the impacts of prevailing uncertainties on its optimal bid price-quantity that will be submitted to the independent system operator (ISO).

Literature Review.
In the literature, several works have investigated the bidding strategy of producers (with and without RES penetration) to optimally participate in the wholesale electricity market. e optimal day-ahead bidding strategy of a large wind power producer to maximize its expected profit under uncertainties was studied in [12]. References [13,14] numerically demonstrated the profitability of the coordinated scheduling of wind and thermal units participating in the wholesale electricity market compared to the nonintegrated one. Authors in [15] proposed a stochastic mixed-integer linear programming optimization model to determine the scheduling of a windthermal producer in the wholesale electricity market under the uncertainties of electricity price and wind power generation. Authors in [16] developed a coordinated bidding strategy for wind-hydro-pumped storage system to maximize the profit in the day-ahead and real-time balancing electricity markets. e work reported in [17] proposed a multiobjective framework to determine the optimal bidding strategy of a wind-thermal-energy storage system in the day-ahead electricity market. In this work, the presented optimization problem tries to simultaneously maximize the profit of coordinated energy sources and minimize the emission of GHG. e paper in [18] studied both price-maker roles in the day-ahead market and pricetaker roles as balancing market in real time. Authors in [19] developed an optimization model to determine the optimal bidding strategy of a GENCO with coordinated windpumped storage-thermal system. e proposed model tries to maximize the economic profit of the GENCO participating in the day-ahead electricity and spinning reserve market, considering the wind generation uncertainty. e paper in [20] developed an intelligent programmed genetic algorithm (GA) equipped with advanced deterministic diversity creating operator to solve GENCO's bidding strategy problem considering carbon credit trading. Reference [21] proposed a stochastic multiobjective bidding strategy model for participation of wind-thermal-photovoltaic GENCO in the wholesale electricity market. Authors in [22] developed a coordinated bidding strategy model for participation of a combined energy system composed of wind farm and compressed air energy storage (CAES) in the day-ahead and ancillary service markets. e authors in [23] proposed a stochastic risk-constrained optimization problem to determine the bidding strategy of a GENCO owing wind-thermal-CAES system.
Although the above-reviewed studies have provided valuable insights into the coordinated bidding strategy of different energy sources, they have not considered the transmission network constraints. In this regard, a networkconstrained linear programming model for the bidding strategy of a hydrothermal system was proposed in [24]. e uncertainties associated with market-clearing prices and forced outage of generators were considered in this work. Authors in [25] presented a multicriteria optimization problem to determine the bidding strategy of hydropower thermal systems. Herein, the uncertainty related to the load demand was modeled by scenario-based optimization. e paper in [26] proposed an optimization problem to determine the bidding strategy of a wind power producer with market power to participate in the day-ahead electricity market with the pay-as-bid payment mechanism. e paper only considered the uncertainty of wind power utilizing the scenario generation technique. A stochastic bilevel optimization problem was proposed for coordinated wind power and gas turbine systems in the real-time market regarding conditional value at risk (CVaR) [27]. Likewise, authors in [28] proposed a stochastic bilevel optimization model to determine the bidding strategy of a strategic wind power producer to participate in the network-constrained electricity market. e upper-level problem maximizes the economic benefit of the wind farm, and the lower-level problem clears the market based on the received bids of the market participants. A comprehensive stochastic optimal bidding strategy for a microgrid is proposed regarding the interdependency of power and gas market prices on the joint day-ahead and real-time markets [29]. In this paper, electricity market prices in the day-ahead and real-time markets, solar and wind productions, and the electrical and thermal demands were considered as uncertain parameters. In the above-reviewed studies, since the transmission network constraints were modeled using simple DC power flow equations, the constraints related to the voltage magnitude and the reactive power flowing in transmission lines have been ignored. Besides, the correlation between uncertain parameters is essential for optimal bidding strategy problems. However, the reviewed studies have not investigated the impact of correlation on the coordinated bidding strategy.

Contributions.
As discussed in the literature review and to the best of the authors' knowledge, the studies in the scope of coordinated bidding strategy have ignored the impact of AC power flow equations and correlation between uncertain parameters on the optimal bidding strategies. Complementary to the above-reviewed studies, this paper proposes a network-constrained bilevel bidding strategy model for a wind-thermal GENCO to maximize its profit. In the presented optimization problem, the uncertainties of wind power generation and electricity demand are modeled using a set of scenarios, considering their mutual correlation. Besides, the nonlinear constraints of AC power flow problem are reformulated as linear equations. erefore, the significant contributions of the paper are briefed as follows: A stochastic bilevel optimization problem is proposed to determine the coordinated bidding strategy of the wind-thermal system. In the upper-level problem, the profit of GENCO in the day-ahead and the balancing market is maximized. On the other hand, the marketclearing optimization problem minimizes the total operation cost in the lower-level problem. It is worth mentioning that the Iberian Peninsula market mechanism is employed to model the revenue of the wind power producer. e uncertain behaviors of wind output power and electricity demand are properly modeled in the optimization problem. Besides, the correlation of uncertain parameters is regarded in the model; that is to say, during a day in summer, when the wind speed increases, the ambient temperature will be decreased.
us, consumers turn off their cooling systems, which will result in the electricity demand decrease. Consequently, wind turbines' production power and electricity demand are negatively correlated. e proposed model's mutual correlation between the uncertainties is modeled using the copula technique. As stated earlier, relevant research works in the area of coordinated bidding strategy studies ignored either the transmission network constraints or employed simple DC power flow equations in the transmission network. To amend this shortcoming, the proposed model utilizes AC power flow equations. Besides, the nonlinear AC power flow constraints are linearized using the piecewise linear approximation technique to improve the accuracy of the solution and mitigate the computation time.

Paper Organization.
e remainder of the paper is structured as follows. e main outlines of the proposed model, including uncertainty modeling and revenue function of the wind power producers in the Iberian Peninsula, are explained in Section 2. Section 3 provides the mathematical formulation of the proposed stochastic bilevel optimization problem and the solution methodology. e proposed coordinated bidding strategy model is implemented in the IEEE RTS 24-bus, and the effectiveness of the proposed model is discussed in Section 4. Finally, the conclusions are presented in Section 5.

Problem Description
is section will discuss the outlines of the developed optimal bidding strategy of the wind-thermal GENCO. First of all, the uncertainty modeling regarding the negative correlation between wind power and demand is presented. en, the revenue function of the wind power producer in the Iberian Peninsula electricity market, including dayahead and balancing market, is described.

Uncertainty Modeling.
e wind-thermal system's proposed coordinated bidding strategy model is exposed to the uncertainties of the wind output power and the electricity demand. As stated earlier, the uncertain parameters are highly correlated. For instance, during a day, when the wind speed increases (and therefore the wind output increases), some customers turn off their cooling systems, resulting in the electricity demand decrease. According to the historical data of Denmark wholesale electricity market [30] the wind speed and electricity demand in summer seasons are negatively correlated with the average correlation factor of −0.56. Note that the wind output power can be calculated using the nonlinear relation with the wind speed as (1). In this equation, V ci, V co, and V r are respectively cutin, cut-out, and rated wind speed and parameters A, B, and C are calculated as follows [31][32][33].
e copula technique is employed to properly consider the mutual correlation between the output power of the wind turbine and electricity demand [34]. e proposed approach determines the joint multivariate distribution according to the marginal cumulative distribution functions (CDF) [35]. Based on this technique, the joint multivariate distribution of random variables (z 1 , z 2 , . . . , z n ) can be presented as follows: (5) where F i (z i ) is the marginal CDF of the random variable z i and C(·) represents the copula function.
us, the joint multivariate distribution is calculated by means of the corresponding marginal CDFs and copula function. More detailed information on the technique can be found in [35]. Copula functions are mainly grouped into two categories: Elliptical copula and Archimedean copula [36]. Elliptical copula consists of Gaussian copula and t-copula, which can be used for symmetrical probability distribution functions (PDF). Archimedean copula includes Gumbel, Frank, and Clayton copulas suitable for asymmetrical PDFs and different tail correlations. e detailed information about these functions can be found in [37].
We utilized a scenario generation and reduction procedure to model the above-mentioned uncertainties. e International Transactions on Electrical Energy Systems procedure employs the copula technique to model the correlation between prevailing uncertainties. Finally, the presented approach determines a set of scenarios for wind power and electricity demand fed into the optimization problem. e following procedure determines the final resulting scenarios.

Revenue Function of Wind Power
Producer. e revenue of wind power producers in the electricity market consists of two terms: (i) revenue from selling energy in the day-ahead electricity market (R DA ) and (ii) revenue/penalty from participating in the real-time balancing electricity market (R RT ). us, the revenue of the wind farm is calculated as follows.
e revenue from selling energy in the day-ahead electricity market can be calculated by multiplying the market-clearing price with the scheduled power (7). In the following equation, λ DA t and P sch.
t are the day-ahead marketclearing price and the scheduled power in the day-ahead electricity market, respectively.
Moreover, due to the uncertainty of wind speed, GENCO's actual real-time wind power deviates from the scheduled power. us, GENCO must participate in the realtime balancing market to compensate for deviation. e revenue/penalty of GENCO for participating in the real-time balancing market can be calculated as follows: (i) When the actual wind power production is higher than the scheduled power and if there is an excess generation (system imbalance is positive), the wind turbine must sell the excess wind power production with a price less than or equal to day-ahead marketclearing price. Furthermore, if there is a deficit in generation (system imbalance is negative), the ISO tends to compensate for generation deficit with excess wind power production. e real-time balancing market price will equate the day-ahead market electricity price in this condition. us, the revenue of the wind power producer when the wind power generation is higher than the scheduled power is equal to where ⌊·⌋ represents the minimum operator, P RT t is the actual real-time wind power production, λ RT+ t is the real-time balancing market price in the state of excess power production of the wind turbine, and λ DN t is the price that ISO purchases from wind turbine excess generation.
(ii) When the actual wind power production is lower than the scheduled power and if the system imbalance is positive, the ISO will sell the excess generation in the power system to the wind turbine owner with the price equal to the day-ahead marketclearing price. Moreover, if the system imbalance is negative, the wind turbine compensates for the power deficit with a price equal to or greater than the day-ahead market-clearing price. us, since the wind turbine will purchase the electricity in the realtime balancing market, the wind turbine revenue is negative, calculated as follows: where ⌊·⌋ represents the maximum operator, λ RT− t is the real-time balancing market price in the state of shortage wind power production, and λ UP t is the price that ISO receives to compensate for the shortcoming wind turbine generation [39].

Problem Formulation
Generally, the power generation units such as wind turbines can participate in the electricity market as price-taker and/or price-maker participants. In the former, the wind turbine submits zero price to the ISO, and consequently, the power production of the corresponding wind turbine will be purchased with the higher priority. In this condition, wind power production is modeled as the negative demand in the optimization problem. Although wind turbine modeling in the optimization problem is plain, it is not practical in a highly wind turbine penetrated system. In this study, the GENCO with wind turbine participates as the market maker participant in the wholesale electricity market. In this regard, we deal with a bilevel optimization problem in which the first level determines the bid price-quantity of the GENCO by maximizing the profit. e market-clearing problem is solved in the second level to determine the market-clearing price and the generation units' scheduling. Figure 1 illustrates the structure of the proposed bilevel optimization problem. e mathematical formulation of the proposed stochastic bilevel bidding strategy model of the wind-thermal GENCO is presented as follows: Y j,s,t � LMP j,s,t P j,t − c j P conv. j,s,t + LMP + j,s,t P j,s,t − P j,t , P j,t ≥ P j,s,t , LMP j,s,t P j,t − c j P conv.
j,s,t + LMP − j,s,t P j,s,t − P j,t , P j,t < P j,s,t , f min e objective function of the upper-level problem (10) is to maximize the expected profit of the wind-thermal GENCO. e profit is calculated as (11)- (13). In (10) Y j,s,t , Y j , and π s are the profit of GENCO j in scenario s at time step t, the expected profit of GENCO j, and the probability of occurrence of scenario s, respectively. In (11), P j,t, P j,s,t and P conv.
j,s,t are the scheduled power of GENCO j at time step t, the available power of GENCO j in scenario s at time step t, and the output power of thermal unit of GENCO j in scenario s at time step t. Besides, LMP j,s,t and c j represent the locational marginal price (LMP) of the corresponding bus in scenario s at time step t and the operation cost of the thermal unit.
Moreover, the objective function of the lower-level problem (14), i.e., market-clearing problem, is to minimize the power purchasing cost of ISO from generation units. Herein, f i,t and f j,t are respectively the bid price of GENCO i and j at time step t. Constraint (15) shows the range of bid price of GENCO j and the bid price of other GENCOs, which equates to the marginal cost of generation units. Constraint (16) shows the limitation in the power production of all GEN-COs. Equality constraints (17) and (18)  Finally, (21) and (22) impose the limitation of power passing through line l and voltage magnitude of bus b. e mathematical formulation of the lower-level problem (market-clearing problem) excluding constraints (19)-(21) forms a linear optimization problem. To enhance the accuracy of the solution and burden of the computational complexity, the nonlinear equations of the problem are linearized using the piecewise linear approximation technique [40]. According to the linearization technique, (19) and (20) are replaced by (23) and (24).
Considering the voltage angle range, |θ o(l),s,t − θ r(l),s,t | < 40°, ω l,s,t cos(θ o(l),s,t − θ r(l),s,t ) represents the piecewise linear approximation; that is where the values of d l,s,t,k and h l,s,t,k are selected so as to coincide with cos(θ o(l),s,t − θ r(l),s,t ) at the breakpoints [41]. It is worth mentioning that the exactness of the above linearization technique is justified in the context of power flow studies [42]. Moreover, equation (21) describes a circle, with the radius of S max l that is linearized by means of a set of segmented lines as [43]: k � 1, 2, . . . , n k ∀l ∈ L, t ∈ T, s ∈ S. (26) In the above equation, n k is the number of segments, and α p k , α q k , and α s are linearization coefficients, calculated as follows: After linearizing the lower-level problem, the described stochastic bilevel optimization problem is solved using the GA. Figure 2 shows the flowchart of determining the optimal bid price-quantity of the wind-thermal GENCO. As shown, the GA is executed after generating the scenarios of uncertain parameters. e algorithm generates the first population, and then the objective function is evaluated. e iterative procedure would continue until convergence is achieved.

Simulation Results
is section presents numerical results of the proposed model on the IEEE RTS 24-bus test system, as shown in Figure 3. e test system includes (i) 32 generation units with a capacity of 12-400 MW, (ii) 43 branches (38 transmission lines and 8 power transformers), and (iii) 17 load points. Besides, a wind farm with an installed capacity of 500 MW is considered in bus 14. e data associated with the test system can be found in [44]. In order to model the uncertainties of wind power and electricity demand, the historical data of Denmark's electricity grid are utilized [30]. Table 1 shows the hourly correlation factor between uncertainties associated with wind power and electricity demand, calculated using Denmark's historical data during 2019. As stated earlier, we need a tractable number of scenarios to model the prevailing uncertainties. First, ten thousand scenarios are produced regarding the calculated correlation factor. en, the produced scenarios are cut to a proper quantity to make the problem tractable (10 scenarios). e expected wind power production and electricity demand are shown in Figure 4. It is worth mentioning that the negative correlation between the power generation of the wind farm and the electricity demand can be perceived. e methodology proposed in the study has been implemented in the MATLAB software on a PC with an Intel Core i7 processor (2.2 GHz) and 8 GB RAM. As explained in the above section, although the lower-level problem is linear, the upper-level problem has nonlinearity. us, to solve the presented nonlinear bilevel optimization problem, we utilized the GA method with proper input data. Although some guidelines for choosing population size and iteration number can be found in [45,46], the literature does not offer a comprehensive method for this purpose. is paper utilizes the empirical values (population size of 200 and iteration number of 150). In the simulation study, to explore the impact of the thermal unit on the revenue of GENCO with wind production, two cases are considered: (i) a GENCO with 500 MW wind farm without the participation of thermal unit and (ii) a GENCO with 500 MW wind farm with the participation of an 80 MW thermal unit.
As the first step of the simulation study, the running time of the presented optimization problem is calculated. e average running time of the optimization problem considering the linearized AC OPF equations is 232 seconds, while the running time of the nonlinear optimization problem is 712 seconds.
us, the presented linearization procedure efficiently mitigates the problem's computational complexity. In the following simulation study, the effectiveness of the proposed coordinated bidding strategy in increasing the profit of GENCO and decreasing the risk of uncertainties is investigated. In this    regard, the optimal bid price and the expected LMP of both cases are illustrated in Figures 5 and 6. As can be seen, GENCO submits bid prices with higher value in the peak-demand hours 9-20 compared to off-peak hours. Besides, the results demonstrate that in case (ii), the GENCO submits the higher value of bid price to ISO with the thermal unit's participation, consequently increasing the LMP of bus 14. e reason is stemmed from the supporting role of the thermal unit in the coordinated bidding strategy of the wind-thermal GENCO.
Moreover, the actual wind power production of the cases is depicted in Figure 7. e results illustrate that the wind farm sells more power (accordingly lower wind curtailment) in case (ii) compared to case (i). In other words, the corresponding wind farm in cases (i) and (ii) produce 4207 MWh and 4547 MWh with the revenue of $4980 and $13240, respectively. Furthermore, Figure 8 shows the power generation of the thermal unit in the dayahead electricity market. Herein, the thermal unit generates higher active power in the peak-demand hours due to the higher value of LMP. Moreover, the thermal unit increases the active power generation to provide the extra demand and play as a supporting role in the bidding strategy of the wind-thermal GENCO. erefore, the    International Transactions on Electrical Energy Systems profit of GENCO with and without the participation of thermal units is $14561 and $4980, respectively. To conclude, the proposed coordinated bidding strategy triples the profit of the wind-thermal GENCO compared to wind-based GENCO.
e hourly profit of GENCO in both cases is illustrated in Figure 9. As can be seen, the participation of the thermal unit increases the profit of GENCO during all hours of the day. Additionally, the outcome signifies that the profit of GENCO during early morning hours, in which the demand and LMP are low, would be negative as a result of penalizing GENCO in the real-time balancing electricity market. On the contrary, the hourly profit of GENCO would be positive in peak hours since the LMP is high.
Last of all, the influence of mutual correlation between prevailing uncertain parameters on the profit of the windthermal GENCO is examined. In this regard, a sensitivity analysis is performed on GENCO's profit with/without regarding the correlation. Figure 10 illustrates the hourly profit of the wind-thermal GENCO. As shown in the figure, the appropriate modeling of correlation between uncertain parameters would increase the profit of the wind-thermal GENCO. Furthermore, the total daily profit of GENCO in four cases, i.e., with and without the participation of thermal unit and with and without consideration of the correlation between uncertainties, is tabulated in Table 2.
e table shows that considering the correlation of uncertainties increases the profit of wind-thermal GENCO and wind GENCO by 35.2% and 19.6%, respectively.

Conclusion
is paper proposed a stochastic bilevel coordinated bidding strategy of a wind-thermal GENCO considering the copulabased correlation of wind power production and electricity demand uncertainties. At the upper level, the GENCO submitted the hourly bid price-quantity to the ISO, intending to maximize the profit. e electricity market was cleared at the lower-level based on the market participants' bids and offers, considering linearized AC power flow equations. Finally, the proposed model was applied to the IEEE RTS 24-bus, and the numerical results illustrated the effectiveness of the proposed coordinated bidding strategy. According to the analysis on the computational efficiency, linearizing the AC power flow equations decreased the running time of the optimization problem by 67.4%, while the differences between the voltage magnitude in all buses and profit of wind-thermal GENCO in linearized and nonlinear AC power flow approaches were negligible. Moreover, the simulation results showed that since the participation of the thermal unit with wind farm has decreased the risk of uncertainties, the wind-thermal GENCO submitted a greater bid price, generated higher wind power, and consequently received higher LMP.
us, the thermal unit in the supporting role for wind farm would increase the profit of wind-thermal GENCO in the day-ahead electricity market (almost triples). To conclude, the numerical results illustrated that modeling the mutual correlation between uncertainties associated with wind power generation and demand using the copula technique would increase the profit of the wind-thermal GENCO by 35.2%.   Variables Y j,s,t : Profit of wind-thermal GENCO j at time step t in scenario s ($) LMP j,s,t : Locational marginal price of the bus of GENCO j at time step t in scenario s ($/MWh) LMP + j,s,t : Real-time balancing market price in the state of excess wind power production of GENCO j ($/MWh) LMP − j,s,t : Real-time balancing market price in the state of shortage wind power production of GENCO j ($/MWh) P conv.

Sets and Indices
j,s,t : Active power production of thermal unit of GENCO j at time step t in scenario s (MW) P j,t : Active power production of wind-thermal GENCO j at time step t (MW) P i,t : Active power production of thermal GENCO i at time step t (MW)

Data Availability
Data are available upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.