This paper presents a power dispatch strategy combining the main grid and distributed generators based on aggregative game theory and the Cournot price mechanism. Such a dispatch strategy aims to increase the electricity under the power shortage situation. Under the proposed strategy, this paper designs a discrete-time algorithm fusing the estimation technique and the Digging method to solve the power shortage problem in a distributed way. The distributed algorithm can provide privacy protection and information safety and improve the power grid's extendibility. Moreover, the simulation results show that the proposed algorithm has favorable performance and effectiveness in the numerical example.

With the fast development of the electricity network, power networks must face a more complicated situation. Additionally, the rapidly increasing scale of the electricity networks makes the traditional centralized algorithm unable to fit the actual power dispatch process ^{[1–4]}. Under such a background, designing an algorithm in a distributed manner becomes a recommendable choice to meet the actual needs. In a power shortage scenario, the priority in the power grid operation lies in increasing the power supply immediately. Otherwise, the best choice is to cut the loads. However, in the actual situation, increasing the power supply is usually not an easy task when the main power source cannot offer enough power. Due to the rapid emergence of distributed generators and energy, a natural idea arises, i.e., encouraging distributed generators to turn up their outputs when a power shortage occurs. Since the distributed generators may belong to different companies or individuals, designing a reasonable strategy for increasing electricity becomes challenging. As one of the most influential and general methods, the price incentive fits the market laws and meets the practical demand. The logic behind it is quite clear, i.e., improving the power selling price for the operators when the power shortage happens. Such a strategy prompts the users to produce more energy. The strategy mentioned above involves several research fields, which are introduced in the following subsection.

In general, the main states of the power operation can be divided into three situations: the regular running state, the power shortage state, and the collapse state. In different operation states of the power grid, distributed algorithms have different applications. Under the regular running state of the large-scale power grid, the main target focuses on a distributed economic dispatch. In this research area, there already exist fruitful results. In the early research, the distributed algorithm is used to solve the conventional economic dispatch problem only subjecting to the supply–demand global constraint. For example, Yu ^{[5]} employed the Laplacian dynamics to ensure that the incremental cost reaches a consensus. Compared with algorithm designs based on dual theory (see ^{[4]}), such a method avoids the update of the Lagrange multipliers and reduces the computation complexity. However, the introduction of the Lagrange multipliers endows the algorithm with the capacity to deal with the global inequality constraints (see ^{[4]}). On the basis of the work in ^{[5]}, Li ^{[6]} introduced the event-triggered scheme to reduce the communication burden caused by the continuous-time algorithm. From then on, researchers began employing many methods to improve the performance and adaptation of the applicable consensus algorithm. For instance, Wen ^{[7]} presented the adaptive consensus-based robust strategy to fit the uncertain communication graph scenario. To increase the convergence of the algorithm in ^{[6]}, He ^{[8]} designed the second-order algorithm. After that, He ^{[9]} developed an ADMM algorithm to solve the economic dispatch problem under the discrete-time communication environment. It needs to be emphasized that the above studies only focused on the dispatch problem with few fundamental constraints. To meet the requirements of the actual system, more constraints need to be considered in this given framework. These constraints include ramp-rate constraints, transmission line limit constraints, power loss constraints, etc. (see ^{[4]}). In recent years, since game theory is a recommendable method to describe the influence of human factors in the power dispatch models, researchers have begun to consider game theory, which extends the scope of model description. From the distributed control aspect, Liu ^{[3]} proposed a non-cooperative distributed coordination control strategy to address the multi-operator energy trading problem. Such a strategy involves electricity trading during the control process and makes the model more reasonable. From the game and optimization aspect, Ye ^{[10]} introduced the Cournot price mechanism to describe the market trading process. This mechanism makes the price change with the total energy demand and matches the law of the market. Based on the work in ^{[10]}, Fu ^{[11]} introduced more constraints in the market trading process. Such an adjustment also changes the algorithm design principle and introduces the multiplier consensus algorithm to deal with the global constraints. A similar technique also can be found in ^{[12, 13]}. Besides, the authors of ^{[13]} also introduced some techniques applicable to the non-smooth case.

This paper aims to design a distributed strategy for electricity trading in an energy shortage environment. Such a strategy is required to match the law of the market and adapt to the complicated communication graph. Furthermore, in the process of implementing the above strategy, there also exist the following challenges:

● The enormous number of distributed generators makes it impossible for the centralized algorithm to dispatch all distributed power simultaneously to compensate for the power shortage.

● How to design a suitable strategy that ensures the benefits of all the generators and maintains the initiative of the power operator (i.e., the pricing power) is the main difficulty in trading strategy design.

● Due to the character of the complicated communication graph, employing a discrete-time algorithm design principle becomes the first choice ^{[4]}.

To solve the above challenges, this paper first formulates the power shortage situation as an aggregative game involving the Cournot price mechanism. In this game, the electricity is supplied by two parts: one is the main power grid, and the other consists of distributed generators. Then, this paper designs a distributed strategy based on the Digging algorithm ^{[14]} and the estimation strategy ^{[15]}. The features of this distributed strategy are concluded as follows:

● The discrete-time algorithm construct offers the power grid dispatch process favorable properties of privacy protection, information safety, and extendibility.

● The employment of the Cournot price mechanism aggregates the distributed generators within a feasible and market-determined framework.

The rest of the paper is arranged as follows. The problem formulation is introduced in the next section. Then, the problem analysis section is presented to analyze the proposed problem and explain the designed algorithm. The simulation results are given to verify the proposed algorithm's performance and effectiveness in the simulation section. The last section concludes the paper.

In this section, we introduce an aggregative game model to describe the electricity energy trading. By solving this game, we aim to design a distributed strategy that encourages operators to supply electricity to the main grid. When the power on the main grid is insufficient (caused by attacks, generator failures, or capacity limitations), such a strategy allows us to adjust the sold price of electricity. After promoting the sold price of electricity, the operators will benefit more by selling more electricity, which improves the motivations of the operators and conforms to the law of the market. The detailed model is presented as:

where

where

where

Explanation for the problem in Equation (1)

Compared with the optimization problem, the problem in Equation (1) has multiple objection functions. Such a problem setting makes the aggregative game in Equation (1) possess the capacity to consider each operator's cost independently rather than the sum of the cost.

In this section, we analyze the proposed problem in Equation (1) based on Lagrange dual theory and game theory. Note that the problem in Equation (1) is a game theory problem, thus some transformations are required such that it can match the framework of the distributed algorithm. For convenience, define the local objective function

Note that

where

Based on the Lagrange dual theory, we can define the Lagrange dual function as follows:

where

Note that the rivals' decisions

Clearly, problem in Equation (6) is still a game problem that has no constraints. To transform the above problem in Equation (6) into a problem suitable for the distributed algorithm framework, some stronger restrictions are required. According to the analysis in ^{[16]} (Theorem 1), the variational solution is the generalized Nash equilibrium ^{[17]} of Equation (6). For such a solution, all the associated optimal Lagrange multipliers

where

We need to emphasize that the variational solution of a game problem is only one of the generalized Nash equilibria. In other words, we cannot obtain all the generalized Nash equilibria by solving different variational inequalities ^{[17]}. Based on Theorem 12.60 stated in ^{[18]}, the conjugate function ^{[19]} (Section 1.5), the constraint in Equation (7) can be replaced by the constraint ^{[20]}, we can transform the problem in Equation (7) into the following unconstrained optimization problem:

where

where ^{[14]}. The function

By following the analysis in ^{[14]} (Lemma 3.13), we can confirm that if the initial condition satisfies

The above result also implies ^{[17]}), we can deduce that, if the algorithm converges, it converges to the solution of Equation (1).

Obviously, the optimal sub-problem within the proposed algorithm is a simple quadratic programming problem with the box constraints, which can be solved by a simple method at every iteration. Note that the algorithm proposed above is a fully distributed algorithm and the time-varying stochastic matrix allows the communication links among the agents to varying from different iterations. Compared to the decaying step-sizes, the constant step-size

^{[14]} and the distributed algorithm presented in ^{[15]}. In this paper, the Digging algorithm is used to calculate the optimal strategy of each operator, while the distributed estimation of the total power of the distributed generators is updated in a distributed way, which has a similar construct to the algorithm proposed in the reference ^{[14]}. Hence, to confirm the convergence result of the algorithm proposed in this paper, the combination of the analysis methods in ^{[14, 15]} is an effective method. First, the small gain theory introduced in ^{[14]} is employed to construct the convergence structure cyclic of the algorithm proposed in this paper. Second, the analysis methods used in ^{[15]} is used to quantify the influences caused by the aggregative variables.

In this section, a simulation based on IEEE 14 bus is presented to verify the performance of the proposed algorithm. ^{[14]} and switch randomly.

The main grid cost coefficients and constraints

0.0037 | 2 | 1 | 50 | 200 | 140 | |

0.0175 | 1.75 | 0.75 | 20 | 80 | 40 | |

0.0625 | 1 | 0.75 | 15 | 50 | 35 | |

0.0083 | 3 | 1.25 | 10 | 35 | 20 | |

0.025 | 2 | 2 | 10 | 40 | 15 | |

0.025 | 1.5 | 1 | 12 | 40 | 23 |

The distributed generator cost coefficients and constraints

0.064 | 1.25 | 0.75 | 8 | 70 | 40 | |

0.0638 | 1.35 | 0.25 | 13 | 50 | 30 | |

0.0622 | 1.5 | 0.5 | 8 | 20 | 18 | |

0.0628 | 2.15 | 1.25 | 5 | 25 | 20 | |

0.0653 | 2 | 1.75 | 7 | 20 | 14 | |

0.0658 | 2 | 2 | 5 | 20 | 12 |

The optimal power output.

The incremental cost of each operator is shown in

The incremental cost of each operator.

The estimation value and the estimation error.

In this case, all the optimal outputs of

The partial derivative and the power mismatch.

The optimal power output without the Cournot price mechanism.

We next employ the algorithm proposed in ^{[11]} to make a comparison. Since the algorithm proposed in ^{[11]} is a continuous-time algorithm and the communication graph is fixed, we assume that the communication graph is a ring, and the abscissa is time instead of iterations. As shown in ^{[11]} asks for a fixed communication graph and continuous-time calculation. Such a requirement may not be suitable for some wireless networks operated in a discrete time.

The optimal power output calculated by the continuous-time algorithm.

In this paper, based on an aggregative game, we design a distributed electricity energy strategy to encourage the operators to supply electricity when the power output of the main grid is in a shortage situation. The simulation results confirm the performance and effectiveness of the proposed strategy/algorithm. In future work, based on the push-sum algorithm analytical framework and the corresponding discrete algorithm, we will further analyze the convergence of the proposed strategy by a rigorous mathematical method.

Made substantial contributions to conception and design of the study and performed data analysis and interpretation: Fu Z, Liu H

Performed data acquisition, as well as provided administrative, technical, and material support: Xu Q, Yu C, Yuan X

Not applicable.

This work was supported by the Science and Technology Project from State Grid Zhejiang Electric Power CO. Ltd (5211JY20001Q).

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2022.

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