Developing Robust Bidding Strategy for Virtual Bidders in Day-Ahead Electricity Markets

Purely ﬁnancial players without any physical assets can participate in day-ahead electricity markets as virtual bidders. They can arbitrage the price difference between day-ahead (DA) and real-time (RT) markets to maximize proﬁts. Virtual bidders encounter various monetary risks and uncertainties in their decision-making due to the high volatility of the price difference. Therefore, this paper proposes a max-min two-level optimization model to derive the optimal bidding strategy of virtual bidders. In this model, the risks of uncertainties associated with the rivals’ strategies and RT market prices are managed by robust optimization. The proposed max-min two-level model is turned into a single-level mixed integer linear programming model through duality theory (DT), strong duality theory (SDT), and Karush-Kuhn-Tucker (KKT) conditions. An illustrative case is designed to demonstrate the advantages of the proposed model over the deterministic model. Moreover, case studies on the IEEE 24-bus test system validate the applicability of the proposed model.

Line l capacity at time t and negative way. ϑ tl Line l capacity at time t and positive way. ρ, η, θ, χ Lagrangian coefficients of the lower-level optimization constraints.

I. INTRODUCTION
V IRTUAL traders, or virtual bidders, are purely financial participants in the electricity market, who can submit their bids/offers into the day-ahead (DA) market without the compulsion to consume/produce the actual power in the realtime (RT) market. In recent years, these transactions which are designed as decrement bids (DECs) or increment offers (INCs), have been considered as part of the electricity market design [1].
Virtual bids contributed around 6% of all transactions in the Midwest Independent Transmission System Operator (MISO) in 2010 and 2011 [2], and are generally employed to decrease the gap between the DA and RT markets' prices and increase the liquidity of the markets. The values of various flexible resources, including virtual bids, were evaluated in [3] through four different two-settlement market-clearing models, and it is confirmed that the errors of deterministic DA scheduling can be amended by virtual bids. The advantages and disadvantages of virtual bids in electricity market environments have been summarized in [4], which stated that besides the benefits of virtual bids, they may increase the risks for market manipulation in some parts of the system.
A model reported in [5] presents the motivations for market participants (MPs) to place virtual bids at the specific buses which are tied to the financial transmission right (FTR) position owned by the MPs. It is shown that deliberate nonoptimal virtual bids to manipulate market prices increase the price difference between DA and RT markets, and thus, create market inefficiencies. To analyze the MP's crossproduct manipulation in three sequential markets, a threestage stochastic game theoretic model has been presented in [6], which applied the numerical simulation to evaluate the influence of this manipulation on price convergence in a two-bus system. An algorithmic trading strategy for virtual bids in power markets using a data-driven approach has been presented in [7]. To optimize the submitting virtual bids, a mixture density network (MDN) model was utilized to predict the price differences between the DA and RT markets. Authors in [8] employed virtual bidding to accommodate the PV solar power producers to optimize their bidding strategies in the electricity market environment, and it is declared that virtual bidding may provide additional risk for risk-taker PV producers. Although multiple works have been done on virtual bids to present the strengths and weaknesses of utilizing these transactions, very limited works has been carried out to study the bidding strategy of this newer type of market participants [9].
In the restructured electricity system, strategic bidding helps MPs to improve their behaviors and maximize their payoffs. The existing approaches for bidding strategy design differ for the price-taker and price-maker MPs. Price-taker MP, whose action does not alter the market outcomes, models the market with the DA market price prediction to optimize its bidding behavior [10], [11]. However, price-maker MP requires a more complicated methodology to design its strategy since its actions impact the market outcomes, thus, it needs to model the rival MPs as well as the market clearing process. [12] applied a MILP model to address the offering strategy problem for a price-maker generator who participates in a DA electricity market in which two distinct methods, based on the nodal and shift factor formulation, were used to model the transmission system. The binary expansion method is employed in [13] to solve the bi-level bidding strategy problem with stepwise offers for the price-maker generator.
In almost all methods existent for the bidding strategy design, there are various uncertainty sources, such as market prices, demand, rivals' strategies, and renewable energy generations, that impact the bidding strategy. The uncertain behavior of rival generators and consumers were modeled probabilistically in [14] and, to find the optimal offering strategy of the price-maker MP, Monte Carlo simulation was employed to compute the expected profit. To derive the optimal bidding strategy for a strategic generation company, a stochastic bi-level optimization problem has been modeled in [15], which modeled the uncertainties of consumers' bids and rival generators' offers through multiple scenarios. In [16], a self-scheduling model is employed to design the bidding strategy of price-maker energy storage and evaluate the potential arbitrage benefits of these resources in the Alberta electricity market using the historical hourly demand and generation price quota curves (DPGCs and GPDCs). To capture the optimal offering decisions of a strategic wind power producer in the DA and balancing markets, a two-stage stochastic model is presented in [17], in which scenario-based modeling is applied to model the uncertainties associated with the wind productions, other players' behaviors, and market price. Optimal bid prices and quantities of a generating company are derived in [18] using a self-organizing hierarchical particle swarm optimization, in which a risk index based on mean-standard deviation ratio (MSR) is optimized, and Monte Carlo simulation is applied to mimic the other MPs' behaviors in the electricity market.
Among all current methods for addressing the uncertainties, robust optimization (RO), which is independent of the probability distribution function (PDF) of the parameters and assumes uncertain intervals around the predicted parameters, has become an appropriate choice for studies with high level of uncertainties and insufficient data for an accurate forecast of PDFs. Recently, the RO method has been widely used for the bidding strategy design problems. A two-stage RO method is utilized in [19] to design the offering strategy of a price-taker virtual power plant (VPP) consisting of a wind power producer, energy storage, and a number of demands taking part in the DA and RT markets. To obtain the optimal strategies of wind power producer and wind-storage aggregator, who are assumed to act as price-makers in the DA market and as deviators in the balancing market, [20,21] presented a multi-stage distributionally RO (DRO) model and its value was confirmed by the different case studies performed on the modified Swiss system and Nordpool. To address the uncertainties associated with wind power generation and loads, and to obtain the optimal behavior of a virtual power plant, a stochastic adaptive RO method has been presented in [22]. The optimal bidding strategy of a hybrid power plant, which acts as a price-maker in the DA market and a price-taker in the balancing market, has been derived in [23]. It addressed the uncertainty of the price quota curve (PQC) with the RO. [24] utilized the bi-level RO model to optimally design a plug-in electric vehicle (PEV) charging station, in which the lower-level problem modeled the strategic behavior of PEV owners. The optimal bidding curve of a price-maker energy storage facility in the DA market was captured in [25], which employed the robustly modeled generation and demand price quota curves to consider the effect of the energy storage power on market prices. The optimal behavior of a pricemaker microgrid aggregator (MGA) using a RO model to address the uncertainties related to renewable generation was presented in [26] and showed that the presented model can improve the MGA profits.
In this paper, we proposed the max-min two-level optimization model for the purely financial player who plays either generation or load in the DA market. The virtual bidder's payoff is maximized in the upper-level subproblem, and the market-clearing procedure is modeled in a lower-level subproblem. In order to avoid the difficulties facing stochastic optimization, such as the expensive computation and the dependency on accurate predictions of the PDFs of uncertain parameters [27], the RO approach is employed in this paper to model the uncertainties of other MPs' offers/bids and RT market prices. Also, the RO approach can guarantee feasibility in all scenarios, while stochastic optimization cannot. The proposed model is turned into its equivalent linear singlelevel problem employing the KKT conditions, duality theory, and strong duality theory (SDT). Thus, the main contributions of this paper are twofold: 1) It is the first work to employ robust optimization to develop a day-ahead market bidding strategy for purely financial virtual bidders. The proposed model allows the virtual bidder to maximize its profit by considering its flexibility of being either generation or load at different locations of the system. Robust optimization is used to handle the uncertainties associated with other players' offer/bids (quantity and price) and RT market LMPs.
2) Based on the KKT conditions, SDT and big-M method, the proposed max-min two-level model is equivalently linearized and transformed into a mixed-integer linear program (MILP), making it solvable by accessible commercial solvers.
The rest of this paper is organized as follows: Section II introduces virtual bidding and its impact on DA market prices. Section III presents the proposed robust two-level and its corresponding MPEC models. Section IV provides an illustrative example. Case study results are discussed in Section V, and Section VI presents the conclusion.

A. VIRTUAL BIDDING
Virtual bidders, also known as virtual arbitragers, may have no physical assets and can participate in energy transactions in the DA market. If a virtual bidder is cleared by the ISO to buy (or sell) energy in the DA market for certain time periods, in the ISO two-stage settlement it will be automatically considered to sell (or buy) the same amount of energy in the RT market for the same time periods. As discussed in [9], virtual bidders can improve the market's ability to manage the forecast errors by increasing the liquidity of the market.
Assuming that a virtual bidder predicts the RT price to be higher than the predicted DA price, there will be an opportunity for the virtual bidder to arbitrage between the DA and RT markets by buying a certain amount of energy in the DA market at a DA market price and selling the same amount of energy in the RT market at a RT market price. As a result of this virtual bidder participation, the DA market price may increase due to the increased load cleared in the DA market. Consequently, the difference between DA and RT prices may become smaller, as illustrated in Fig. 1. Therefore, virtual bidder participation in energy markets may reduce the price gap between DA and RT markets, which is considered an improvement in market convergence.

B. MODEL STRUCTURE
The upper-level subproblem of the proposed two-level approach represents the profit maximization of a virtual bidder, whose decisions (virtual bid quantity and price (V bid ti , α bid ti )) are then passed to the lower-level subproblem. The lower-level subproblem represents the quasi market where energy and market prices are cleared on an hourly granularity on a daily basis. The market results (i.e., cleared virtual quantity and market price (V DA ti , λ DA tn )) are fed back to the upper-level subproblem, which provides a closed loop response of the virtual bidder decision on market price (Fig. 2).
To optimize its decision, the virtual bidder needs to consider various parameters, including the quantities and prices of other generators'/loads' offers/bids, as well as RT market prices. All these parameters play a critical role in the virtual bidder's ultimate payoff. For instance, the DA market price, which is utilized by virtual bidder to estimate its DA profit, may alter as a result of various rivals' offers/bids. Furthermore, the RT price assists the virtual bidder in evaluating the DA profit vs. the RT profit and deciding whether to be virtual generation or virtual demand in the DA market. As these parameters are unknown to the virtual bidder, they need to be forecasted or estimated. However, making a precise prediction is practically impossible. Therefore, to consider the risks of the forecasted uncertainty sources, a robust optimization approach is employed in this paper, which is widely applied by the risk-averse market participants [28]. In this approach, a confidence interval needs to be introduced around an uncertain parameter, then the worst-case scenario of uncertain circumstances within this permissible limit is assessed [25]. Therefore, the proposed max-min two-level optimization model can be formulated as follows. (1))

Maximize the Profit (Model
Subject to: As it is seen in Model (1), the objective function of the virtual bidder is maximized regarding to its main vari- , Ug ti , Ud ti and minimized with respect to the uncertain parameters = { λ RT tn , P G tjb , P D tdk , λ G tjb , λ D tdk }. Constraints (1b) and (1c) set the maximum bounds for the virtual bids (generation/demand). Constraint (1d) guarantees that virtual generation and demand cannot be submitted to the DA market simultaneously. Uncertain parameters are limited in (1f) -(1j) by means of corresponding confidence intervals. The robust parameters ζ RT n , σ G jb , σ D dk , τ G jb , and τ D dk are determined by the virtual bidder and used as known parameters to measure the length of uncertain range around the predicted values. Note that the correlation between uncertain variables can be reflected in the corresponding bounds in this model. Furthermore, the model is flexible to take into account size-varying bounds of uncertain variables for different time periods.
It is worth mentioning that this paper focuses on the most influential uncertainties (such as predicted RT price, rivals' offers/bids) for the bidding strategy problem. Other events with low probabilities, such as unplanned outages of generating units and transmission branches, are ignored in this work.
Model (2) represents the lower-level subproblem which is linear since the ISO takes α bidG ti and α bidD ti and V bidG ti and V bidD ti as parameters. Therefore, it can be substituted by its KKT conditions. Combining these equivalenced constraints in the upper-level subproblem results in a Mathematical Problem with Equilibrium Constraint (MPEC), whose formulation is as follows.

Min Max
Complementarity constraints related to inequality constraints are stated by (3h)-(3q) which are nonlinear equations, which can be linearized using the Fortuny-Amat transformation (Big M method) described in [29], [30]. Thus, each of the equations of 0 ≤ V ti ⊥d ti (x) ≥ 0 can be rewritten as follows.
where M ti is a large number and ω ti is a binary variable. Therefore, the equivalent model will be as Model (4).

B. EQUIVALENT MILP FORMULATION (Model (4))
Objective Function (3a) (4a) Subject to: Constraints (3b)-(3g) (4b) VOLUME 8, 2021 Linearized form of (3h)-(3q) based on Big M method (4c) (4) is the objective function, which is expressed explicitly with regard to uncertainties ( ). To linearize the objective function, at the first step, it needs to be described implicitly based on , which can be done using the SDT [15]. Due to the linearity of the inner problem, SDT can provide an objective function that has a zero duality-gap with the primal objective function value at the optimal point [31]. Doing some mathematical simplification, the objective function (4a) can be implicitly expressed with respect to the uncertain variables as follows (equation (4a)):

Now, the only nonlinear equation in Model
Therefore, Model (4) represents the single level nonlinear max-min problem. In order to remove the nonlinearities in the objective function, duality theorem is used here. Since the internal optimization problem (which is with regard to uncertain set) is linear, the dual form of that can be replaced. This procedure is fully illustrated in [32]. Employing this approach to Model (4) leads us to the following linear maximization form (Model (5)).

IV. ILLUSTRATIVE EXAMPLE
The considered test system is illustrated in Fig. 3. This system includes 5 generators, 3 loads, and 6 transmission lines. It is assumed that the virtual bidder attends to submit its bids from two locations (bus B and bus E) to the DA market.
For the sake of simplicity, it is assumed that the problem is solved for one period, and the corresponding forecasted RT prices ($/MWH) for different buses are shown in Table 1. Forecasted generators'/loads' offers/bids are summarized in Table 2 and Table 3   ters to zero), the virtual bidder's strategy would be 28.14MW generation at the price of $57.57/MWH at bus B, and 200MW generation at the price of $20/MWH at bus E. As it is shown in Table 4, if we test the Deterministic model results in the worst-case scenario, none of these virtual bids are cleared since, in this case, these bid prices are higher than the LMP of the system at the corresponding nodes.
On the contrary, the bidding strategy of the virtual bidder is completely different when he/she applies the proposed model (Model (5)). As this model considers the occurrence of the worst-case scenario, its solution will be optimal in this scenario. The worst-case scenario happens when the L 1 and G 5 are the marginal MPs at buses B and E, respectively. As the robustness parameters are 0.1, the LMPs will be $66/MWH at bus B and $18/MWH at bus E, in the worstcase scenario. Therefore, virtual bidder bids as a demand at bus B at the price of $66/MWH and as a generator at bus E at the price of $18/MWH to be cleared in the DA market in this situation. As a result, the total profit of the virtual bidder is $404 using the proposed robust model at the worst-case scenario, while the profit would be zero when bids obtained from the Deterministic model are used.

V. CASE STUDY A. DATA AND CASES SETUPS
The proposed approach is implemented on the IEEE 24-bus Reliability Test System [33] (Fig. 4). This system includes 24 buses, 32 generators, 17 demands, and 38 transmission lines. A virtual bidder is assumed to bid from 5 different locations (buses #6, #11, #14, #16, #22). Suppose the maximum bid quantity that virtual bidder can bid in the DA market is VOLUME 8, 2021  60MW, which is determined according to the proxy amount owned by the virtual bidder [7]. Forecasted Real-Time LMP at different locations and periods are presented in Fig. 5. Offer quantities and prices of other generation units are represented in Table 5, which are assumed to be the same for all periods. Forecasted loads' bid quantities are depicted in Fig. 6, and their corresponding predicted bid prices is shown in Fig. 7. Note that this predicted bid price profile is considered the same for all loads.
We designed 9 different cases to present the effectiveness of the proposed model. The first case is the Deterministic case in which all robustness parameters are zero. In Cases 2 -6, just one robustness parameter is assumed to be non-zero, and in other Cases (Cases 7 -9), all robustness parameters are non-zero. Cases 7 -9 are designed to present the benefit of the proposed robust model in the highly uncertain situation. Table 6 summarizes the designed cases.

B. RESULTS AND DISCUSSION
We solved the designed cases, explained in the previous section, with the proposed Model (5). Moreover, for comparison purpose, the Deterministic model results were tested at the worst-case situations. As it is shown in Fig. 8, the total profit of virtual bidder is always higher than the profit this MP can obtain from the Deterministic model. This is because the Deterministic model results are not applicable in the worst-case scenario and most of the time, they are not 336 VOLUME 8, 2021   cleared in the DA market, which leads to lower profit. Therefore, a risk-averse virtual bidder would prefer to apply the Robust-based solution in situations with uncertain sources.
All tests were performed on a computer with a 3.2 GHz Intel Core i7 CPU and 32GB of RAM. The models were implemented in AIMMS 4.75.1.0 [34] and solved using  Table 7.
To present the influence of the virtual bids on the DA market prices, DA LMPs at two selected buses where the virtual bidder places its bids (buses 6 and 22), are shown in Fig. 9. Note that these prices are captured in the worst-case scenario. As seen from Fig. 9, the predicted RT LMP at bus 6 is higher than the DA LMP before placing the virtual bids. Using the deterministic model, virtual bids cause a reverse divergence between RT and DA LMPs at multiple hours, which results in a negative profit for the virtual bidder. However, there is a VOLUME 8, 2021 reasonable convergence between RT and DA LMPs when the robust optimization results are applied by the virtual bidder. The same situation applies to LMPs at bus 22, except that the predicted RT price is smaller than the DA LMP before virtual bids.
A sensitivity analysis has been done here to find the most critical uncertain parameter which can highly affect the total profit. Therefore, the Profit Change is calculated using equation (6) for designed Cases 2 -6. In each of the cases, only one of the uncertainty parameters is considered. In equation (6), R d is the profit of Deterministic result in the forecasted scenario, and R t is the profit of the Deterministic/Robust models' results testing in the worst-case scenario.
As it is seen in Fig. 8, the profit obtained from the Deterministic result in the forecasted scenario (Case 1) is $175,635, while the profit of the virtual bidder is $133,974 when applying the Robust-based results in the worst-case scenario (Case 2). Thus, the profit change is 23.72% for this designed case. Fig. 10 compares the profit changes calculated for Cases 2 -6 for both deterministic-based and Robust-based results tested at the worst-case scenarios. It is obvious that the higher the profit change is, the greater the impact of the corresponding parameter on the total profit. Therefore, as shown in Fig. 10, RT LMP has the greatest influence on the total profit of the virtual bidder.
In order to observe the performance of the proposed model, this model has been tested with different levels of uncertainty (Cases 7 -9). As shown in Fig. 11, the difference between the profits attained from the proposed model and deterministic model results will rise as the level of uncertainty increases. Note that, in these tests, the worst-case scenario was utilized to evaluate the results of the robust and deterministic models.  Therefore, the outcome of deterministic model results may change when the uncertainty level changes.
The improvement in profit, which is calculated by equation (7), represents the advantages of applying the proposed model specifically for the risk-averse virtual bidders who consider the higher confidence interval for the uncertain parameters. Fig. 11 shows that the improvement in profit reaches 50% when the virtual bidder chooses 0.3 for the robustness parameters in his/her decision-making process. It clearly demonstrates the benefits of the proposed model. In equation (7), R r is the profit of the robust-based model, and R d is the profit of the Deterministic model results testing at the worst-case scenario.

VI. CONCLUSION
In this paper, a max-min two-level optimization model is presented to optimize the bidding strategy for a risk-averse virtual bidder taking part in the DA market. Duality theorem, KKT optimality conditions, SDT, and the big-M method are employed to translate the two-level problem into a MILP problem. As the lower-level subproblem of the model represents the quasi DA market, a virtual bidder can mimic the market clearing process and can appropriately make bidding decisions to its best interest. Therefore, through the bid price, a virtual bidder can effectively compromise between the amount of cleared virtual bids and the affected price difference between the DA and RT markets considering the uncertainties of other MPs' strategies and RT market LMPs. Numerical results and sensitivity analysis show that RT LMP is the most critical uncertain parameter that the virtual bidder needs to consider in his/her decision-making procedure. Moreover, as compared to using the deterministic model, a risk-averse virtual bidder can always make more profit at the worst-case scenario employing the proposed model, and the improvement in profits increases dramatically as the uncertainty level increases.