Optimal Bidding Strategy for Profit Maximization of Generation Companies under Step-Wise Bidding Protocol

Deregulation of electricity industry has completely transformed the present power system. The generation companies (Gencos) are now interested in maximizing their profit by optimally strategizing their bids. In an oligopolistic market environment, only a few independent power suppliers can increase their own profit through strategic bidding. The profit depends to a great extent on imperfect knowledge of rival companies. In this work, a new approach is developed for power suppliers to bid optimally for enhancement in their profits. The probability density function is used to determine the rival’s behavior. An optimization routine has been established and Grey Wolf Optimizer (GWO) is used to find the optimal solution to the bidding problem. The approach is validated on four generators test system. The results are also compared with the application of Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Biogeography Based Optimization (BBO) algorithms. Comparison of the results shows that GWO is an effective solution provider.

of suppliers by solving a set of differential equations that specify the necessary conditions for bidders to maximize their expected payoffs. Ramping limits were ignored in this work. The explicit analysis of gaming and price spikes in the electricity market was presented in [14]. The author formulated a prisoner's dilemma matrix game and introduced the notion of opportunistic tacit collusion to explain strategic bidding behavior in which suppliers withhold generation capacity from the market to drive up prices. Wang Xian et al. [15] presented a mixed nonlinear complementarity problem by combining the Karush-Kuhn-Tucker conditions of all strategic generating companies. The results of this work showed that the generating companies can exercise their market power by over production under congestion. Binary expansion approach to solve optimal bidding problem was presented in [16]. The author solved the problem of strategic bidding under uncertainty in short-term electricity markets. In [17], author proposed a methodology to determine the optimal bidding strategy of a retailer in the short term electricity markets. GA was used to optimize the parameters defining the best purchasing strategy. Mostafa Kazemi et al. [18] presented a combined scheduling and bidding algorithm for constructing the bidding curve of an electric utility that participated in the day-ahead energy markets. The uncertainty involved regarding electricity markets was modeled using information-gap decision theory. S. Jalal et al. [19] proposed a mathematical model for large consumers to alter pool prices to gain more benefit in deriving bidding strategies. Uncertainty was modeled by a stochastic complimentarity model. Modeling of Risk management was not done in this work. In [20], a risk constrained bidding model for generation companies competing in a poolbased electricity market was presented. The dynamic programming algorithm was used to formulate strategic bidding problem. The bidding model took into account the uncertainty in system demand. The risk management was also integrated into the bidding model. Fco. Alberto Campos et al. [21] proposed a novel methodology to obtain real bidding curve for the spinning reserve market. The reserve cost curve was constructed by considering the day ahead market opportunity cost. This methodology was used in one of the prime power producers of Spain. BBO approach was used to maximize profit of generation companies in [22]. In [22], a generation company maximized its own profit while also maximizing profit of rival companies. The method predicted bid prices of rivals stochastically and then generated bid prices of the generation company whose profit was maximized.
Given this background, the work presented in this paper aims at maximizing the profit of a generating company while taking social welfare into account. The GWO [23] is used to secure the purpose to obtain bidding strategy for maximizing the profit of Genco. GWO is a new Meta heuristic algorithm inspired by grey wolves. It mimics the leadership hierarchy and hunting mechanism of grey wolves in nature. The algorithm provides very competitive results compared to other well known Meta heuristics algorithms. A comparison is also has been done between GWO and other heuristic algorithms (GA, PSO, BBO) in this paper. The remaining part of the paper is presented as follows; in section II and III, problem description and proposed solution algorithm along is presented. In section IV, simulation results are presented and finally, in section V, the conclusion and future scope of the work are presented.
II. PROBLEM DESCRIPTION Consider a system consists of independent N+1 generators. To maximize the profit, an optimal bidding strategy has to be developed for generator A, with N rivals in the market. Suppose that each generator has only one registered unit. In the day-ahead market, each generator bids for every one hour trading period under step wise block bidding protocol, uniform market clearing price (MCP) and sealed auction system, using the pay-as bid mechanism.
Each Generator can bid a maximum of I blocks of output for each trading period. Bidding in multiple blocks reduces the risk of financial loss. Probability Distribution Function (PDF) is used to predict the bidding prices of rivals, which is based on statistical analysis of historical bidding data. The distribution of the rivals' block bidding prices can be given by normal PDF as: (1) Where n i C is the i th block price of n th rival. n i  , n i  are the mean values and standard deviation, respectively, of the n th rival of i th block.
In this work, bidding strategies are to be developed for one period (one hour) auction only and hence Inter-temporal operating constraints for a generator, such as minimum up and down times and the maximum number of start-up and shutdowns allowed, have not been taken into account.
The profit of j th Generator at the considered hour is given by: Here j  and TC j are the profit and total production cost for j th Generator, respectively. i is the number of bid blocks for j th Generator. C ji and P ji are the price and quantity offered at block i for j th Generator, respectively and the product of these two gives market revenue. Total production cost can be calculated at a considered hour from (3) as Here A j, B j and C j are the generator cost coefficients. P j is the total quantity produced by j th Generator at a considered time, where Each Generator is presumed to be aware of its rivals' attempt to maximize their profits. In the proposed bidding strategy formulation approach, a Generator maximizes its own profit along with rivals' profits. Thus, this profit maximization problem is multi-objective. Instead of different maximization objectives, a single objective function is formulated from a combination of objectives to be optimized by associating appropriate weights to represent their significance.
Here 0  is the weight associated with the sum of profits and j  is the weight representing the importance of individual Generator's profit. Subject to constraints (i) Operating constraints: Regulatory Constraint:  Limitation on bid price  ji C Price cap (8)  Limitation on block bid quantity max min ji ji ji P P P   (9) The optimization problem defined in (5)-(9) can be solved for the optimal block bid price of the i th block of the j th Generator represented by ji C .
Using the PDF described in (1), rivals' bidding price can be determined using historical bidding data. Formulating the optimal bidding strategy for Generator A, with objective function (5) and constraints (6)-(9), becomes a stochastic optimization problem, to be solved by GWO algorithm.
III. PROPOSED SOLUTION ALGORITHM The optimization problem formulated in the previous section has been solved by using Monte Carlo (MC) simulation, a technique which obtains a probabilistic approximation of a mathematical problem by using statistical sampling. It performs stochastic simulation using random numbers, and repeatedly calculates the equation to arrive at a solution. These MC simulations are incorporated with GWO, a modern heuristic algorithm based on the behavior of Grey Wolf, to obtain the optimal bidding strategy for a Genco.

A. MONTE CARLO APPROACH
 Generate a large number of random samples of block bid prices for all the rivals, considering their PDFs.  Obtain large trial outcomes, by solving the optimization problem with sample values of block bid prices for all the rivals.  Calculate the expectation value, taking average of all the trial outcomes.
 The detailed algorithm for solving the optimal bidding problem for Genco A, with N rivals, is given below.
1. Specify the maximum number of MC simulations, M . 2. Initialize the simulation counter, m=0 .
3. Generate random sample values of bid prices of rivals n i C (n = 1,2,….,N and i = 1,2,….,I) based on their PDF, as given in (1) under price cap constraints. 4. Obtain mean and standard deviation for each block of Genco A, from bid prices of rivals. It is assumed that Genco A decides its bid prices considering rivals' bidding behavior. 5. Generate bid prices ji C (with price cap constraints) using normal PDF for each block of Genco A.
6. Use GWO to search the optimal dispatch quantity ( ji P ) for each block of all the Gencos that maximizes (5) and store those corresponding prices and quantities as ji C and ji P , respectively.
9. Calculate the expected value of the optimal bid price, i.e. the average of m ji C and optimal bid quantity, i.e. the average of m ji P . Maximized profit for each Genco is the profit, corresponding to these average price and quantity of each block.

B. GREY WOLF OPTIMIZATION
A new population based GWO algorithm is discussed in this section. The algorithm is based on the leadership hierarchy of grey wolves in nature. Grey wolves follow the leadership hierarchy, as shown in Fig. 1 according to which each wolf is assigned a particular task to perform. Alpha wolf is at the top of hierarchy and instructions passed from alpha are followed by other wolves according to their position on hierarchy. Beta wolves come second in the hierarchy after alpha, follows the orders given by alpha and they help alpha in searching and hunting the prey. Beta wolves dominate delta and omega wolves. They work as an advisor to alpha and become the leader when alpha dies. Delta wolves assist alpha and beta wolves in hunting and maintaining the discipline among omega wolves. Omega wolves come at last in hierarchy and they aren't much help to other wolves as they do not participate in hunting procedure necessarily. They are allowed to eat after other wolves. The group size is 5-12 on average. The important features of hunting mechanism of grey wolves are as follows: 1. Tracking, chasing, and approaching the prey 2. Pursuing, encircling, and harassing the prey until it stops moving 3. Attack towards the prey α β δ ω The mathematical modeling of behavior of grey wolves can be done by considering the fittest solution as the alpha ( ), second best as the beta (  ), third best as the delta (  ) and rest of the candidate solutions are assumed to be omega (  ). Only   , and  participate in hunting. The  wolves only follow these three wolves.
Encircling the prey is a significant task in hunting procedure, the following equations represents the encircling behavior of grey wolves: Where t indicates the current iteration, A Grey wolves have the ability to recognize the location of prey and encircle them. The hunt is usually guided by the alpha. The beta and delta might also participate in hunting occasionally. However, in an abstract search space there is no idea about the location of the optimum (prey). In order to mathematically simulate the hunting behavior of grey wolves, suppose that the alpha (best candidate solution) beta, and delta have better knowledge about the potential location of prey. Therefore, the first three best solutions obtained so far are saved and oblige the other search agents (including the omegas) to update their positions according to the position of the best search agent. The following formulas are proposed in this regard.
Recently GWO has been applied in many fields namely AGC controller design [24], robust generation control strategy [25], computational simulations [26], combined economic load dispatch problem [27], surface wave estimation [28] and training of multi-layer perceptrons [29]. The application of GWO to formulate strategic bidding is explored in this paper. Following section presents simulation results on 4 generators test system.

IV. RESULTS
This section presents the simulation results of proposed Monte Carlo and GWO approach. A computer program has been developed using MATLAB 2013 and run on a Pentium IV CPU, 2.69 GHz, and 1.84 GB RAM computer. The objective of the optimization routine is to maximize the profit of a specific Genco. A 4 generator system has been considered for validating this simulation approach. The data for the generating units have been taken from [22]. The optimization routine has been addressed by two different load levels. The application of Monte Carlo approach is for forecasting the bid price for participating Gencos. Different bid price predicted for all four Gencos by Monte Carlo simulations are given in Fig. 2.  Table 3 shows dispatch calculated by the optimization routines of GA, PSO, BBO and GWO. Table 4 shows the profit calculated by corresponding Gencos. It is observed that the profit earned by the Genco 1 is higher as compared to other Gencos. This result shows that the objective of optimization routine has been achieved. The comparative analysis of the profit calculated by different algorithms shows that GWO outperforms over rest of the algorithms. V. OPTIMIZATION PERFORMANCE To compare the optimization performance of these algorithms, population sizes, no of iterations are kept same for all. In this work these parameters are chosen as 100 and 1000.Stopping criterions of optimization process are chosen on the basis of execution of optimization process till maximum no. of iterations or the values of objective functions in successive run should have the difference 1e-3. Keeping these parameters, optimization process is executed with all four algorithms. It is observed from Fig. 3 that the optimization process handled by GA got trapped in local minima and the value of the calculated profit for Genco 1 is 9843.609 $. This profit is 35% less than the profit calculated by GWO. Moreover the execution time of the optimization routine also puts a question mark on this approach. It is also observed that the convergence point of BBO and GWO are quite close but the values obtained from the optimization process of GWO is more. Fig. 4 shows the convergence characteristics of the values of profit obtained from the optimization processes by GA, PSO and BBO. The profits calculated by these algorithms are less than by 32.16%, 30.23% and 0.5% respectively as compared with GWO. It is observed that GA and PSO algorithm stopped at local minima. However, a marginal difference has been observed in the values of objective functions in GWO and BBO.

VI. CONCLUSION
In this work, a method based on GWO algorithm is proposed to solve the optimal bidding strategy in an imperfect environment of electricity market under step wise bidding protocol and uniform market clearing price (MCP) environment. A problem which consists of four Gencos is used to verify the feasibility of the proposed method. Rivals behavior is predicted by historical data and normal probability density function and the same is applied to simulate it. The results are found for two different load levels, 1500 MW and 1600 MW. The proposed method gives promising results in comparison to other heuristic algorithms (GA, PSO and BBO). In future, an attempt will be incorporated to include inter-temporal operating constraints of generators.