Game Design and Analysis for Price based Demand Response: An Aggregate Game Approach

Abstract: In this paper, an aggregate game approach is proposed for the modeling and analysis of energy consumption control in smart grid. Since the electricity user's cost function depends on the aggregate load, which is unknown to the end users, an aggregate load estimator is employed to estimate it. Based on the communication among the users about their estimations on the aggregate load, Nash equilibrium seeking strategies are proposed for the electricity users. By using singular perturbation analysis and Lyapunov stability analysis, a local convergence result to the Nash equilibrium is presented for the energy consumption game that may have multiple Nash equilibria. For the energy consumption game with a unique Nash equilibrium, it is shown that the players' strategies converge to the Nash equilibrium non-locally. More specially, if the unique Nash equilibrium is an inner Nash equilibrium, then the convergence rate can be quantified. Energy consumption game with stubborn players is also investigated. Convergence to the best response strategies for the rational players is ensured. Numerical examples are provided to verify the effectiveness of the proposed methods.

• The paper models price-based smart grid demand response as an aggregate game and proposes Nash equilibrium seeking strategies using an average consensus protocol.
• It applies the approach to HVAC systems, showing convergence of player actions to a neighborhood of the Nash equilibrium, even in the presence of stubborn players.
• The paper analyzes local and exponential convergence properties of the proposed strategies using singular perturbation and Lyapunov stability analysis.

The paper addresses energy consumption control in smart grids using an aggregate game-theoretic approach. It focuses on price-anticipating electricity users aiming to minimize their costs, which depend on the aggregate energy consumption. The paper introduces strategies for Nash equilibrium seeking, particularly when users lack direct access to the aggregate energy consumption.

Key aspects and contributions include:

• Aggregate Game Formulation: The energy consumption control problem is modeled as an aggregate game, where each user's cost function depends on their energy consumption and the total energy consumption of all users. The cost function for user $i$ is defined as $C_{i}(l_{i},\bar{l})=V_{i}(l_{i})+P(\bar{l})l_{i}$, where $V_{i}(l_{i})$ is the load curtailment cost and $P(\bar{l})$ is the price function dependent on the aggregate energy consumption $\bar{l}$.

  $C_{i}(l_{i},\bar{l})$ = Cost of user $i$

  $l_i$ = Energy consumption of user $i$

  $\bar{l}$ = Aggregate energy consumption of all users

  $V_{i}(l_{i})$ = Load curtailment cost for user $i$

  $P(\bar{l})$ = Price function dependent on the aggregate energy consumption

• Average Consensus Protocol: An average consensus protocol is employed to enable users to estimate the aggregate energy consumption through neighboring communication. This addresses the challenge of users not knowing the total energy consumption directly.
• Nash Equilibrium Seeking: The paper proposes Nash seeking strategies based on the average consensus protocol. It considers scenarios with potentially multiple isolated Nash equilibria and derives local convergence results using singular perturbation analysis and Lyapunov stability analysis.
• Heating, Ventilation, and Air Conditioning (HVAC) Systems: The energy consumption control problem is specifically investigated for a network of HVAC systems. Under the assumption of a unique Nash equilibrium, it is shown that players' actions converge to a neighborhood of this equilibrium. Furthermore, exponential convergence is achieved when the unique Nash equilibrium is an inner Nash equilibrium.
• Stubborn Players: The paper analyzes the energy consumption game with stubborn players who maintain a constant energy consumption. The results demonstrate that the actions of rational players converge to a neighborhood of their best response strategies in the presence of stubborn players.
• Nash Seeking Strategy Design: The Nash seeking strategy for player $i$ is designed as:

  $$\dot{D}_i=-D_i-\sum_{j \in \mathcal{N}_i}(D_{i}-D_{j})-\sum_{j \in \mathcal{N}_i}(_{i}-_{j})+Nl_i$$

  $\dot{}_{i}=\sum_{j \in \mathcal{N}_i}(D_{i}-D_{j})$

  $$\dot{l}_i=-\bar{k}_i(\frac{\partial V_i}{\partial l_i}+P(D_i)+l_i\frac{\partial P(D_i)}{\partial D_i})$$

  $D_i$ = Player $i$'s estimation on the aggregate energy consumption

  $\mathcal{N}_i$ = Neighboring set of player $i$

  = Small positive parameter $k_i$ = Fixed positive parameter = Intermediate variable $V_i$ = Load curtailment cost $P$ = Pricing Function

• Assumptions:
  • The electricity users can communicate with their neighbors via an undirected and connected graph. Furthermore, the total number of the electricity users, $N$, is known to all the electricity users.
  • There exists isolated, stable Nash equilibrium on which $\frac{\partial C_i}{\partial l_i}(\mathbf{l}^*)=0$, $\frac{\partial ^2C_i}{\partial l_i^2}(\mathbf{l}^*)>0, \forall i\in \mathds{N}$ where $\mathbf{l}^*$ denotes the Nash equilibrium.

  • The matrix $B$ is strictly diagonally dominant.

    $$B=\left[ \begin{array}{cccc} \frac{\partial ^{2C_1}{\partial} l_1^{2}(\mathbf{l}^*)} & \frac{\partial ^{2C_1}{\partial} l_1 \partial l_2}(\mathbf{l}^*) & \cdots & \frac{\partial ^{2C_1}{\partial} l_1 \partial l_N}(\mathbf{l}^*) \ \frac{\partial ^{2C_2}{\partial} l_1 \partial l_2}(\mathbf{l}^*) & \frac{\partial ^{2C_2}{\partial} l_2^{2}(\mathbf{l}^*)} & & \vdots \ \vdots & & \ddots & \ \frac{\partial ^{2C_N}{\partial} l_1 \partial l_N}(\mathbf{l}^*) & \cdots & & \frac{\partial ^{2C_N}{\partial} l_N^{2}(\mathbf{l}^*)} \ \end{array} \right]$$

• Convergence Analysis:
  • The paper presents a theorem stating that under certain assumptions, there exists a positive constant $^*$ such that for every $0<<^*$, $(\mathbf{l}(t),\mathbf{D}(t),\widetilde{\mathbf{}(t))$ converges exponentially to $(\mathbf{l}^*,\mathbf{1}\sum_{i=1}^{N}l_i^*,\widetilde{\mathbf{}^e(\mathbf{l}^*))$ under the proposed updating law given that $||\mathbf{l}(0)-\mathbf{l}^*||$, $||\mathbf{D}(0)-\mathbf{1}\sum_{i=1}^{N}l_i^*||$, $||\widetilde{\mathbf{}(0)-\widetilde{}^e(\mathbf{l}^*)||$ are sufficiently small.

  • The energy consumption game is a potential game with a potential function being $Q(\mathbf{l})=\sum_{i=1}^{N}_i_i^2(l_i-\hat{l}_i)^2+\sum_{i=1}^{N}a(\sum_{j=1,j\neq i}^{N}l_j)l_i+\sum_{i=1}^N(al_i^{2}+p_0l_i)$.

    $Q(\mathbf{l})$ = Potential function $_i$ and $_i$ are thermal coefficients $\hat{l}_i$ is the energy needed to maintain the indoor temperature of the HVAC system. $a$ = non-negative constant $p_0$ = constant

• Numerical Validation: Numerical examples are provided to validate the effectiveness of the proposed methods, demonstrating the convergence of energy consumptions to the Nash equilibrium and the behavior of rational players in the presence of stubborn players.