A Leader–Follower Game on Congestion Management in Power Systems

Abstract

Since the beginning of power system restructuring and creation of numerous temporal power markets, transmission congestion has become a serious challenge for independent system operators around the globe. On the other hand, in recent years, emission reduction has become a major concern for the electricity industry. As a widely accepted solution, attention has been drawn to renewable power resources promotion. However, penetration of these resources impacts on transmission congestion. In sum, these challenges reinforce the need for new approaches to facilitate interaction between the operator and energy market players defined as the generators (power generation companies) in order to provide proper operational signals for the operator. The main purpose of this chapter is to provide a combination of a leader–follower game theoretical mechanism and multiattribute decision-making for the operator to choose his best strategy by considering congestion-driven and environmental attributes. First the operator (as the leader) chooses K strategies arbitrarily. Each strategy is constituted by emission penalty factors for each generator, the amount of purchased power from renewable power resources, and a bid cap that provides a maximum bid for the price of electrical power for generators who intend to sell their power in the market. For each of the K strategies, the generators (as the followers) determine their optimum bids for selling power in the market. The interaction between generation companies is modeled as Nash-Supply Function equilibrium (SFE) game. Thereafter, for each of the K strategies, the operator performs congestion management and congestion-driven attributes and emission are obtained. The four different attributes are congestion cost, average locational marginal price (LMP) for different system buses, variance of the LMPs, and the generators’ emission. Finally, the operator’s preferred strategy is selected using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). The proposed procedure is applied to the IEEE reliability 24-bus test system and the results are analyzed.

Appendix When \(a_{i} = 0\) for at Least One Generator

Assume that $a_{i} = 0$ when $i = 1, \ldots ,g_{0}$, and $a_{i} > 0$ when $i = g_{0} + 1, \ldots ,g,\,1 \le g_{0} \le g$. Taking the limit for (9) gives:

$$\lambda_{t} = \mathop {\text{Lim}}\limits_{{(a_{1} , \ldots ,a_{{g_{0} }} ) \to (0, \ldots ,0)}} \frac{{d + \sum\nolimits_{i = 1}^{g} {\frac{{\beta_{it} }}{{a_{i} }}} }}{{\sum\nolimits_{i = 1}^{g} {\frac{1}{{a_{i} }}} }}$$

(47)

Separating the terms related to the $g_{0}$ generators with $a_{i} = 0$, and the $g - g_{0}$ generators with $a_{i} > 0$, gives:

$$\begin{aligned} \lambda_{t} & = \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{d + \sum_{{i = g_{0} + 1}}^{g} {\frac{{\beta_{it} }}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{{\beta_{it} }}{{a_{i} }}} }}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{1}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{1}{{a_{i} }}} }} = \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{d + \sum_{{i = g_{0} + 1}}^{g} {\frac{{\beta_{it} }}{{a_{i} }}} }}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{1}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{1}{{a_{i} }}} }} \\ & \quad + \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{\sum_{i = 1}^{{g_{0} }} {\frac{{\beta_{it} }}{{a_{i} }}} }}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{1}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{1}{{a_{i} }}} }} = 0 + \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{\sum_{i = 1}^{{g_{0} }} {\frac{{\beta_{it} }}{{a_{i} }}} }}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{1}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{1}{{a_{i} }}} }} \\ & = \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{\frac{{\beta_{1t} }}{{a_{1} }}}}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{1}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{1}{{a_{i} }}} }} + \cdots + \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{\frac{{\beta_{{g_{0} t}} }}{{a_{{g_{0} }} }}}}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{1}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{1}{{a_{i} }}} }} \\ & = \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{\beta_{1t} }}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{{a_{1} }}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{{a_{1} }}{{a_{i} }}} }} + \cdots + \mathop {\text{Lim}}\limits_{{\left( {a_{1} , \ldots ,a_{{g_{0} }} } \right) \to \left( {0, \ldots ,0} \right)}} \frac{{\beta_{{g_{0} t}} }}{{\sum_{{i = g_{0} + 1}}^{g} {\frac{{a_{{g_{0} }} }}{{a_{i} }}} + \sum_{i = 1}^{{g_{0} }} {\frac{{a_{{g_{0} }} }}{{a_{i} }}} }} \\ & = \frac{{\beta_{1t} }}{{0 + g_{0} }} + \cdots + \frac{{\beta_{{g_{0} t}} }}{{0 + g_{0} }} = \frac{{\beta_{1t} + \cdots + \beta_{{g_{0} t}} }}{{g_{0} }} \\ \end{aligned}$$

(48)

The calculated $\lambda_{t}$ will be substituted in (10) and the rest of the calculation procedure will be the same as what we developed for the case of $a_{i} > 0;i = 1, \ldots ,g$.