Consumer Theory-Based Primary Frequency Regulation in Multi-Microgrid Systems within a P2P Energy Management Framework

Abstract

This paper presents a novel primary frequency regulation strategy for multi-microgrid (MMG) systems, utilizing consumer theory within a peer-to-peer (P2P) energy management framework. By coordinating photovoltaic (PV) systems and energy storage systems (ESS), the proposed method ensures a rapid and effective response to frequency deviations. Unlike conventional approaches, this strategy minimizes the curtailment of renewable energy sources by prioritizing the use of ESS, allowing excess energy to charge the ESS for later use during under-frequency events. This not only enhances energy efficiency but also maximizes renewable energy utilization. Simulations demonstrate that the proposed scheme achieves lower frequency deviations and faster stabilization compared to traditional droop and virtual inertia methods. These results highlight the potential benefits of integrating consumer theory-based models into primary frequency regulation, significantly enhancing system stability and efficiency in power systems with high levels of renewable energy penetration.

1. Introduction

In recent years, there has been a notable shift in the traditional unidirectional flow architecture of power systems. The centralized generation model, characterized by large power plants located far from consumption centers, is transitioning toward a decentralized framework with the emergence of microgrids (MGs) interconnected with distribution networks [1]. These MGs integrate components such as loads, distributed generation (DG) units, and energy storage systems (ESS), enabling coordinated operation in both connected and isolated modes. This transformation, driven by advances in power electronics, introduces a new paradigm that offers numerous benefits to power networks. These benefits include enhanced reliability, resilience, and robustness. Additionally, this new model improves energy efficiency, reduces environmental impact, integrates renewable energy sources, empowers consumers, and lowers operating costs, particularly in systems reliant on traditional fossil fuels and in rural areas [2,3].

The advantages of MGs are especially relevant in weak or low-inertia networks, such as those found in islands or remote areas. These systems are particularly sensitive to abrupt changes in demand or generation [4,5]. In the event of a contingency, the insufficient dynamic response of synchronous machines to balance the system can cause the frequency to reach values outside safe operating limits. Microgrids, thanks to the fast dynamic response of their interconnected elements through power electronics interfaces [6], are positioned as an alternative to respond quickly to contingencies, providing complementary frequency and voltage regulation services and contributing to the stability and reliability of the power supply. These benefits are more noticeable when multiple microgrids (MMG) with different characteristics participate as they can offer these services in various locations depending on the system requirements.

Frequency regulation in low-inertia systems with a high penetration of inverter-based resources has become a highly relevant issue. Without an adequate control algorithm, these technologies do not contribute to system stability as they fail to add to the system’s equivalent inertia. Several regulation schemes to address this challenge are presented in [6,7,8]. These schemes can be divided into three stages based on the time scale: primary, secondary, and tertiary regulation.

Primary regulation responds to real-time imbalances, acting within the first seconds of an event to stabilize frequency at a value that may not always correspond to the nominal frequency, thus allowing for a non-zero steady-state error. Secondary regulation acts over the medium term, e.g., tens of seconds to minutes, to eliminate this steady-state error and restore the primary control reserves. Tertiary regulation, managed by the market operator, deals with long-term control reserves, e.g., hours to days, considering the integration of new energy resources and system optimization.

Traditionally, research at this last time scale focuses on the energy management system (EMS), which uses optimization problems and various algorithms to minimize operating costs while considering the interests of different market actors [2,3,9,10]. Approaches to EMS can be categorized into centralized or decentralized schemes, with the latter offering greater reliability in the face of communication network failures and requiring lower computational resources. In this category, researchers often use distributed optimization methods based on game theory or consensus algorithms to develop energy markets. For example, studies such as [10,11,12,13] have explored various aspects of power system operation and energy trading. However, these works do not specifically address primary frequency regulation issues in low-inertia power systems and the ancillary services that agents can offer to the grid.

Regarding primary frequency regulation in MGs, droop control, demand management, and virtual inertia emulation are commonly proposed strategies in the literature. Several studies have incorporated droop-type logic into their frequency regulation frameworks. Ref. [14] proposes two operational scenarios for primary frequency regulation in isolated grids, utilizing a droop loop for normal operation and an emergency mode for rapid system balancing. Ref. [15] introduces a distributed online optimization algorithm for power management in MGs, integrating frequency droop into the response of distributed generators. Ref. [16] suggests a modified droop control scheme for ESS, accounting for the specific characteristics of storage technologies. In [17], a distributed EMS based on a two-level optimization problem is proposed, where the lower level implements adaptive droop control and includes a cost function for storage systems based on marginal price and state of charge. While these studies demonstrate the effectiveness of droop control and its variants in primary frequency regulation, they lack selectivity over ESS and photovoltaic systems (PV). This lack of selectivity can lead to unnecessary curtailment of solar energy when ESS have sufficient capacity to balance the system during over-frequency events.

In the field of demand-side management to provide primary frequency regulation, ref. [18] proposes a dynamic demand response control strategy for isolated MGs, focusing on the use of electric water heaters (EWHs) as fast response resources. The authors highlight the importance of achieving accurate frequency regulation within the responsiveness range of the EWHs group, proposing a dynamic strategy to achieve this regulation in real-time. On the other hand, reference [19] proposes an emergency demand response scheme for the autonomous operation of MGs based on local frequency measurements. The authors suggest the active participation of MG loads to guarantee post-isolation stability, considering the frequency behavior and the energy available in the storage units, with electric vehicles connected to the system considered as an additional element that can respond to changes in frequency. While these strategies may be effective for primary frequency regulation in the face of minor imbalances, their performance can be limited if there are insufficient loads capable of rapidly adjusting their consumption to balance the system.

Furthermore, studies such as [20,21,22,23,24,25] implement a virtual inertia droop strategy aimed at emulating the behavior of a synchronous generator in response to changes in the power balance of the system, thereby contributing to the equivalent inertia of the system. This approach improves dynamic performance compared to conventional droop loops. However, primary regulation schemes based on virtual inertia droop do not allow for a selective and flexible response between energy generation and storage systems to address specific load balance needs, restricting the efficiency and adaptability of such schemes.

In addition, the current literature does not address compensation for owners of MGs that provide the ancillary service of primary frequency regulation to a low-inertia, low-reliability grid. The lack of a fair compensation incentive may discourage MGs from participating in this crucial service. The aforementioned limitations of current approaches to primary frequency regulation in multiple MMGs motivate the proposal of an alternative mechanism that meets the specific needs of these interconnected systems.

In this work, a primary regulation scheme is proposed that applies concepts of consumer theory to model the behavior of the different elements of interconnected MGs within the framework of an EMS based on a peer-to-peer (P2P) market, applied to a system of MMGs. The proposal allows for a fast and coordinated response to imbalances in the system load, in addition to generating signals that could be used for the remuneration of the primary frequency regulation service to the different agents. The use of consumer theory provides a simple yet effective modeling approach that leverages well-established economic principles to describe the response of ESS and PV systems to price signals. This approach, when applied to primary frequency regulation, helps to balance the grid more effectively, as demonstrated by the simulation results.

Compared to conventional droop and virtual inertia control strategies, our consumer theory-based model shows good coordination in frequency regulation by avoiding unnecessary curtailment of PV power during over-frequency events. This advantage is crucial as it allows excess energy to be used to charge ESS, which can then be utilized during under-frequency events. This not only improves energy efficiency but also maximizes the utilization of renewable energy, contributing to a more sustainable power system.

The main contributions of this work are as follows:

• Proposal of a primary regulation alternative that allows a coordinated response among agents within the framework of a P2P market, enhancing energy efficiency, and maximizing renewable energy utilization.
• Modeling of storage and PV systems using consumer theory.
• Adaptation of the EMS based on the P2P market scheme proposed in [26] to MMG systems.

The article is structured as follows: Section 2 details the primary frequency regulation strategy using droop control and simulated inertia, describes the P2P market scheme from [26] for pre-event operational conditions, and defines consumer theory concepts for ESS and PV system modeling. Section 3 presents the modeling of ESS and PV systems based on consumer theory for effective integration into the energy management system. Section 4 outlines the adaptation of the P2P energy management scheme to an MMG system. Section 5 introduces a new primary frequency regulation strategy. Section 6 presents simulations and results, followed by conclusions in Section 7. For a more comprehensive understanding, Appendix A presents a table summarizing the notation used throughout the paper.

2. Background

2.1. Droop Control for Primary Frequency Regulation

Droop control is a technique used in electrical power systems to maintain the network frequency within acceptable limits after sudden disturbances. This control is implemented by adjusting the output power of the generators linearly to the deviation of the frequency from its nominal value. The linear relationship between the change in frequency and the change in generator active power is given by the following equation:

$$\Delta P_d=\left\{\begin{array}{cc}{K}_d·(\Delta f-BM)\hfill & \mathrm{if}\Delta f>BM\hfill \\ K_d·(\Delta f+BM)\hfill & \mathrm{if}\Delta f<-BM\hfill \\ 0\hfill & \mathrm{if}-BM\le \Delta f\le BM\hfill \end{array}\right.,$$

(1)

where $\Delta P_d$ represents the per unit active power variation, $K_d$ is the droop slope coefficient controlling the magnitude of power variation per unit frequency deviation, $\Delta f$ is the measured frequency deviation from the nominal frequency, and $BM$ is the dead band, a range of frequency deviations where no power variation occurs. This prevents the system from reacting to minor fluctuations.

2.2. Virtual Inertia in Primary Frequency Regulation

The implementation of the virtual inertia control loop aims to emulate the behavior of a synchronous machine in response to variations in system frequency [21]. This helps to decrease the rate of change of frequency deviations, known as RoCoF (rate of change of frequency). The power variation generated by this loop is defined in Equation (2).

$$\Delta P_i=-2H_{sim}·{\displaystyle \frac{df}{dx}}=-K_i·{\displaystyle \frac{df}{dx}}.$$

(2)

The primary regulation loop considering droop and inertia, as presented in the literature, is shown in Figure 1. In this diagram, f represents the network frequency measurement, $f_{nom}$ is the nominal frequency, $P_{sp}$ is the active power setpoint, $\overline{P}$ and $\underline{P}$ are the maximum and minimum active power of the generation or storage system, and $P_{cmd}$ is the final reference for the primary frequency regulation.

2.3. Consensus Equation

Consensus algorithms are commonly found in multi-agent systems, enabling distributed agents to agree on certain state variables essential for coordinated actions. These algorithms ensure that despite possible differences in initial conditions and local objectives, all agents converge to a common decision.

In multi-agent systems, consensus algorithms promote information sharing and collective decision-making. Each agent updates its state based on the states of its neighbors. The consensus equation is typically expressed as follows:

$$x_n(t+1)=x_n\left(t\right)+\sum _{m\in {\omega }_n}{a}_{nm}(x_n\left(t\right)-x_m\left(t\right)),$$

(3)

where $x_n\left(t\right)$ is the state of agent n at time t, ${\omega }_n$ represents the set of neighbors of agent n, and $a_{nm}$ are the weights that quantify the influence of neighbor m on agent m.

2.4. P2P Market Scheme

In the EMS based on multi-agent systems (MAS) with a P2P market scheme as described in [26], agents agree on the amount of energy and the associated trading price according to their preferences in a bilateral way with their neighbors. To establish the negotiation process among the participants, the consensus + innovation algorithm is employed. Figure 2 illustrates the P2P market scheme. A fully connected communication network is established, where all agents share information with each other.

A set $\Omega$ with N agents is considered. The power injected by agent n is defined as the sum of the power traded with each agent that composes its neighborhood set ${\omega }_n$:

$$p_n=\sum _{m\in {\omega }_n}{p}_{nm}.$$

(4)

For agents that are conventional synchronous machines, a quadratic cost function is used:

$$c_n\left(p_n\right)={\displaystyle \frac{1}{2}}{a}_n{p}_n^2+b_n{p}_n+d_n,$$

(5)

where $a_n$, $b_n$, and $d_n$ are positive coefficients.

For agents that are considered non-dispatchable, such as photovoltaic systems or fixed loads, although it is possible to model them as agents whose marginal cost is zero ($a_n=b_n=0$), a virtual quadratic cost function is assigned to them, which does not affect the optimal solution of the algorithm.

The trading costs of agent n are defined as follows:

$$\widetilde{c_n}({\mathit{\rho }}_{\mathit{n}},{\mathit{\lambda }}_{\mathit{n}})=-\sum _{m\in {\omega }_n}{\lambda }_{nm}{p}_{nm}.$$

(6)

Here, ${\lambda }_{nm}$ represents the trading price offered by agent n to its neighbor m over the power $p_{nm}$. The vectors ${\lambda }_{\mathbf{n}}$ and ${\rho }_{\mathbf{n}}$ are composed of the variables ${\lambda }_{nm}$ and $p_{nm}$, respectively. The negative sign in (6) reflects the revenue obtained by the agent from the sale of power when $p_{nm}>0$ or the cost for power purchase when $p_{nm}<0$.

The total cost function for agent n is obtained by combining the individual cost functions:

$$c_n({\mathit{\rho }}_{\mathit{n}},{\mathit{\lambda }}_{\mathit{n}})={\displaystyle \frac{1}{2}}{a}_n{p}_n^2+b_n{p}_n+d_n-\sum _{m\in {\omega }_n}{\lambda }_{nm}{p}_{nm}.$$

(7)

With the purpose of optimally allocating resources, an optimization problem is formulated, minimizing the aggregate cost function of the whole system (8), where both the cost functions of the synchronous generators and the virtual ones associated with the non-dispatchable agents are considered. This is done by considering boundary constraints (9), as well as the power balance (10). In this approach, ${\omega }_n$ is defined as the set of neighboring agents of agent n, and ${\Omega }_C$ and ${\Omega }_G$ as the subsets of $\Omega$ containing the consumers and generators, respectively.

$$\begin{array}{cc}\hfill & \min \sum _{n\in \Omega }{c}_n({\mathit{\rho }}_{\mathit{n}},{\mathit{\lambda }}_{\mathit{n}})\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \underline{P_n}\le p_n\le \overline{P_n}\forall n\in \Omega ,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & p_{nm}+p_{mn}=0\forall (n,m)\in (\Omega ,{\omega }_n),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & p_{nm}\ge 0\forall (n,m)\in ({\Omega }_G,{\omega }_n),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & p_{nm}\le 0\forall (n,m)\in ({\Omega }_C,{\omega }_n).\hfill \end{array}$$

(12)

For non-dispatchable agents, also referred to as must-take agents, the same value is assumed for the maximum and minimum power, which is represented in (13).

$$\underline{P_n}=\overline{P_n}.$$

(13)

In the context of a P2P market where there is a negotiation process between agents, it is important to note that once the market clearing is achieved, the reciprocity constraint (10) is satisfied, and a bilateral trading price agreement is reached. Consequently, there is reciprocity over the dual variables ${\mathit{\lambda }}_n$ ensuring the following relationship:

$${\lambda }_{nm}-{\lambda }_{mn}=0\forall (n,m)\in (\Omega ,{\omega }_n).$$

(14)

To solve the centralized optimization problem in a fully distributed way, each agent applies the consensus equation over the dual variables ${\lambda }_{nm}$ and, taking into account a dualization over constraint (10), an additional term called innovation is included in the consensus equation as explained in Section 2.5. This approach is useful to decompose the centralized optimization problem into local optimization problems over the vector ${\mathit{p}}_n$ composed by the primal variables for each agent as follows:

$$\begin{array}{cc}\hfill & \min c_n({\mathit{\rho }}_{\mathit{n}},{\mathit{\lambda }}_{\mathit{n}})\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \underline{P_n}\le p_n\le \overline{P_n},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & p_{nm}\ge 0\forall (n,m)\in ({\Omega }_G,{\omega }_n),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & p_{nm}\le 0\forall (n,m)\in ({\Omega }_C,{\omega }_n).\hfill \end{array}$$

(18)

This local optimization problem is solved by each agent over decision variables ${\mathit{p}}_n$, taking the value of ${\mathit{\lambda }}_n$ obtained with the consensus + innovation equation. The next subsection provides a more detailed explanation of the consensus + innovation algorithm.

2.5. Consensus + Innovation Algorithm

Within the proposed scheme, the agents iteratively negotiate and reach an agreement on a bilateral trading price for energy, using the consensus algorithm with an additional term called “innovation”. This term aims to meet the (10) constraint of power reciprocity between the agents in the global optimization problem, thus seeking a balance between generation and demand.

As shown in Figure 3, the algorithm is divided into three main stages: updating the dual variables, updating the primal variables, and evaluating the convergence criteria.

2.5.1. Dual Variables Update

Once ${\mathit{\lambda }}_{\mathit{n}}$ and ${\mathit{\rho }}_{\mathit{n}}$ have been initialized for all n in $\Omega$ at iteration $k=0$, the agents update their local vector of prices ${\mathit{\lambda }}_{\mathit{n}}$ by applying the consensus + innovation equation:

$${\lambda }_{nm}^k={\lambda }_{nm}^{k-1}-{\beta }^k({\lambda }_{nm}^{k-1}-{\lambda }_{mn}^{k-1})-{\alpha }^k(p_{nm}^{k-1}+p_{mn}^{k-1})\forall (n,m)\in (\Omega ,{\omega }_n),$$

(19)

where ${\beta }^k$ and ${\alpha }^k$ are sequences of positive values representing the weight of the consensus factor and the innovation factor, respectively, at iteration k.

The consensus term, ${\beta }^k({\lambda }_{nm}^{k-1}-{\lambda }_{mn}^{k-1})$, promotes price convergence among neighboring agents, ensuring that all agents gradually align through iterative negotiation to a common trading price to satisfy (14). This term manages the negotiation process between agents, facilitating an agreement over bilateral trading prices, which may differ during the negotiation process or in the initialization step.

The innovation term, ${\alpha }^k(p_{nm}^{k-1}+p_{mn}^{k-1})$, introduces an adjustment based on the power transacted in the previous iteration. The purpose of this adjustment is to satisfy the power reciprocity constraint (10), ensuring that the power exchanged between the agents is equivalent in both directions, such that ${\lambda }_{nm}^k$ diverges if this condition is not met. This mechanism guarantees that the trading prices accurately reflect the negotiated agreements, thereby maintaining the stability and consistency of the power exchange among the agents.

Then, the dual slack variables corresponding to the boundary constraints of (16) are updated:

$${\overline{{\mu }_n}}^k=\max (0,{\overline{{\mu }_n}}^{k-1}+{\eta }^k(p_n^{k-1}-\overline{P_n})),$$

(20)

$${\underline{{\mu }_n}}^k=\max (0,{\underline{{\mu }_n}}^{k-1}+{\eta }^k(\underline{P_n}-p_n^{k-1})),$$

(21)

where ${\eta }^k$ is a sequence of positive numbers.

2.5.2. Primal Variables Update

The Lagrangian of (15) is formulated for the update of the vector ${\mathit{\rho }}_{\mathit{n}}$.

$$L_n({\mathit{\rho }}_{\mathit{n}},{\mathit{\lambda }}_{\mathit{n}},\underline{{\mu }_n},\overline{{\mu }_n})={\displaystyle \frac{1}{2}}{a}_n{p}_n^2+b_n{p}_n+d_n-\sum _{m\in {\omega }_n}{\lambda }_{nm}{p}_{nm}+\overline{{\mu }_n}(p_n-\overline{P_n})-\underline{{\mu }_n}(p_n-\underline{P_n}).$$

(22)

Considering the Lagrangian (22), the Karush–Kuhn–Tucker optimality conditions are obtained,

$$\begin{array}{c}\hfill a_n{p}_n+b_n-{\lambda }_{nm}+{\overline{\mu }}_n-{\underline{\mu }}_n=0.\end{array}$$

(23)

Then, for each trade, the optimal power reference for the current iteration that agent n is disposed to negotiate with its neighbor m is determined based on the price conditions and slack variables of the k-th iteration

$$\begin{array}{c}\hfill p_n^{\left(m\right)k}={\displaystyle \frac{{\lambda }_{nm}-{\overline{\mu }}_n+{\underline{\mu }}_n-b_n}{a_n}}.\end{array}$$

(24)

Therefore, $p_n^{\left(m\right)k}$ is an intermediary variable used in the primal update process to represent the potential power exchange that agent n would optimally trade with neighbor m at iteration k, considering only the bilateral interaction between n and m. This value is calculated assuming no other neighbors influence the decision, effectively isolating the trading relationship between n and m.

However, to consider the influence of other neighbors in the update of exchange power, a weighting factor $f_{nm}^k$ is included in the equation for the gradient descent update of each power exchange interaction as a step size parameter. This is defined as follows:

$$\begin{array}{cc}\hfill p_{nm}^k& =\max (0,p_{nm}^{k-1}+f_{nm}^k(p_n^{\left(m\right)k}-p_n^{k-1}))\forall n\in {\Omega }_G,\hfill \\ \hfill p_{nm}^k& =\min (0,p_{nm}^{k-1}+f_{nm}^k(p_n^{\left(m\right)k}-p_n^{k-1}))\forall n\in {\Omega }_C.\hfill \end{array}$$

(25)

The weighting factor $f_{nm}^k$ balances the influence of each neighbor m in the update process of the power exchange $p_{nm}$ for agent n. This factor ensures that the power exchange updates consider the relative importance of each neighbor’s contribution to the overall power exchange decision, taking into account the magnitude of the power exchanged with each neighbor in the previous iteration:

$$\begin{array}{c}\hfill f_{nm}^k={\displaystyle \frac{\left|p_{nm}^{(k-1)}\right|+{\delta }^k}{{\sum }_{l\in {\omega }_n}(\left|p_{nl}^{(k-1)}\right|+{\delta }^k)}}.\end{array}$$

(26)

To ensure that the denominator in the calculation of $f_{nm}^k$ does not become zero or excessively small, which could otherwise lead to numerical instability or disproportionate updates, a small positive constant ${\delta }^k$ is included in the calculation of the weighting factor.

2.5.3. Convergence Criteria Evaluation

As a final step, it is evaluated if the changes in the variables $p_{nm}$, ${\lambda }_{nm}$, and ${\mu }_n$ between iterations k and $k-1$ are less than the thresholds established by ${ϵ}_p$, ${ϵ}_{\lambda }$, and ${ϵ}_{\mu }$, respectively, to determine if the convergence of the algorithm is reached or if the iterative process should be continued.

$$\begin{array}{c}\hfill \left|p_{nm}^k-p_{nm}^{k-1}\right|\le {ϵ}_p,\end{array}$$

(27)

$$\begin{array}{c}\hfill \left|{\lambda }_{nm}^k-{\lambda }_{nm}^{k-1}\right|\le {ϵ}_{\lambda },\end{array}$$

(28)

$$\begin{array}{c}\hfill \left|{\mu }_n^k-{\mu }_n^{k-1}\right|\le {ϵ}_{\mu }.\end{array}$$

(29)

When the convergence criteria are satisfied, the market clearing prices ${\mathit{\lambda }}_{\mathit{n}}^{\mathit{t}}$ and the traded powers ${\mathit{\rho }}_{\mathit{n}}^{\mathit{t}}$ for the dispatch interval t are obtained. In this way, the algorithm can be applied for real-time dispatch in different time intervals, depending on the operating conditions of the system in each period.

2.6. Consumer Theory

Consumer theory, a fundamental branch of microeconomics, studies how individuals make consumption decisions based on their preferences and budget constraints, assuming that they are rational agents seeking to maximize their profit or satisfaction within their economic constraints [27]. Two key concepts in this framework are elasticity of demand, which measures the sensitivity of quantity demanded to changes in price, reflecting the responsiveness of the consumer to market signals, and diminishing marginal profit, which states that the additional benefit of consuming one more unit decreases as the quantity consumed increases [28]. These concepts are useful for modeling agents’ response to price signals and provide a sound theoretical basis for understanding consumption decisions and the interaction between supply and demand in the market.

3. Modeling ESS and PV Based on Consumer Theory

In this section, consumer theory is employed for its simplicity and effectiveness in modeling the dynamic behavior of energy systems within a MMG environment operating within a P2P market framework. This theory provides a straightforward yet powerful framework for capturing the responses of MG components to market price signals. Elasticity models the response of ESS to price changes, while diminishing marginal benefit illustrates the decreasing additional gain from generating more power in PV systems. ESS and PV systems are chosen due to their prevalence in microgrids. Leveraging these consumer theory principles provides a practical representation of MG dynamics, ensuring effective integration within a P2P energy management framework. A graphical representation of this modeling is shown in Figure 4 and detailed in Section 3.1 and Section 3.2.

3.1. ESS Modeling

In the framework of an EMS based on a P2P market, we propose modeling the behavior of storage systems in response to varying energy trading prices ($\lambda$) using the concept derived from consumer theory of supply/demand elasticity. This concept is defined as the variation in the demand or supply of a good or service as a function of its price [28]. Considering that ESS can act both as producers and consumers depending on the market price, a combined supply/demand function is proposed. In this function, the power injected or consumed by the system is zero when the trading price ($\lambda$) equals the valuation of the stored energy (${\lambda }_E$), which is directly related to the state of charge (SOC). The elasticity of the ESS ($E_{ESS}$) determines the influence of the price on the variation of exchanged power ($\Delta p_{ESS}/\Delta \lambda$). Thus, if the market price exceeds the valuation of the stored energy, the ESS will inject power proportional to the price difference weighted by its elasticity. Similarly, if the price is lower, the ESS will demand power to charge itself, as depicted in Figure 4a.

As shown in Figure 4a, active power will be limited to the maximum possible rate of charge ($\overline{P_{ESS}^{ch}}$) and discharge ($\overline{P_{ESS}^{dch}}$) of the ESS. Additionally, when the state of charge ($SOC$) reaches the maximum level ($SO{C}_{max}$), the maximum rate of charge is restricted to zero. Conversely, if the $SOC$ reaches the minimum level ($SO{C}_{min}$), the maximum rate of discharge is zero. Equations (30) and (31) model the proposed scheme:

$$\begin{array}{cc}\hfill p_{ESS}\left(\lambda \right)& =E_{ESS}·(\lambda -{\lambda }_E))\hfill \\ \hfill -\overline{P_{ESS}^{ch}}\le & p_{ESS}\le \overline{P_{ESS}^{dch}},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \overline{P_{ESS}^{ch}}=0& \mathrm{if}SOC=SO{C}_{max}\hfill \\ \hfill \overline{P_{ESS}^{dch}}=0& \mathrm{if}SOC=SO{C}_{min}.\hfill \end{array}$$

(31)

3.2. PV Systems Modeling

To effectively model the behavior of photovoltaic (PV) systems, it is crucial to consider their unique characteristics. Unlike dispatchable units, PV systems constantly operate at their maximum available power point (MPPT), regardless of energy trading prices. This behavior stems from their reliance on solar radiation, which inherently lacks an associated cost [20,29,30]. Thus, traditional cost models, such as the quadratic function, are inadequate for capturing their operational dynamics. Despite their non-dispatchable nature, PV systems offer significant advantages. Locally generated and clean energy from PV systems reduces reliance on the grid, minimizes pollutant emissions, and contributes to overall energy sustainability. These benefits underscore the importance of accurately representing PV system behavior in energy modeling and policy frameworks.

To model the benefit perceived by the MG when generating energy with PV systems, the concept of diminishing marginal benefit from consumer theory is used. This concept establishes that the benefit or satisfaction of producing one more unit of the good or service decreases as the quantity produced increases [28], as illustrated in Figure 4b. In this case, the diminishing marginal benefit ($b_{PV}$) applies to the power generated by the PV system as follows:

$$\begin{array}{cc}\hfill b_{PV}\left(p_{PV}\right)& =-{\displaystyle \frac{1}{2}}·\alpha ·p_{PV}^2+\beta ·p_{PV},\hfill \\ \hfill \alpha & =2·{\displaystyle \frac{B_p}{{\overline{P_{PV}}}^2}},\hfill \\ \hfill \beta & =2·{\displaystyle \frac{B_p}{\overline{P_{PV}}}},\hfill \end{array}$$

(32)

where $\overline{P_{PV}}$ represents the maximum power of the generation system, and $B_p$ represents the perceived benefit of generating such power.

The total profit perceived by the MG when generating PV energy is composed of the decreasing benefit and the additional revenue from exporting or not consuming power from the grid, which is directly related to the commercialization price:

$$\begin{array}{cc}\hfill u_{PV}(p_{PV},\lambda )& =-{\displaystyle \frac{B_p}{{\overline{P_{PV}}}^2}}·p_{PV}^2+\left({\displaystyle \frac{2·B_p}{\overline{P_{PV}}}}+\lambda \right)·p_{PV.}\hfill \end{array}$$

(33)

Hence, the optimal PV generation power, which maximizes total profit, is obtained by differentiating Equation (33) with respect to $p_{PV}$ and setting the derivative to zero, yielding the following:

$$\begin{array}{cc}\hfill p_{PV}\left(\lambda \right)& ={\displaystyle \frac{\beta +\lambda }{\alpha }}\hfill \\ \hfill \underline{P_{PV}}\le & p_{PV}\le \overline{P_{PV}}.\hfill \end{array}$$

(34)

4. EMS Based on the P2P Market for MMG

This section details the modeling of MGs based on their associated load and the models of their constituent elements, ESS and PV discussed in Section 3. The previously discussed P2P market in Section 2.4 is revisited to integrate the MGs as agents. The agents shown in Figure 2 can be conventional generators, loads, or MGs. These agents interact in a dynamic P2P market where MGs aim to maximize local renewable generation and operate their ESSs based on market price signals, while conventional generators optimize their cost functions.

The EMS works as an real-time P2P market, establishing operational conditions and dispatching resources for specific time intervals, typically within the order of minutes. To solve this multi-agent system with the new MG agents, adapting the EMS proposed by Sorin et al. [26] is essential. This adaptation ensures the efficient integration and optimal operation of MG components by determining the optimal dispatch and market clearing prices in a decentralized way. The obtained clearing prices and dispatches for the time interval in which the resource allocation problem is solved will serve as inputs for the proposal of primary frequency regulation strategy based on consumer theory, wich will be explained later in Section 5.

The original algorithm by Sorin et al. is designed for systems with multiple agents categorized as either consumers or producers. While suitable for certain contexts, it does not address the integration of agents such as MGs, which are capable of assuming both consumption and generation roles, as shown in Figure 5. Therefore, there is a need to extend this algorithm to accommodate the flexibility of MGs, enabling them to dynamically adjust their behavior based on system requirements and market conditions. This extension ensures a more comprehensive and adaptable EMS for integrating diverse energy agents within the grid ecosystem. To adapt the algorithm to a MMG system, it is necessary to model the behavior of the MGs in response to variations in the energy trading price. These variations may motivate the MMGs to act as consumers or producers, depending on market conditions and their own objectives.

4.1. MG Modeling

A MG is defined as an active agent within the distribution system, integrating various interconnected components such as loads, distributed generation units, and ESS. The distinguishing characteristic of MGs is their ability to operate autonomously or connected to the main power grid [31,32].

In this paper, Equation (35) is used to model the active power of MGs as a function of the trading price. This equation is derived from the approaches in Section 3, considering a MG composed of PV systems, storage systems, and non-dispatchable loads.

$$p_{MG}\left(\lambda \right)=\sum _{i\in MG}{p_{ESS}}_i\left(\lambda \right)+\sum _{i\in MG}{p_{PV}}_i\left(\lambda \right)+\sum _{i\in MG}{P_L}_i.$$

(35)

To keep the equations simple and practical, the power calculated for each MG component based on the trading price $\lambda$ will be projected over its respective maximum and minimum power limits interval. This means that the power output or consumption for each element (ESS, PV, or load) is confined to its operational boundaries, ensuring the values remain realistic and feasible. This adjustment reflects the actual capabilities and limitations of the MG components, providing a more accurate depiction of their behavior in response to market price signals.

4.2. MG Integration into P2P Market

Considering that MGs consist of multiple elements, their total power output is derived from the sum of the powers of all individual components, dependent on the commercialization price, as detailed in the previous subsection. To incorporate MGs into the P2P market approach described in Section 2.4, they are modeled as must-take agents. These agents have the same maximum and minimum total power exchanged, but in this special case, this value is variable throughout the negotiation process with their neighbors.

The behavior of MGs in the proposed market scheme is determined by their internal optimization processes and constraints, which translate into the maximum and minimum total power exchanged, calculated using (35). To incentivize MGs to respond effectively to market conditions, $\lambda$ is set to the maximum value of all prices traded with neighboring agents for that iteration k, ensuring responsiveness to the highest price signals within their neighborhood, as shown below:

$${\underline{p_n}}^k={\overline{p_n}}^k=p_{MG}\left(\max \left({\lambda }_n^k\right)\right)\forall n\in {\Omega }_{MG}.$$

(36)

The rationale for selecting the maximum of the negotiated prices is that if the MG is consuming energy, it is the price at which it satisfies its total energy demand. Conversely, if it is exporting energy, the maximum price corresponds to the most favorable market condition with its neighbors, representing the benefit perceived by this activity. It is important to note that once the market converges, the market prices among different agents converge to the same value, making it indistinct to select the maximum market price between agents.

This means that, in each iteration, maximum and minimum power values must be updated for all agents, in contrast to the formulation in [26], where these values were kept constant. Therefore, agents that do not belong to the MG set will need to maintain constant maximum and minimum power values throughout all iterations. Considering that these parameters are now updated in each iteration, we redefine them as follows:

$$\begin{array}{cc}\hfill {\underline{p_n}}^k& ={\underline{p_n}}^{k-1}\forall n\in ({\Omega }_G\cup {\Omega }_C),\hfill \\ \hfill {\overline{p_n}}^k& ={\overline{p_n}}^{k-1}\forall n\in ({\Omega }_G\cup {\Omega }_C).\hfill \end{array}$$

(37)

For agents that do not belong to the MG set and have a dispatchable nature, the maximum and minimum power values are updated independently and are not required to be the same.

Therefore, it is necessary to include the maximum and minimum power values of each agent in the initialization of the algorithm and incorporate an intermediate step between dual and primal updates to update maximum and minimum powers using Equations (36) and (37).

For updating slack variables, the maximum and minimum power of the previous iteration are considered. Thus, (20) and (21) become the following:

$${\overline{{\mu }_n}}^k=\max (0,{\overline{{\mu }_n}}^{k-1}+{\eta }^k(p_n^{k-1}-{\overline{P_n}}^{k-1})),$$

(38)

$${\underline{{\mu }_n}}^k=\max (0,{\underline{{\mu }_n}}^{k-1}+{\eta }^k({\underline{P_n}}^{k-1}-p_n^{k-1})).$$

(39)

Moreover, to frame the optimization problem with a convex set, it is assumed that the MG follows a virtual quadratic cost function, similar to the strategy employed in [26] for non-dispatchable agents. This way, the global optimization problem remains as proposed in Equations (8) to (9), but now considers the ${\Omega }_{MG}$ subset of the $\Omega$ agents grouping MGs, and the variable power limits in each iteration. Thus, constraint (9) becomes the following:

$${\underline{p_n}}^k\le p_n\le {\overline{p_n}}^k\forall n\in \Omega .$$

(40)

The local optimization problem for agent n is shown below:

$$\begin{array}{cc}\hfill & \min c_n({\mathit{\rho }}_{\mathit{n}},{\mathit{\lambda }}_{\mathit{n}})\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\underline{p_n}}^k\le p_n\le {\overline{p_n}}^k,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & p_{nm}\ge 0\forall (n,m)\in ({\Omega }_G,{\omega }_n),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & p_{nm}\le 0\forall (n,m)\in ({\Omega }_C,{\omega }_n).\hfill \end{array}$$

(44)

Therefore, Equation (25) for the update of $p_{nm}$ as a function of agent type is reformulated as follows:

$$\begin{array}{cc}\hfill p_{nm}^k& =\max (0,p_{nm}^{k-1}+f_{nm}^k(p_n^{\left(m\right)k}-p_n^{k-1}))\forall n\in {\Omega }_G,\hfill \\ \hfill p_{nm}^k& =\min (0,p_{nm}^{k-1}+f_{nm}^k(p_n^{\left(m\right)k}-p_n^{k-1}))\forall n\in {\Omega }_C,\hfill \\ \hfill p_{nm}^k& =p_{nm}^{k-1}+f_{nm}^k(p_n^{\left(m\right)k}-p_n^{k-1}))\forall n\in {\Omega }_{MG},\hfill \end{array}$$

(45)

where $p_n^{\left(m\right)k}$ and $f_{nm}^k$ are calculated with Equations (24) and (26), respectively.

5. Proposed Strategy for Primary Frequency Regulation

Following the previously discussed proposals, this section presents an alternative approach for primary regulation in MMG, which differs from the droop strategy and the inertia simulation presented in Section 2. This approach considers the energy management system (EMS) mentioned in Section 4 as an algorithm for real-time economic dispatch, determining the system operating conditions before the frequency event and the trading price for the t interval. Additionally, the modeling of PV generation and ESS described in Section 3, based on consumer theory, is taken into account. These models can be integrated into the primary frequency regulation scheme for agents to adjust their injected/demanded power not only based on frequency deviation but also based on individual benefits. Two additional terms are proposed in the model of the PV and ESS systems to modify the instantaneous power in the event of frequency deviations.

5.1. Primary Frequency Regulation on PV Systems

Considering that PV systems are operating at their maximum power point, they can only support the system during over-frequency events by decreasing their injected power. To promote the participation of PV systems in primary frequency regulation, a compensation price ${\lambda }_c$ is proposed for the curtailment of the system production with respect to the maximum available power. This term is added to (33), obtaining the following:

$$U_{PV}(p_{PV},{\mathit{\lambda }}_{\mathit{n}}^t,{\lambda }_c)=-{\displaystyle \frac{B_p}{{\overline{P_{PV}}}^2}}·p_{PV}^2+\left({\displaystyle \frac{2·B_p}{\overline{P_{PV}}}}+\max \left({\mathit{\lambda }}_{\mathit{n}}^t\right)\right)·p_{PV}+{\lambda }_c·(\overline{P_{PV}}-p_{PV}).$$

(46)

With this modification, Equation (34) for the optimal power of the PV system becomes the following:

$$p_{PV}({\mathit{\lambda }}_{\mathit{n}}^t,{\lambda }_c)={\displaystyle \frac{\beta +\max \left({\mathit{\lambda }}_{\mathit{n}}^t\right)-{\lambda }_c}{\alpha }}.$$

(47)

5.2. Primary Frequency Regulation on ESS

Energy storage systems have the ability to react to both over- and under-frequency events, depending on the operating conditions at the time of the event. To incentivize their participation in primary frequency regulation, an additional price ${\lambda }_i$ is proposed in (30):

$$p_{ESS}({\mathit{\lambda }}_{\mathit{n}}^t,{\lambda }_i)=E_{ESS}·(\max \left({\mathit{\lambda }}_{\mathit{n}}^t\right)+{\lambda }_i-{\lambda }_E)),$$

(48)

where ${\lambda }_i$ can be interpreted as either a discount (for negative values) or a bonus (for positive values) relative to the maximum perceived market price $\max \left({\mathit{\lambda }}_{\mathit{n}}^t\right)$. This mechanism encourages ESSs to adjust their active power dispatch from the baseline level established prior to the event.

5.3. Determination of Compensation and Incentive Prices

To adjust the incentive prices ${\lambda }_i$ and compensation ${\lambda }_c$ in response to frequency events, a scheme based on the frequency deviation $\Delta f$ with respect to the nominal frequency is proposed. This scheme introduces a proportionality factor $K_p$ and a deadband $BM$ to control the agents’ response.

The proportionality factor, $K_p$, determines the magnitude of the variation of ${\lambda }_i$ and ${\lambda }_c$ relative to the frequency variation. The deadband $BM$ establishes a range of frequency deviation within which no price adjustments are applied, thus avoiding unnecessary responses to small fluctuations, as seen in Figure 6. The behavior is governed by Equations (49) and (50).

$${\lambda }_c=\left\{\begin{array}{cc}{K}_p·(\Delta f-BM)\hfill & \mathrm{if}\Delta f>BM\hfill \\ 0\hfill & \mathrm{if}\Delta f\le BM,\hfill \end{array}\right.$$

(49)

$${\lambda }_i=\left\{\begin{array}{cc}-K_p·(\Delta f-BM)\hfill & \mathrm{if}\Delta f>BM\hfill \\ -K_p·(\Delta f+BM)\hfill & \mathrm{if}\Delta f<-BM\hfill \\ 0\hfill & \mathrm{if}-BM\le \Delta f\le BM.\hfill \end{array}\right.$$

(50)

It is important to note that the maximum discount that ESSs can receive is limited by the maximum trading price received by agent n in the considered time interval. Primary frequency regulation is managed locally by each component of the MG, enabling quick and effective response times. The EMS operates as a real-time P2P market with a timescale of minutes, establishing operational conditions and trading prices for primary frequency regulation during each time interval t. Once these reference values are determined, the updates to the incentive and compensation prices are performed locally by each system through real-time frequency measurements. This ensures that primary frequency regulation does not impose a significant computational load on the local controllers of the ESS and PV systems and does not depend on communication between agents. This approach is designed to function efficiently within real-time constraints, stabilizing the grid within seconds.

6. Simulations and Results

In this section, the results of the proposed primary regulation scheme are compared with the traditional approaches of droop and virtual inertia, both implemented locally in the ESS and PV systems of the MGs. For this purpose, the simplified local distribution system of Puerto Carreño, Colombia, consisting of a biomass generation plant (5.5 MW) and three radial feeder circuits (1.3 MW each), as shown in Figure 5, is used. The scenario of high MG penetration in the system is considered; therefore, each circuit is modeled as an independent MG with a fixed load of 1.3 MW, a storage system of 0.2 MW, and a PV generation system of 0.3 MW. The parameters of the PV and ESS systems of the different MGs are presented in

Table 1. Detailed parameters of MG components and cost functions for agents in the MMG system.

PV
MG	$\underline{{\mathit{P}}_{\mathit{PV}}}$ [MW]	$\overline{{\mathit{P}}_{\mathit{PV}}}$ [MW]	${\mathit{B}}_{\mathit{p}}$ [$]		${\mathit{K}}_{\mathit{d}}$	BM [Hz]	${\mathit{K}}_{\mathit{i}}$
$M{G}_1$	0.0	0.3	0.3		20	0.3	7
$M{G}_2$	0.0	0.3	0.3		20	0.3	7
$M{G}_3$	0.0	0.3	0.3		20	0.3	7
ESS
MG	$\overline{{\mathit{P}}_{\mathit{ESS}}^{\mathit{ch}}}$ [MW]	$\overline{{\mathit{P}}_{\mathit{ESS}}^{\mathit{dch}}}$ [MW]	${\mathit{\lambda }}_{\mathit{E}}$ [$]	${\mathit{E}}_{\mathit{ESS}}$ [MW/$]	${\mathit{K}}_{\mathit{d}}$	BM [Hz]	${\mathit{K}}_{\mathit{i}}$
$M{G}_1$	0.2	0.2	0.26	3	20	0.3	7
$M{G}_2$	0.2	0.2	0.30	3	20	0.3	7
$M{G}_3$	0.2	0.2	0.36	3	20	0.3	7
Cost Functions
Agent		${\mathit{a}}_{\mathit{n}}$ [$/MW2]		${\mathit{b}}_{\mathit{n}}$ [$/MW]		${\mathit{d}}_{\mathit{n}}$ [$]
$G_o$		0.04		0.2		0.0
$M{G}_1$		0.052		0.5		0.0
$M{G}_2$		0.052		0.5		0.0
$M{G}_3$		0.052		0.5		0.0

. The table also includes the coefficients for real and virtual cost functions.

Due to the small size of the case study, the influence of the grid on short-term frequency dynamics and intra-machine power oscillations is considered negligible, allowing loads and generation to be modeled at a single bus. The system inertia constant is set to 4 seconds, following typical values for steam turbines reported in [33].

The simulation is divided into two stages. In the first stage, the initial conditions prior to the contingency are determined using the algorithm presented in Section 4. These conditions include the trading prices and the powers traded between agents. Once these initial operating conditions are determined, we move on to the second stage, where different events are simulated to evaluate the effectiveness of the primary frequency regulation strategy proposed in this work in comparison with the traditional droop strategy.

6.1. Operational Conditions Determination

The algorithm parameters and convergence criteria were chosen after a tuning process. This process involved various simulations under different operational conditions to ensure stable and optimal behavior of the system. Our approach aligns with the methodology used in [26]. Through iterative simulations and adjustments, we identified parameter values that consistently yielded reliable and efficient performance. The final parameters were determined as follows:

$$\begin{array}{cccccccc}\hfill {\beta }^k& =0.03;\hfill & \hfill {\alpha }^k& =0.01;\hfill & \hfill {\eta }^k& =0.005;\hfill & \hfill {\delta }^k& =1;\hfill \\ \hfill {ϵ}_{\lambda }& =0.0001;\hfill & \hfill {ϵ}_P& =0.0001.\hfill \end{array}$$

Additionally, all variables were initialized to zero. Figure 7 illustrates the evolution of price and power negotiation between the agents over three time periods. In each period, the load of each MG increases, starting with 0.8 MW in the first period, 1 MW in the second, and reaching 1.3 MW in the last one.

Power dispatches resulting from the EMS for each time period are detailed in

Table 2. Agent dispatches across different time intervals.

Agent		${\mathit{t}}_0$ [MW]	Total ${\mathit{t}}_0$ [MW]	${\mathit{t}}_1$ [MW]	Total ${\mathit{t}}_1$ [MW]	${\mathit{t}}_2$ [MW]	Total ${\mathit{t}}_2$ [MW]
	PV	0.3000		0.3000		0.3000
$M{G}_1$	ESS	0.03098	−0.4690	0.08903	−0.6109	0.1694	−0.8305
	LOAD	−0.8000		−1.0000		−1.3000
	PV	0.3000		0.3000		0.3000
$M{G}_2$	ESS	−0.0890	−0.5890	−0.0309	−0.7309	0.0494	−0.9505
	LOAD	−0.8000		−1.0000		−1.3000
	PV	0.3000		0.3000		0.3000
$M{G}_3$	ESS	−0.2000	−0.6999	−0.2000	−0.9000	−0.1306	−1.1306
	LOAD	−0.8000		−1.0000		−1.3000
$G_o$		1.7580	1.7580	2.2419	2.2419	2.9117	2.9117

. For the final time interval $t_2$, the agents converge to a trading price of USD 0.316471. This price and the final dispatches will be taken into account as initial operational conditions for the proposed frequency regulation scheme.

The total iterations needed for the convergence of each case and the total time it takes, are reported in

Table 3. Total iterations and time taken by the EMS to achieve convergence.

Time Interval	Total Iterations	Time of Convergence [s]
$t_o$	594	0.619
$t_1$	140	0.133
$t_2$	127	0.176

.

The algorithm was implemented using Python 3.12 and executed on a system with an AMD Ryzen 7 3700U processor at 2.3 GHz and 20 GB of RAM. The results in Table 3 demonstrate that the algorithm can be effectively used as a real-time EMS within a time window of minutes. The convergence times and the number of iterations required for each time interval provide a practical indication of its applicability, especially for small systems.

6.2. Analysis of Frequency Regulation Schemes

Considering the operational conditions obtained in the previous step for $t_2$, a load shedding event is simulated, representing a 1 MW load reduction in $M{G}_3$. The system’s performance in response to this event is compared by analyzing the synchronous generator, PV systems, ESS, and system frequency response. This comparison applies both the traditional primary frequency regulation scheme for ESS and PV, using the parameters mentioned at the beginning of this section, and the proposed scheme with $K_p=1.5$. The simulations are conducted using the DigSilent dynamic simulation software, utilizing typical library models for the different systems. The local controllers for ESS and PV are programmed to implement the proposed primary frequency regulation strategy.

Figure 8 and Figure 9a,b illustrate the comparison between the conventional frequency regulation strategy, the proposed strategy, and the system response without frequency regulation control on PV systems and ESS under a load shedding event.

Figure 8 shows the frequency deviation and the power output of the synchronous generator ($P_{G_o}$) in different scenarios. It is evident that the frequency deviation in the proposed strategy is lower compared to the conventional strategy, indicating better frequency stability and faster recovery. Additionally, the synchronous generator’s power output stabilizes more quickly in the proposed strategy, reflecting a more efficient and balanced system response.

Figure 9a highlights the behavior of PV systems under different frequency regulation strategies. In the conventional strategy, all devices, including PV systems, reduce their active power output in response to the frequency deviation. In contrast, the proposed strategy ensures that PV systems do not curtail their active power unless the ESS alone is insufficient to balance the system. This selective curtailment prevents unnecessary reduction in PV generation, optimizing the use of renewable energy sources.

Figure 9b displays the response of ESS in different scenarios. In the conventional strategy, ESS devices reduce their active power output along with PV systems. In the proposed strategy, ESS devices are primarily responsible for balancing the system during frequency deviations. PV systems only contribute to frequency regulation when support from the ESS is insufficient. This hierarchical approach ensures that ESS is fully utilized before curtailing PV generation.

Overall, the proposed primary frequency regulation strategy based on consumer simulations demonstrates several advantages over the conventional approach. By prioritizing the full potential of ESS before involving PV systems in frequency regulation, the proposed strategy minimizes the curtailment of renewable energy sources during over-frequency events. This ensures that excess energy can be efficiently stored in ESS, which can then be utilized during under-frequency events. Consequently, this approach not only enhances energy efficiency but also maximizes the utilization of renewable energy, contributing to a more sustainable power system. Additionally, the lower frequency deviation and faster stabilization indicate that the proposed strategy effectively maintains system stability.

Another key advantage of the proposed frequency regulation scheme is its ability to operate independently from inter-agent communication. The compensation and incentive prices for frequency regulation are updated locally by each component of the MG, ensuring coordinated operation without the need for direct communication between agents. Communication between agents is only required within EMS, which operates as an real-time market, setting the operational conditions for the specific time interval t in which the proposed frequency regulation scheme operates. This independence significantly enhances the system’s robustness and reliability, ensuring that the scheme can run in real-time based only on local measurements.

These results highlight the potential benefits of integrating consumer theory-based models into primary frequency regulation strategies, particularly in systems with a high penetration of renewable energy sources. Consumer theory, despite its simplicity, allows for a better-coordinated response, as demonstrated in the simulations. By improving energy efficiency, maximizing renewable energy utilization, and enhancing system stability, the proposed approach offers a compelling alternative to traditional frequency regulation methods.

7. Conclusions

This study presents a new approach to regulate primary frequency in multi-microgrid systems using consumer theory within a peer-to-peer energy management framework. Unlike traditional methods, this approach utilizes energy storage systems (ESS) and photovoltaic (PV) systems more selectively for frequency regulation. The proposed strategy significantly reduces frequency deviations, demonstrating better stability compared to traditional droop control and virtual inertia methods. By prioritizing ESS for frequency regulation, the strategy minimizes unnecessary reductions in PV power, ensuring PV systems only reduce their output when ESS alone cannot balance the system, thereby optimizing the use of renewable energy.

This hierarchical approach ensures that ESS resources are fully utilized before involving PV systems, resulting in a more balanced and efficient response to frequency changes. Additionally, incorporating consumer theory into the energy management system makes the frequency regulation strategy more flexible and adaptable, allowing it to scale for different multi-microgrid system configurations. Overall, the proposed primary frequency regulation method offers a practical and effective alternative to traditional approaches, especially in power systems with high levels of renewable energy.

Future work could focus on refining compensation and incentive mechanisms to improve the economic appeal and acceptance of these strategies among various stakeholders and more complex power systems.