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We focus on the application of bi-level programming in electricity markets (power market) in this chapter. Competitive strategic bidding optimization of electric power plants (companies) is becoming one of the key issues in electricity markets. This chapter presents a strategic bidding optimization technique developed by applying the bi-level programming. By analyzing the strategic bidding behavior of power plants, we understand that this bidding problem includes several power plants and only one market operator respectively known as multiple leaders and single follower. The problem can be considered as a bi-level multi-leader optimization problem which is introduced in Chap. 5. We therefore build a bi-level multi-leader (BLML) decision (programming) model for this bidding problem in day-ahead electricity markets. In the BLML decision model, each power plant is allowed to choose its biddings to maximize its individual profit, and the market operator can find its minimum purchase electricity fare that is determined by the output power of each unit and the uniform marginal prices.

In this chapter, we first give the background of this bi-level programming application in Sect. 12.1. Section 12.2 conducts bidding strategy analysis in competitive electricity markets that is used for modeling. Section 12.3 presents a BLML competitive electricity markets model. A real data-based case study is shown in Sect. 12.4 to illustrate and test the bi-level programming model for competitive strategic bidding optimization in electricity markets. Experimental results on a strategic bidding problem for a day-ahead electricity market have demonstrated the validity of the proposed decision model. Section 12.5 summaries this application.

1 Background

Throughout the world, electric power industries are undergoing enormous restructuring from nationalized monopolies to individual organizations in a competitive market (Huang and Pai 2002) with the support of digital eco-systems. Because of the significance of electricity energy to national economies and society (Guerrero et al. 2008), electricity markets must be operated under extensive conditions of absolute security and stabilization. The research on electricity markets has attracted many researchers, owners and managers from electricity entities. The competitive mechanism of day-ahead markets is a very important research issue in electricity market studies, which can be described as follows: each power plant submits a set of hourly (half-hourly) generation prices and the available capacities for the following day. According to this data and an hourly (half-hourly) load forecast, a market operator allocates the generation output for each unit.

As no determinate operation model for electricity markets exists, the marketing procedure of electric power industries varies from country to country. Generally speaking, there are three kinds of running models in electricity markets: the power pool model, wholesale competitive model, and retail competitive model. These models adopt three kinds of electric power trading methods: long term contract, day-ahead market, and facility service. Among them, the day-ahead market is the most competitive and active imposing great influence on profits for each participant in the market. Specifically, each power plant submits a set of generation prices and other related data, based on which the market operator makes a generating plan for the following day. To optimize this procedure, many models and algorithms have been proposed.

This chapter applies the bi-level optimization approach for dealing with strategic bidding optimization in electricity markets. We propose both a strategic bidding model for power plants and a generation output dispatch model for a market operator in a day-ahead electricity market. Since there are several power plants considered as leaders; and there is only one market operator as the follower, this decision problem is a bi-level multi-leader optimization model which was introduced in Chap. 5. Based on these two models, a specific BLML decision model, which includes ramp rate constraints for competitive electricity markets, is proposed. A real data-based case study on competitive strategic bidding problem in an electricity market is then presented.

2 Bidding Strategy Analysis in Competitive Electricity Markets

In an auction-based day-ahead electricity market, each power plant will try to maximize its own profit by strategic bidding. Normally, each power plant submits a set of hourly (half-hourly) generation prices and available capacities for the following day. Based on this data and an hourly-load (half-hourly-load) forecast, a market operator will allocate generation output. In this section, under the analysis of bidding strategy optimization problems, we build a competitive strategic bidding model for power plants and a generation output dispatch model for a market operator in a day-ahead electricity market.

2.1 Strategic Pricing Model for Power Plants

In the upper level, each power plant is concerned with how to choose a bidding strategy, which includes generation price and available capacity. Many bidding functions have been proposed. For a power system, the generation cost function generally adopts a quadratic function of the generation output, i.e. the generation cost function can be represented as:

$$ C_{j} \left( {P_{j} } \right) = a_{j} P_{j}^{2} + b_{j} P_{j} + c_{j} $$
(12.1)

where \( P_{j} \) is the generation output of generator \( j \), and \( a_{j} ,b_{j} ,c_{j} \) are coefficients of the generation cost function of generator \( j \).

The marginal cost of generator \( j \) is calculated by:

$$ \lambda_{j} = 2a_{j} P_{j} + b_{j} $$
(12.2)

It is a linear function of its generation output \( P_{j} \). The rule in a goods market may expect each power plant to bid according to its own generation cost. Therefore, we adopt this linear bid function. Suppose that the bidding for the jth unit at time \( t \) is:

$$ R_{tj} = \alpha_{tj} + \beta_{tj} P_{tj} $$
(12.3)

where \( t \in T \) is the time interval, \( T \) is time interval number, \( j \) represents the unit number, \( P_{tj} \) is the generation output of unit \( j \) at time \( t \), and \( \alpha_{tj} \) and \( \beta_{tj} \) are the bidding coefficients of unit \( j \) at time \( t \).

According to the justice principle of the same quality, the same network, and the same price, we adopt a uniform marginal price (UMP) as the market clearing price. Once the energy market is cleared, each unit will be paid according to its generation output and UMP. The payoff of the ith power plant is:

$$ F_{i} = \mathop \sum \limits_{t = 1}^{T} (\mathop \sum \limits_{{j\, \in \,G_{i} }} UMP_{t} P_{tj} - \mathop \sum \limits_{{j\, \in \,G_{i} }} (a_{j} P_{tj}^{2} + b_{tj} P_{tj} + c_{tj} )) $$
(12.4)

where \( G_{i} \) is the suffix set of the units belonging to the ith power plant. Each power plant wishes to maximize its own profit \( F_{i} \). In fact, \( F_{i} \) is the function of \( P_{tj} \) and \( UMP_{t} \), and \( UMP_{t} \) is the function of all units’ bidding \( \alpha_{tj} \), \( \beta_{tj} \) and output power \( P_{tj} \), which will impact on each other.

Therefore, we establish a strategic pricing model for power plants as follows:

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{\alpha_{tj} ,\beta_{tj} \in \,\,G_{i} }} F_{i} & = F_{i} \left( {\alpha_{t1} ,\beta_{t1} , \ldots ,\alpha_{tN} ,\beta_{tN} ,P_{t1} , \ldots ,P_{tN} } \right) \\ & = \mathop \sum \limits_{t = 1}^{T} \left( {UMP_{t} P_{ti} - \mathop \sum \limits_{{j\, \in \,G_{i} }} \left( {a_{j} P_{tj}^{2} + b_{tj} P_{tj} + c_{tj} } \right)} \right) \\ & \quad \quad \;\;i = 1, \ldots ,L \\ \end{aligned} $$
(12.5)

where \( L \) is the number of power plants, \( P_{ti} = \sum\nolimits_{{j\, \in \,G_{i} }} {P_{tj} } \), \( t = 1, \ldots ,T \).

The profit calculated for each power plant will consider both \( P_{ti} \) and \( UMP_{t} \), which can be computed by a market operator, according to the market clearing model.

2.2 Generation Output Dispatch Model for Market Operator

A market operator actually represents the consumer electricity purchase from power plants, under the conditions of security and stabilization. The objective of a market operator is to minimize the total purchase fare, while encouraging power plants to use a bid price as low as possible. It is reasonable that the lower the price, the more the output. Thus, the function value of a market operator’s objective will be calculated according to the bidding price. Most previous strategic bidding models do not include ramp rate constraints, without which the solution for generating dispatch may not be a truly optimal one. We should consider the ramp rate constraints in the real world when modeling a generating dispatch. However, if a model includes ramp rate as a constraint, the number of decision variables involved in the problem will increase dramatically, which requires a more powerful solution algorithm. Based on the analysis above, we build a market operator’s generation output dispatch model as follows:

$$ \begin{aligned}&{\mathop {{{\min}} }\limits_{{P_{tj} }} } \quad {f = f(\alpha_{t1} ,\beta_{t1} , \ldots ,\alpha_{tN} ,\beta_{tN} ,P_{t1} , \ldots ,P_{tN} ) = \mathop \sum \limits_{t = 1}^{T} \sum\limits_{j = 1}^{N} {R_{tj} } P_{tj} } \\ &{{\text{s}}.{\text{t}}.} \quad{\mathop \sum \limits_{j = 1}^{N} P_{tj} = P_{tD} } \\& \quad\quad \, {P_{j{{min}}} \le P_{tj} \le P_{j{{max}}} } \\& \quad \quad \, {- D_{j} \le P_{tj} - P_{t - 1,j} \le U_{j} ,\quad t = 1,2, \ldots , T}\end{aligned} $$
(12.6)

where \( t \in T \) is the time interval, \( T \) is the time interval number, \( j \) represents the unit number, \( P_{tj} \) is the generation output of unit \( j \) at the time \( t \), and \( \alpha_{tj} \) and \( \beta_{tj} \) are the bidding co-efficients of unit \( j \) at the time \( t \), \( P_{tD} \) is the load demand at the time \( t \), \( P_{j{\text{min}}}\) is the minimum output power of the \( j \)th unit, \( P_{j{\text{max}}} \) is the maximum output power of the \( j \)th unit, \( D_{j} \) is the maximum downwards ramp rate of the \( j \)th unit, and \( U_{j} \) is the maximum upwards ramp rate of the \( j \)th unit.

After receiving all power plants’ bid data, a market operator determines the output power of each unit and \( UMP_{t} \) in time slot t. \( UMP_{t} \) can be calculated according to the following steps:

  • [Begin]

  • Step1: calculate output power of each unit j for all time slots t using formula (12.6);

  • Step2: compute bidding R tj corresponding to the generation output \( P_{tj} \);

  • Step3: account \( UMP_{t} = {\max}_{j = 1}^{N} R_{tj}. \)

  • [End]

3 BLML Decision Model in Competitive Electricity Markets

From the analysis above, we know that in an auction-based day-ahead electricity market, each power plant tries to maximize its own profit by strategic bidding, and each market operator tries to minimize its total electricity purchase fare. The decision of one will influence the other. This is a typical bi-level decision problem, which has multiple leaders and only one follower, with power plants as leaders and a market operator as a follower.

By combining the strategic pricing model defined in (12.5) with the generation output dispatch model defined in (12.6), we establish a BLML decision model for competitive strategic bidding-generation output dispatch in an auction-based day-ahead electricity market as follows:

$$ \begin{aligned} \mathop {\max }\limits_{{\alpha _{{tj}} ,\beta _{{tj}} \in \,G_{i} }} & \quad F_{i} = F_{i} \left( {\alpha _{{t1}} ,\beta _{{t1}} , \ldots ,\alpha _{{tN}} ,\beta _{{tN}} ,P_{{t1}} , \ldots ,P_{{tN}} } \right) \\ & \quad = \mathop \sum \limits_{{t = 1}}^{T} \left( {UMP_{t} P_{{ti}} - \mathop \sum \limits_{{j\, \in \,G_{i} }} \left( {a_{j} P_{{tj}}^{2} + b_{{tj}} P_{{tj}} + c_{{tj}} } \right)} \right) \\ {\text{s}}.{\text{t}}. & \quad \alpha _{{t\min }} \le \alpha _{{tj}} \le \alpha _{{t\max }} , \\ & \quad \beta _{{t\min }} \le \beta _{{tj}} \le \beta _{{t\max }} , \\ & \quad \quad t = 1,2, \ldots ,~T,\quad j = 1,2, \ldots ,N,\quad j = 1,2, \ldots ,L~ \\ & \quad \quad \mathop {\min }\limits_{{P_{{tj}} }} \quad f = f(\alpha _{{t1}} ,\beta _{{t1}} , \ldots ,\alpha _{{tN}} ,\beta _{{tN}} ,P_{{t1}} , \ldots ,P_{{tN}} ) = \mathop \sum \limits_{{t = 1}}^{T} \sum\limits_{{j = 1}}^{N} {R_{{tj}} } P_{{tj}} \\ & \quad\quad{\text{s}}.{\text{t}}. \quad \mathop \sum \limits_{{j = 1}}^{N} P_{{tj}} = P_{{tD}} , \\ &\quad\quad P_{{j\min }} \le P_{{tj}} \le P_{{j\max }} , \\ & \quad \quad - D_{j} \le P_{{tj}} - P_{{t - 1, j}} \le U_{j} , \\ & \quad\quad t = 1,2, \ldots ,~T. \\ \end{aligned} $$
(12.7)

where \( \alpha_{tj} \) and \( \beta_{tj} \) are the bidding coefficients of unit j at time t, \( \alpha_{t\hbox{min} } \), \( \alpha_{t\hbox{max} } \), \( \beta_{t\hbox{min} } \), \( \beta_{t\hbox{max} } \) are the lower and upper limits for \( \alpha_{tj} \) and \( \beta_{tj} \) respectively, \( L \) is the number of power plants, \( P_{ti} = \sum\nolimits_{{j\, \in \,G_{i} }} {P_{tj} } \), \( P_{j\hbox{min} } \) is the minimum output power of the \( j \) th unit, \( P_{j\hbox{max} } \) is the maximum output power of the \( j \)th unit, \( D_{j} \) is the maximum downwards ramp rate of the \( j \)th unit, and \( U_{j} \) is the maximum upwards ramp rate of the \( j \)th unit.

This model describes strategic bidding problems in competitive electricity markets from a bi-level angle. In this model, there are multiple leaders (power plants) but only one follower (a market operator). This kind of problem has been studied in Chap. 5, and we will use the developed BLML-PSO algorithm proposed in Sect. 5.6 of Chap. 5 to solve it.

4 A Case Study

In this section, we will use a real world competitive strategic bidding example to illustrate the application of bi-level decision technology on an electricity market.

4.1 Test Data

In order to test the effectiveness of the proposed BLML decision model and the BLML-PSO algorithm when solving the model defined by (12.7), a typical competitive strategic bidding case consisting of three companies with six units and twenty-four time intervals is chosen. The generation cost function can be calculated by using formula (12.1), where the cost coefficients \( a_{j} ,b_{j} ,c_{j} \) of unit j and other technical data are given in Table 12.1. The load demands for each time interval t are given in Table 12.2.

Table 12.1 Technical data of units
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Table 12.2 Load demands in different time intervals
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In Table 12.1, Units 1 and 2 belong to the first power plant, Units 3 and 4 belong to the second power plant, and Units 5 and 6 belong to the third power plant.

To simplify computation, the limit of strategic bidding coefficients does not vary by different time slots and we suppose:

$$ \alpha_{{t{ \hbox{min} }}} = 7, \alpha_{{t{ \hbox{max} }}} = 9,\beta_{{t{ \hbox{min} }}} = 0.0002, \beta_{{t{ \hbox{max} }}} = 0.007, $$
$$ t = 1,2, \ldots T, \quad j = 1,2, \ldots , N. $$

4.2 Experiment Results

This example is run by the BLML-PSO algorithm developed in Sect. 5.6. The running results are listed in Tables 12.3, 12.4, 12.5, 12.6 and 12.7, where \( \alpha_{tj} \) and \( \beta_{tj} \) are the bidding coefficients of unit \( j \) at time \( t \), \( P_{tj} \) is the generation output of unit \( j \) at time \( t \), UMP t is the uniform marginal price at time \( t \).

Table 12.3 Running results for α tj from the example
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Table 12.4 Running results for β tj from the example
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Table 12.5 Running results for UMP t from the example
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Table 12.6 Running results for P tj from the example
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Table 12.7 Objective values for the decision makers
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Under these solutions, the objective values for both the leaders and the follower are listed in Table 12.7.

4.3 Experiment Analysis

By the BLML-PSO algorithm developed in Sect. 5.6, solutions are reached for both the power plants and the market operator to help them make strategic decisions. We conclude the BLML decision model and the BLML-PSO algorithm in the experiment as follows:

  1. 1.

    The BLML decision model can effectively model strategic bidding problems from electricity markets. By considering the gaming and bi-level relationships between several power plants and a market operator, the BLML decision model can better reflect the features of such real-world strategic bidding problems in electricity markets and format these problems practically.

  2. 2.

    The BLML-PSO algorithm is quite effective for solving strategic bidding problems defined by the BLML decision model. By making several power plants and a market operator decide sequentially, the hierarchical relation between them is fully considered. By moving the choice by power plants as close as possible to their rational reactions, the Nash equilibrium solution can be obtained.

5 Summary

Competitive strategic bidding optimization of power plants in electricity markets is in a practical sense important and it is technically implementable. This chapter applies a BLML decision model and BLML-PSO algorithm to handle the competitive strategic bidding decision-making problem in electricity markets. The proposed solution method can achieve a generalized Nash equilibrium for the BLML decision problem in an electricity market by providing power plants with competitive strategic bidding within the prevailing network security constraints.