Competitive Equilibrium in Microgrids With Dynamic Loads

TL;DR

This paper addresses decentralized energy management in microgrids with dynamic loads by formulating a competitive equilibrium that incorporates voltage constraints through a LinDistFlow-based model. It proves, via duality, that under convexity assumptions, the competitive equilibrium is equivalent to a social welfare maximization and to a Nash equilibrium of a standard game, with locational prices reflecting energy balance and grid constraints. The authors derive conditions for when all prosumers share the same price and for when locational prices decay to zero over time, indicating eventual supply sufficiency. Numerical validations include an EV charging scenario and a synthetic 3D-load microgrid, demonstrating tight alignment between competitive equilibrium and social welfare, and illustrating price decay behavior with realistic network dynamics. These results support practical, decentralized microgrid operation by providing rigorous market mechanisms that respect network constraints while enabling efficient energy trading among prosumers.

Abstract

In this paper, we consider microgrids that interconnect prosumers with distributed energy resources and dynamic loads. Prosumers are connected through the microgrid to trade energy and gain profit while respecting the network constraints. We establish a local energy market by defining a competitive equilibrium which balances energy and satisfies voltage constraints within the microgrid for all time. Using duality theory, we prove that under some convexity assumptions, a competitive equilibrium is equivalent to a social welfare maximization solution. Additionally, we show that a competitive equilibrium is equivalent to a Nash equilibrium of a standard game. In general, the energy price for each prosumer is different, leading to the concept of locational prices. We investigate a case under which all prosumers have the same locational prices. Additionally, we show that under some assumptions on the resource supply and network topology, locational prices decay to zero after a period of time, implying the available supply will be more than the demand required to stabilize the system. Finally, two numerical examples are provided to validate the results, one of which is a direct application of our results on electric vehicle charging control.

Figures (6)

• Figure 1: Diagram of the microgrid in Examples 1 and 2.
• Figure 2: Net supply $a_i(t)$ in Example 1.
• Figure 3: Locational prices $\lambda_i^\ast(t)$ in Example 1.
• Figure 4: Competitive equilibrium and social welfare maximization solution in Example 1.
• Figure 5: The $\tilde{v}_i(t)= \sum_{k =1}^n R_{ik} p_k^\ast(t)$ in Example 1.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Remark 1
• Theorem 1
• proof
• Proposition 1
• proof
• Remark 2
• Remark 3
• Definition 3