Secondary voltage and frequency regulation for islanded microgrids: A model predictive coordinated control approach

Abstract

The hierarchical control strategy enables seamless integration of power electronics-based distributed generators (DGs) into microgrids (MGs). This work presents a distributed coordinated secondary voltage and frequency control strategy of islanded MGs, utilizing distributed model predictive control (DMPC) approach. The Proposed approach is entirely distributed in nature, with plug and play features and requires minimal communication network with information exchange between immediate neighbors. This technique facilitates efficient information exchange and obviates the need for centralized control. Droop control is the primary controller for all the DGs, which leads to the deviations in DG voltage and operating frequency from standard nominal values. Secondary control enables the restoration of DG voltage and frequency to their nominal values. Input–output feedback linearization (IOFL) is used to transform nonlinear dynamics of inverter-interfaced DGs into first-order linear dynamics between input and output. Compared to some published works, the single-integrator dynamics eliminates the need to compute the derivative of the output voltage. Furthermore, DMPC has been used for all DG units as secondary controller, to achieve voltage and frequency regulation. This approach takes advantage of distributed architecture with optimal control actions and enhances reliability and flexibility of MG operations. Moreover, the proposed approach ensures that voltage and frequency of overall system are restored to its reference value while maintaining accuracy in real power sharing. Effectiveness of proposed control scheme is validated with simulations conducted on islanded MG test system using MATLAB/Simulink toolbox.

Keywords

Cooperative control microgrid secondary control model predictive control

Introduction

Concept of microgrid (MG) has emerged as a solution to the unpredictable performance of renewable energy sources, caused by their intermittent nature which can significantly impact grid stability (Olivares et al., 2014; Zhang et al., 2023). Typically an MG comprises of distributed generators (DGs) including wind energy, photovoltaic (PV), and energy storage systems (ESS) which work in cohesion and facilitate the optimized use of renewable energy sources (Shan et al., 2021). Typically, MGs operate in grid-connected mode, wherein main grid decides voltage and frequency, and control algorithms strive to achieve these references and ensure stability. However, in islanded operational mode, stability of MG is susceptible to disturbances like load changes (Jamali et al., 2023). In general, hierarchical control strategy is employed to attain control objectives of an islanded MG, which includes primary, secondary, and tertiary control levels (Bagheri Rouch and Fakharian, 2022; Li et al., 2018). Primary control in DG is implemented using droop control strategy.

In conventional power systems, synchronous generators possess inertia, which causes frequency deviations in response to load changes and helps to maintain system synchronism. However, inverter interfaced generators lack the inertia. Hence, in inverter interfaces, DG’s intentional droop in frequency and voltage is introduced and the scheme is called droop control strategy. This helps to keep the MG synchronized without any communication between the DGs. Droop control, which emulates the operating performance of synchronous generators, is typically applied to every DG for primary control in an MG for frequency and output voltage stabilization. Primary control of DG is implemented by inner current, outer voltage control loop, and a power controller (Lou et al., 2016; Singh and Maulik, 2023). Inherent limitation of droop control approach is voltage and frequency deviates from nominal values during load increase (Panda and Subudhi, 2023). Hence secondary control is introduced to restore deviations in MG parameters that occur as a result of primary control. Frequency deviation, inaccuracy in power sharing, and voltage adjustment need to be managed and mitigated by secondary control (Li et al., 2017; Simpson-Porco et al., 2015). Optimizing power flow and managing economic dispatch are the key responsibilities of tertiary control.

A centralized control strategy is conventionally utilized in secondary control of an MG. It relies on information being passed from every single DG to one central computing unit. The central computing unit decides the operating point for each DG and sends the corresponding control signals. Wen et al. (2016) presents a frequency regulation of source-grid-load system, where a centralized MPC approach is applied for allocating the active power to each generator in a power grid. Centralized approach compromises system’s overall reliability, even make it more prone to failures because of single point vulnerability (Cady et al., 2015; Zhang et al., 2023). To address the aforementioned drawbacks, a multi-agent system (MAS)-based distributed control approach has recently been reported in the literature.

The distributed control scheme operates by coordinating local control decisions of DGs with the information from the neighboring DGs. This control technique enhances both the reliability and scalability of the system due to its sparse communication network requirement and decentralized calculation of the control signals (Cady et al., 2015; Guo et al., 2014; Gurmu and Hu, 2025; Liu et al., 2015). A surge in research focused on multi-agent-based cooperative theory has emerged and is yielding significant results in the domain of voltage and frequency restoration in MG (Arora et al., 2024; Babayomi et al., 2020; Bidram et al., 2013; Simpson-Porco et al., 2015). For islanded MGs, Simpson-Porco et al. (2015) proposes a distributed strategy for voltage and frequency control that combines proportional, integral control, and distributed averaging. While this method offers a adjustable trade-off in voltage regulation and reactive power sharing, it overlooks critical need for synchronization of DGs. In Bidram et al. (2013), fully distributed secondary voltage control is proposed, in which highly nonlinear dynamics of DG is tackled using feedback linearization and a linear second-order tracking synchronization problem is formed. Subsequently, the controller gains are found by solving the algebraic Riccati equation. To overcome the uncertainties in the system and communication links, secondary control schemes in distributed manner have been presented in the literature (Ahmed, 2023; Lee et al., 2022; Panda and Subudhi, 2023; Yu et al., 2022).

Other aspects of recent research in distributed control of MG is adaptive and event triggered secondary control schemes, which have been used to handle uncertainties in the system and reduce the communication burden (Ashrafi et al., 2025; Chen et al., 2024a; Ma et al., 2025; Zhao et al., 2023). A distributed adaptive secondary control scheme for frequency regulation in networked MGs is proposed by Kandasamy et al. (2024) using a leader–follower consensus mechanism. To enhance dynamic performance of secondary control of an MG, finite-time convergence algorithms (Chen et al., 2024b; Li et al., 2018; Ning et al., 2020; Pilloni et al., 2017), and for uncertainties sliding-mode control (Abianeh et al., 2021; Alfaro et al., 2021) have been proposed. While these control methods are reliable, they have certain limitations compared to prediction-based algorithm. For instance, they do not explicitly handle constraints such as voltage and frequency limits and lack predictive capabilities. This serves as a motivation for adopting MPC as a secondary controller. Due to the limitations of both droop control and centralized control approaches, there is a clear need for a distributed secondary control strategy in MGs. This is where MPC comes into play. One of the key motivations for adopting MPC is its ability to effectively handle physical constraints such as voltage and frequency limits. In addition, MPC’s predictive and feedback-based nature makes it well-suited for managing MG dynamics, as it helps minimize system disturbances and enhances overall stability.

MPC is a control strategy that optimizes future system behavior by solving an optimization problem at each time step. It effectively handles constraints while finding the control action for the next time step. MPC controller has been designed to address voltage and frequency regulation in an MG. In the published results Lou et al. (2018), Panda and Subudhi (2023) and Bidram et al. (2013) have used MPC as a secondary controller along with feedback linearization. The linearized dynamics used is of second order, with the output voltage and its derivative as the state variables. Both of these state variables are necessary to compute the control action. While the output voltage is directly measurable, the derivative of the output voltage must be synthesized or estimated. However, the published results do not mention how this derivative is obtained. Moreover, the derivative of the output is not a physical variable and hence cannot be measured with the sensor. In contrast, the present work employs a first-order linearized map between input and output, which eliminates the need to calculate the derivative of the output voltage signal.

While MPC controllers have been designed to address voltage and frequency regulation in an MG, they have also successfully addressed challenges like seamless transition from islanded operation to resynchronization with the main grid (Fachini et al., 2024). The study in Kiani et al. (2024) proposes a learning-based MPC scheme for voltage restoration in islanded MGs, integrating Gaussian process regression to predict load uncertainties.

In summary, to ensure the performance of the MG with minimal communication network distributed control is emerging as a key solution. While droop control is invariably the choice for primary control, for secondary control strategies like MPC, finite time convergence control and sliding mode control have successfully been deployed. The key objectives associated with MG secondary control are: (1) effective voltage and frequency restoration while ensuring accurate power sharing via a sparse communication topology, (2) secondary control should achieve fast convergence of voltage and frequency, (3) a robust control scheme is desirable to handle the various uncertainties in DG and MG parameters, and (4) DG’s secondary controllers for islanded MGs in remote areas should have limited tunable parameters and be able to adjust to the variety of operating environments. Model predictive control (MPC) strategy is gaining popularity for its ability to meet the objective optimally and simultaneously handle system uncertainties and the constraints (Aragon et al., 2021; Lou et al., 2016; Shabbir et al., 2022).

In the present work, the problem of voltage and frequency deviation from their nominal values is addressed using distributed model predictive control (DMPC)-based secondary control of DGs. The problem is formulated as a tracker synchronization task, wherein the output voltage and frequency of each DG aim to follow the given reference values. Although the reference is available to only one or a few nodes, the connected nature of the communication graph ensures that all nodes eventually converge to the specified reference values. Since the dynamics of the inverter interfaced DGs is a nonlinear dynamics, the predictive control problem evolves as a nonlinear MPC (NMPC). Furthermore, the optimization to be solved to find the controller also becomes nonconvex. This significantly complicates the solution procedures and requires nonlinear programming (NLP) solvers which are computationally expensive. However the issue has been addressed using input–output feedback linearization (IOFL). Using IOFL, the input–output map is linearized, and hence, the problem can now be addressed using Linear MPC. In linear MPC, to find the controller, convex optimization problem is solved using QP solvers. Thus IOFL is applied to get first-order linear model between input and output and some internal dynamics. It simplifies the control approach, as there is no need to handle complex nonlinear dynamics, which further reduces the complexity of the optimal control computation. The proposed DMPC approach operates in an online mode, where each DG unit independently computes its optimal control actions at every control interval, using local measurements and communication with neighboring units. The control strategy follows a receding horizon principle, allowing each DG to iteratively predict and adjust its inputs based on the latest system state and exchanged neighbor data (Foss and Heirung, 2013; Reigstad and Uhlen, 2021). Key contributions of the present work are as follows:

The proposed method employs IOFL to represent the DG’s nonlinear dynamics into a simplified first-order (single integrator) linear dynamics between input and output. This has an advantage over published IOFL technique that yields second-order dynamics and necessitate to compute output voltage derivative—a requirement that introduces noise sensitivity. By eliminating this dependency, the approach enhances system robustness while reducing implementation complexity.

This work proposes a DMPC-based secondary controller for MG voltage and frequency regulation and ensures voltage and frequency restoration to nominal values, with particular emphasis on achieving accurate proportional power sharing among DGs according to their respective ratings. For the frequency regulation, the designed MPC controller makes use of first-order model dynamics. Compared to the published results where PI controller has been used, the MPC gives better performance being predictive and optimal controller. The DMPC framework simultaneously ensures frequency restoration to nominal values while maintaining power sharing.

The proposed method introduces a fully distributed predictive control framework with plug and play capability, allowing each DG to make optimal decisions based on local and neighboring data. It requires minimal communication infrastructure while ensuring coordinated secondary control.

Hierarchical control framework of MG

MG is a highly distributed system in which the DGs are spread across a wide area. To control the MG, a hierarchical, three-level control structure is adopted. Each level of the hierarchical control architecture of the MG has a specific objective. The principal objectives of the primary control are proportional power sharing between each DG and systematic voltage/frequency stabilization. The secondary control eliminates the inherent deviations introduced by the primary control in preparation for grid connection. At the top of the hierarchy, tertiary control manages power flows between the MG and the main grid, as well as the economic power flow within the MG (Ashtari Mahini et al., 2024; Khosravi et al., 2020).

Primary control

Primary control is referred as the local control for a DG. It operates at the lowest level of the hierarchy and is the quickest to respond to the changes. The droop control scheme is virtually implemented in each DG of the MG to achieve proportional power sharing, formulated as follows:

\begin{matrix} v_{m} = V_{n} - nQ \\ ω = ω_{n} - mP \end{matrix}

(1)

where $V_{n}$ and $ω_{n}$ represent the nominal levels of voltage and frequency of an MG. Voltage and frequency produced through primary control at DG terminal is represented by $v_{m}$ and ω. According to the DG ratings, droop coefficients mentioned as m and n are selected. P and Q represent the instantaneous active and reactive power on DG terminal. Droop coefficients m and n for a DG are selected as:

\begin{matrix} m = \frac{ω_{\max} - ω_{\min}}{P_{\max} - P_{\min}} \end{matrix}

(2)

\begin{matrix} n = \frac{V_{\max} - V_{\min}}{Q_{\max} - Q_{\min}} \end{matrix}

(3)

where maximum and minimum permitted voltage is denoted by $V_{\max}$ and $V_{\min}$ , respectively, and maximum and minimum allowable frequency range of the MG is given as $ω_{\max}$ and $ω_{\min}$ . $P_{\max}$ , $P_{\min}$ and $Q_{\max}$ , $Q_{\min}$ denote the permitted active and reactive power limits of specific DG (Wang et al., 2020).

Secondary control

Unlike synchronous generators, the inverter interfaced generators are inertia-less. Thus, an intentional droop in frequency and voltage is introduced to keep the MG synchronized without any communication between the DGs. Due to the nature of droop control, there will always be variations from the rated voltage and frequency. The idea of secondary control is to mitigate the deviations in voltage magnitude and network frequency. Secondary control shifts the entire droop curve with an additional offset in vertical direction, which can be expressed as follows:

\begin{matrix} v_{m} = V_{n} - nQ + Δ v \\ ω = ω_{n} - mP + Δ ω \end{matrix}

(4)

Conventionally, secondary control is attained through centralized control strategies. Centralized control suffers from the requirement to collect vast amounts of data from each DG through communication link and process it in a centralized computing unit. However, in the present work, cooperative control is used where the offsets are decided based on the information shared among neighboring units.

Large-signal dynamical model of inverter-interfaced DG

A large signal dynamical model of inverter-interfaced DG serves as a foundation for the design of control technique. As seen in Figure 1, each DG contains prime DC source that is typically interfaced with an MG system by voltage source inverter (VSI), an inductor–capacitor (LC) filter, and resistor–capacitor (RC) output connector. Primary control of each DG consists of three control loops: external power control, inner current control, and voltage control loop. Output voltage magnitude and operating frequency references are determined by power controller on the basis of droop coefficients selected for real and reactive powers. Based on voltage reference generated by the power controller, the voltage controller tries to achieve this target value. To do so, the voltage controller generates a current reference signal, which is then used by the current controller (Bidram et al., 2013; Pogaku et al., 2007).

Figure 1.

Inverter-interfaced DG.

The dynamic model of each DG is developed in its own direct-quadrature (d-q) reference frame. It is assumed that the reference frame of the $i^{th}$ DG is rotating with frequency $ω_{i}$ . Reference frame of one of the DG is chosen as a common reference frame which is rotating with the frequency $ω_{com}$ . Angle between the common reference frame and the $i^{th}$ DG is represented by $δ_{i}$ , which obeys the following differential equation:

{\overset{\cdot}{δ}}_{i} = ω_{i} - ω_{com}

(5)

All reference frames rotate at common angular frequency because of the frequency droop features given in (1). Droop technique is applied in power controller, and it provides voltage reference values $v_{odi}^{*}$ and $v_{oqi}^{*}$ to voltage controller. Power controller also provides reference operating frequency $ω_{i}$ to the PWM generator for the operation of inverter bridge of the DG. For extracting fundamental components of active ( $P_{i}$ ) and reactive ( $Q_{i}$ ) power, two low-pass filters are used, each having a cutoff frequency $ω_{c}$ . Following differential equations represents the power controller (Bidram et al., 2013; Pogaku et al., 2007):

\begin{matrix} \overset{\cdot}{P_{i}} = - ω_{ci} P_{i} + ω_{ci} (v_{odi} i_{odi} + v_{oqi} i_{oqi}) \end{matrix}

(6)

\begin{matrix} \overset{\cdot}{Q_{i}} = - ω_{ci} Q_{i} + ω_{ci} (v_{oqi} i_{odi} - v_{odi} i_{oqi}) \end{matrix}

(7)

Direct and quadrature component of output voltage $v_{oi}$ and current $i_{oi}$ seen in Figure 1 are represented in above equation as $v_{odi}, v_{oqi}, i_{odi}$ , and $i_{oqi}$ . Each DG’s output voltage magnitude is aligned with d-axis of its reference frame through primary control strategy, and thus:

\begin{matrix} v_{odi}^{*} & = V_{ni} - n Q_{i} \\ v_{oqi}^{*} & = 0 \end{matrix}

(8)

The primary control strategy includes a voltage controller. The following differential and algebraic equations represent the behavior of the voltage controller:

\begin{matrix} {\overset{\cdot}{ϕ}}_{di} & = v_{odi}^{*} - v_{odi} \end{matrix}

(9)

\begin{matrix} {\overset{\cdot}{ϕ}}_{qi} & = v_{oqi}^{*} - v_{oqi} \end{matrix}

(10)

\begin{matrix} i_{ldi}^{*} & = F_{i} i_{odi} - ω C_{fi} v_{oqi} + K_{PV i} (v_{odi}^{*} - v_{odi}) + K_{IV i} ϕ_{di} \end{matrix}

(11)

\begin{matrix} i_{lqi}^{*} & = F_{i} i_{oqi} + ω C_{fi} v_{odi} + K_{PV i} (v_{oqi}^{*} - v_{oqi}) + K_{IV i} ϕ_{qi} \end{matrix}

(12)

where ω represents the nominal angular frequency, and $ϕ_{di}$ and $ϕ_{qi}$ are the auxiliary state variables defined for PI controller implementation with proportional and integral gains as $K_{PV i}$ and $K_{IV i}$ , respectively. Figure 1 shows additional parameters in the aforementioned equations.

Standard proportional–integral (PI) controller-based current controller provides the reference to bridge inverter. Following differential equations define the model of the current controller:

\begin{matrix} {\overset{\cdot}{γ}}_{di} & = i_{ldi}^{*} - i_{ldi} \end{matrix}

(13)

\begin{matrix} {\overset{\cdot}{γ}}_{qi} & = i_{lqi}^{*} - i_{lqi} \end{matrix}

(14)

\begin{matrix} v_{idi}^{*} & = - ω L_{fi} i_{lqi} + K_{PCi} (i_{ldi}^{*} - i_{ldi}) + K_{ICi} γ_{di} \end{matrix}

(15)

\begin{matrix} v_{iqi}^{*} & = ω L_{fi} i_{ldi} + K_{PCi} (i_{lqi}^{*} - i_{lqi}) + K_{ICi} γ_{qi} \end{matrix}

(16)

where $i_{ldi}$ and $i_{lqi}$ are the direct and quadrature parts of $i_{li}$ as shown by Figure 1. Auxiliary state variables $γ_{di}$ and $γ_{qi}$ are defined for PI controllers with $K_{PCi}$ and $K_{ICi}$ as proportional and integral gains.

Following differential equations model, the output LC filter and output connector circuit are depicted in Figure 1:

\begin{matrix} {\overset{\cdot}{i}}_{ldi} & = - \frac{R_{fi}}{L_{fi}} i_{ldi} + ω_{i} i_{lqi} + \frac{1}{L_{fi}} v_{idi} - \frac{1}{L_{fi}} v_{odi} \end{matrix}

(17)

\begin{matrix} {\overset{\cdot}{i}}_{lqi} & = - \frac{R_{fi}}{L_{fi}} i_{lqi} - ω_{i} i_{ldi} + \frac{1}{L_{fi}} v_{iqi} - \frac{1}{L_{fi}} v_{oqi} \end{matrix}

(18)

\begin{matrix} {\overset{\cdot}{v}}_{odi} & = ω_{i} v_{oqi} + \frac{1}{C_{fi}} i_{ldi} - \frac{1}{C_{fi}} i_{odi} \end{matrix}

(19)

\begin{matrix} {\overset{\cdot}{v}}_{oqi} & = - ω_{i} v_{odi} + \frac{1}{C_{fi}} i_{lqi} - \frac{1}{C_{fi}} i_{oqi} \end{matrix}

(20)

\begin{matrix} {\overset{\cdot}{i}}_{odi} & = - \frac{R_{ci}}{L_{ci}} i_{odi} + ω_{i} i_{oqi} + \frac{1}{L_{ci}} v_{odi} - \frac{1}{L_{ci}} v_{bdi} \end{matrix}

(21)

\begin{matrix} {\overset{\cdot}{i}}_{oqi} & = - \frac{R_{ci}}{L_{ci}} i_{oqi} - ω_{i} i_{odi} + \frac{1}{L_{ci}} v_{oqi} - \frac{1}{L_{ci}} v_{bqi} \end{matrix}

(22)

Large-signal dynamic model of inverter-based $i_{th}$ DG is represented by (5)–(22), and it is expressed as:

\begin{matrix} ẋ_{i} & = f_{i} (x_{i}) + k_{i} (x_{i}) D_{i} + g_{i} (x_{i}) u_{i} \\ y_{i} & = h_{i} (x_{i}) \end{matrix}

(23)

where $x_{i}$ represents state vector, and $D_{i}$ represent the known disturbance as given below. $f_{i} (x_{i})$ , $g_{i} (x_{i})$ , and $k_{i} (x_{i})$ may be extracted from equations (5) to (22):

\begin{matrix} x_{i} & = {[δ_{i} P_{i} Q_{i} ϕ_{di} ϕ_{qi} γ_{di} γ_{qi} i_{ldi} i_{lqi} v_{odi} v_{oqi} i_{odi} i_{oqi}]}^{T} \end{matrix}

(24)

\begin{matrix} D_{i} & = {[ω_{com} v_{bdi} v_{bqi}]}^{T} \end{matrix}

(25)

Secondary voltage control approach works by selecting the nominal voltage $V_{ni}$ for the individual DG in a way that the output voltage at every single DG terminal approaches to reference $v_{odi}$ → $v_{ref}$ . Thus, the output is set as $y_{i}$ = $v_{odi}$ , and the input is set to $u_{i}$ = $V_{ni}$ . Secondary frequency control approach sets $ω_{ni}$ in a way that each DG’s frequency approaches to $ω_{ref}$ .

Secondary control based on DMPC

Graph theory and tracking synchronization

In an islanded MG with n DGs, where DGs can share information to their neighbors via a communication network forms a network. By considering DGs as agents, islanded MG can be considered as an MAS. This MAS can be modeled as a directed graph, commonly referred to as digraph. Digraph is typically expressed by $G = (V, E, A)$ , where $V = (v_{1}, v_{2}, \dots, v_{n})$ represents the set of n-nodes, E ⊂ V × V is the set of edges, and $A = [a_{ij}]$ represents the associated adjacency matrix. For an MG graph, DG units represented as nodes and communication links denoted as edges of the digraph. When node j obtain information through node i, corresponding edge is denoted as $(v_{j}, v_{i})$ . Weight of edge $(v_{j}, v_{i})$ is denoted as $a_{ij}$ , and $a_{ij} = 1$ if $(v_{j}, v_{i})$ ∈ E, otherwise $a_{ij} = 0$ . When $(v_{j}, v_{i})$ ∈ E, node i is referred as the neighbor node of j. Typically, node j gather data through its immediate neighboring nodes represented as $N_{j}$ ={ $i | (v_{i}, v_{j})$ ∈ E}. In case of digraph if node i is the immediate neighbor of node j, then node j will take data by node i, but the vice-versa is not necessarily true. Flow of information is encoded by graph’s Laplacian matrix, which is defined by $L = D - A$ , here D represents the in-degree matrix, $D = diag {d_{i}}$ with $d_{i} = Σ_{j \in N_{i}} a_{ij}$ . Under cooperative control, if node i represents the value $y_{i}$ , all nodes are expected to converge to the reference given to the leader node, that is, $y_{i} = y_{j} = y_{ref}$ , $\forall i, j \in N$ .

IOFL

The dynamical model of the DGs is nonlinear. In the present work, the IOFL approach has been used to develop a linear model between the input and the output. The approach is discussed below. Consider a affine nonlinear system:

\begin{matrix} ẋ & = f (x) + g (x) u \\ y & = h (x) \end{matrix}

(26)

where $f, g$ , and h are sufficiently smooth in a domain $D \subset ℝ^{n}$ . The time derivative of output ẏ can be given as:

\begin{matrix} ẏ & = \frac{∂h}{∂x} (x) f (x) + \frac{∂h}{∂x} (x) g (x) u \end{matrix}

(27)

If $\frac{∂h}{∂x} (x) g (x) u \neq 0$ for any $x \in D_{0}$ , then the relative degree of the system is one on $D_{0}$ . It denotes that input u appears in differential equation at first derivative. If $\frac{∂h}{∂x} (x) g (x) u = 0$ , we keep differentiating until the input u appears. To define the higher-order derivatives, it is convenient to introduce Lie derivative. The Lie derivative of h with respect to f is given as:

\begin{matrix} L_{f} h (x) = \frac{∂h}{∂x} (x) f (x) \end{matrix}

(28)

The notation is convenient for repeated derivatives given as:

\begin{matrix} L_{f}^{k} h (x) & = L_{f} L_{f}^{k - 1} h (x) = \frac{\partial}{∂x} (L_{f}^{k - 1} h (x)) f (x) \end{matrix}

(29)

\begin{matrix} L_{g} L_{f}^{k} h (x) & = \frac{\partial}{∂x} (L_{f}^{k} h (x)) g (x) \end{matrix}

(30)

According to the Lie derivative, if $L_{g} h (x) = 0$ , we keep on taking derivatives until $L_{g} L_{f}^{r - 1} h (x) \neq 0$ that denotes input u appears in the equation $y^{r}$ of the output. If $L_{g} L_{f}^{i} h (x) = 0$ for $i = 1, 2, …r - 2$ and $L_{g} L_{f}^{r - 1} h (x) \neq 0$ , then the nonlinear system is said to have a relative degree r. If the system having the relative degree r, then $y^{(r)} = L_{f}^{r} h (x) + L_{g} L_{f}^{r - 1} h (x) u$ , and hence, the system is input–output linearizable, with the input feedback control as:

\begin{matrix} u = \frac{1}{L_{g} L_{f}^{r - 1} h (x)} [- L_{f}^{r} h (x) + v] \end{matrix}

(31)

This gives the input–output linear map as $y^{(r)} = v$ . This structure essentially forms a chain of r integrators, which can be easily controlled by choosing a suitable auxiliary control input v. However, when $r \neq n$ , it means that some of the system’s states specifically $n - r$ of them are not controlled by control input u. These unaffected states are known as zero dynamics of the system (Farrell and Polycarpou, 2006; Khalil, 2015). It is commonly assumed that the system’s zero dynamics are stable—an assumption adopted by many existing studies and approaches. In the present work, the linear input–output model is considered to be of first order, and the rest of the internal dynamics is assumed to be stable, as supported by previously published results (Huang et al., 2022; Zhao et al., 2024).

DMPC-based voltage restoration

This section presents a linear DMPC-based approach which synchronizes the output voltage $v_{o}$ for individual DG with the provided reference voltage. DG’s output voltage is aligned with the d-axis of the reference frame, and thus, direct axis voltage $v_{od}$ of the DGs is to be synchronized with the reference voltage. The primary aim of secondary control is the selection of appropriate $V_{ni}$ value, the control input, so as to mitigate effects of droop control. Dynamics of voltage and current control loop is quicker than power control loop. Thus, ignoring faster dynamics (8) can be written as:

\begin{matrix} v_{odi} & = V_{ni} - n_{i} Q_{i} \\ v_{oqi} & = 0 \end{matrix}

(32)

Differentiating (32) gives:

\begin{matrix} {\overset{\cdot}{v}}_{odi} = {\overset{\cdot}{V}}_{ni} - n_{i} {\overset{\cdot}{Q}}_{i} = u_{i} \end{matrix}

(33)

here $u_{i}$ is auxiliary control input. With the choice of auxiliary control input, input ( $u_{i}$ )–output ( $v_{odi}$ ) dynamics is a first-order linear dynamics. Even though complete dynamics of DG is nonlinear, the input–output map is linearized. This technique is called the IOFL (Khalil, 2015). Using this first-order linear dynamics, first, the appropriate value of $u_{i}$ , to synchronize output voltage, is found, and subsequently, the actual control action $V_{ni}$ is evaluated from $u_{i}$ . Nonlinear dynamics of DGs transformed to a linear state-space model through IOFL approach is given as:

\begin{matrix} \overset{\cdot}{z_{i}} & = A_{i} z_{i} + B_{i} u_{i} \\ y_{i} & = C_{i} z_{i} \end{matrix}

(34)

where $z_{i} = [v_{odi}]$ , $A_{i} = [0]$ , $B_{i} = [1]$ , and $C_{i} = [1]$ . Each DG communicates with their nearest neighbors only via sparse communication network. Few DGs act as leader nodes, having direct access of reference $v_{ref}$ . So, the key objective of secondary control is to determine distributed $u_{i}$ in (34) in order for $y_{i}$ → $v_{ref}$ $\forall i$ . To accomplish this goal, a linear DMPC controller is used in this work.

A discrete-time model is established to support iterative nature of the MPC algorithm. Continuous time state-space system given in (34) can be formulated in discrete time model using the Forward Euler discretization technique. In this method, derivative is approximated as:

\begin{matrix} \frac{z_{i} (k + 1) - Z_{i} (k)}{T} \end{matrix}

(35)

and thus, the discrete-time state-space model is written as:

\begin{matrix} z_{i} (k + 1) & = A_{i} z_{i} (k) + B_{i} u_{i} (k) \\ y_{i} (k) & = C_{i} z_{i} (k) \end{matrix}

(36)

here, $A_{i} = 1$ and $B_{i} = T$ . In MPC approach, the output of the system is predicted using the model up to the time $T_{P}$ called prediction horizon. Based on the predicted output values of the system, the desired control actions are calculated over a time interval $T_{U}$ , known as the control horizon. The future evolution of the output $y_{i}$ of all DG units can be evaluated for $T_{P}$ steps ahead by considering the current state $Z_{i} (k)$ and the discrete time model given in (36) as (Bordons et al., 2020; Lou et al., 2018):

\begin{matrix} y_{i} (k + 1 | k) & = C_{i} A_{i} z_{i} (k) + C_{i} B_{i} u_{i} (k) \end{matrix}

(37)

\begin{matrix} y_{i} (k + 2 | k) & = C_{i} A_{i}^{2} z_{i} (k) + C_{i} A_{i} B_{i} u_{i} (k) + C_{i} B_{i} u_{i} (k + 1) \end{matrix}

(38)

\begin{matrix} y_{i} (k + T_{P} | k) & = C_{i} A_{i}^{T_{P}} z_{i} (k) + C_{i} A_{i}^{T_{P} - 1} B_{i} u_{i} (k) \\ + \dots + C_{i} A_{i}^{T_{P} - T_{U}} B_{i} u_{i} (k + T_{U} - 1) \end{matrix}

(39)

Equations (37)–(39) provide prediction of the system output over a finite prediction horizon $T_{P}$ at time step k. Each predicted output is calculated by repeatedly applying the system model, where the effect of the current state is propagated through powers of $A_{i}$ , and the influence of future control inputs is accumulated using weighted combinations of $A_{i}$ and $B_{i}$ . This structure represents how future outputs depend on both the initial condition and upcoming control actions, and it serves as the foundation for constructing the prediction model used in MPC optimization. The given equations (37)–(39) can be combined into a single equation and can be rewritten in matrix form:

\begin{matrix} Y_{i} (k + 1, T_{P} | k) = F_{i} z_{i} (k) + G_{i} U_{i} (k, T_{U} | k) \end{matrix}

(40)

where $Y_{i} (k + 1, T_{P} | k)$ = ${[y_{i} (k + 1 | k) y_{i} (k + 2 | k) \dots y_{i} (k + T_{P} | k)]}^{T}$ $\in ℝ^{T_{P}}$ ; $U_{i} (k, T_{U} | k) = {[u_{i} (k) u_{i} (k + 1) \dots u_{i} (k + T_{U} - 1)]}^{T} \in ℝ^{T_{U}}$ . $F_{i}$ and $G_{i}$ in (38) are given as:

\begin{matrix} F_{i} = {[C_{i} A_{i} C_{i} A_{i}^{2} \dots C_{i} A_{i}^{T_{P}}]}^{T} \\ G_{i} = [\begin{matrix} C_{i} B_{i} \\ C_{i} A_{i} B_{i} & C_{i} B_{i} \\ ⋮ & ⋮ & ⋱ \\ C_{i} A_{i}^{T_{P} - 1} B_{i} & C_{i} A_{i}^{T_{P} - 2} B_{i} & \dots & C_{i} A_{i}^{T_{P} - T_{U}} B_{i} \end{matrix}] \end{matrix}

Now, the optimal control sequence at $k_{th}$ instance is calculated by minimizing error among predicted output and reference value. Key objective of tracking synchronization in an MAS is to eliminate local neighboring discrepancies as well as to ensure convergence to the reference provided to the leader node. It is determined that the subsequent receding-horizon optimization index is:

\begin{matrix} \min J_{i} & = ∥ \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k) - Y_{i} (k + 1, T_{P} | k) ∥_{Q_{i}}^{2} \\ + ∥ Y_{r} (k) - Y_{i} (k + 1, T_{P} | k) ∥_{W_{i}}^{2} + ∥ Δ U_{i} (k, T_{U} | k) ∥_{R_{i}}^{2} \end{matrix}

(41)

$N_{i}$ is the number of neighboring DG units for the $i_{th}$ DG unit. If a DG has direct access to the reference, then $w_{i} > 0$ , and $w_{i} = 0$ otherwise. Assumption is made that reference remains practically constant within prediction horizon; thus, the reference vector can be defined as follows: $Y_{r} (k) = {[1 1 \dots 1]}^{T} v_{ref} = I_{T_{P}} v_{ref}$ where $v_{ref}$ is the reference voltage at each sampling time and $I_{T_{P}}$ is the vector of length $T_{P}$ with all entries 1. The future incremental control action can be expressed in vector form at the sampling instant k as $Δ U_{i} (k, T_{U} | k) = {[Δ u_{i} (k), Δ u_{i} (k + 1), \dots, Δ u_{i} (k + T_{U} - 1)]}^{T}$ and $Δ u_{i} (k + h) = u_{i} (k + h) - u_{i} (k + h - 1)$ , $h = 0, \dots, T_{U} - 1$ . The first component of the optimization index (41) penalizes the state disagreement over the next $T_{P}$ steps between every pair of states in neighborhood, while second component penalizes any discrepancies between each of the states and provided reference for $T_{P}$ steps. The third component represents the control efforts for the $T_{U}$ time steps ahead. $Q_{i}, W_{i}$ , and $R_{i}$ are described as real, symmetric, and positive definite weighting matrices, each corresponding to the three components mentioned in (41) and $∥ x ∥_{Q_{i}}^{2} = x^{T} Q_{i} x$ . The weighting matrices are be chosen as $Q_{i} = q_{i} I, W_{i} = w_{i} I$ , and $R_{i} = r_{i} I (q_{i}, w_{i}, r_{i} > 0)$ , respectively. The control effort is now calculated as $u_{i} (k + h) = u_{i} (k - 1) + \sum_{r = 0}^{h} Δ u_{i} (k + r)$ , equation (40) for the perspective of incremental control can be written as:

\begin{matrix} Y_{i} (k + 1, T_{p} | k) = N_{i} Δ U_{i} (k, T_{U} | k) + M_{i} (k) \end{matrix}

(42)

here, $N_{i} = G_{i} α_{i}$ and $M_{i} (k) = F_{i} z_{i} (k) + G_{i} β_{i} u_{i} (k - 1)$ and $α_{i}$ and $β_{i}$ are given as follows:

\begin{matrix} α_{i} = [\begin{matrix} 1 & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 1 & \dots & 1 \end{matrix}] \\ β_{i} = [\begin{matrix} 1 \\ ⋮ \\ 1 \end{matrix}] \end{matrix}

Using (42), the cost function in (41) transforms on an unconstrained quadratic programming (QP) problem in terms of the variable $Δ U_{i}$ , which must be solved at every sampling instant,

\begin{matrix} \min_{Δ U_{i} (k, T_{U} | k)} J_{i} = \\ Δ U_{i}^{T} (k, T_{U} | k) H_{i} Δ U_{i} (k, T_{U} | k) - L_{i}^{T} (k) Δ U_{i} (k, T_{U} | k) \end{matrix}

(43)

here, $H_{i}$ is the positive definite matrix, given as $H_{i} = N_{i}^{T} (Q_{i} + W_{i}) N_{i} + R_{i}$ and:

\begin{matrix} L_{i} (k) = 2 N_{i}^{T} [W_{i} (M_{i} (k) - Y_{r}) + Q_{i} (M_{i} (k) \\ - \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k))] \end{matrix}

Explicit solution that minimizes the cost function is given as $\partial J_{i} ∕ \partial Δ U_{i} (k, T_{U} | k) = 0$ :

\begin{matrix} 2 H_{i} Δ U_{i} (K, T_{U} | K) - L_{i} (K) = 0 \end{matrix}

\begin{matrix} Δ U_{i} (k, T_{U} k) = (1 ∕ 2) H_{i}^{- 1} L_{i} (k) \end{matrix}

(44)

As per receding horizon concept of MPC, first component of optimal sequence applied to the procedure allowing that control action to be represented as $u_{i} (k) = u_{i} (k - 1) + τ_{i} Δ U_{i} (k, T_{U} | k)$ , where $τ_{i} = [1 0 \dots 0]$ . Final control action is given as:

\begin{matrix} u_{i} (k) & = u_{i} (k - 1) + τ_{i} H_{i}^{- 1} N_{i}^{T} [W_{i} (M_{i} (k) - Y_{r} (k)) + Q_{i} (M_{i} (k) \\ - \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k))] \end{matrix}

(45)

The flowchart for the secondary control of MG based on DMPC is depicted in Figure 2. The following is illustration of the corresponding steps:

$Step 1 :$ Each DG, functioning as an agent, is represented by its discrete time model (36), and the relevant information in the form of local states $z_{i}$ and voltages of its immediate neighbors is collected at the current time k.

$Step 2 :$ According to the optimization index in (43), each DG computes its optimal control sequence. The first component of this sequence is then used as a control effort $u_{i} (k)$ from (45) at current time k.

$Step 3 :$ The secondary control input $V_{ni}$ to be applied is evaluated as described in (43), and the resulting nominal control value is then added to the primary droop controller.

$Step 4 :$ Repeat the previous steps after increasing the time from k to $k + 1$ .

Figure 2.

Flowchart of DMPC-based secondary control of microgrid.

The block diagram for DMPC-based secondary voltage control is illustrated in Figure 3. Control input $V_{ni}$ is derived using a feedback law and can be expressed as follows:

\begin{matrix} V_{ni} = \int (u_{i} + n_{i} \overset{\cdot}{Q_{i}}) dt \end{matrix}

(46)

where $\overset{\cdot}{Q_{i}}$ is given in (7) as:

\begin{matrix} \overset{\cdot}{Q_{i}} = - ω_{c} Q_{i} + ω_{c} (v_{oqi} i_{odi} - v_{odi} i_{oqi}) \equiv T_{i} (X_{i}) \end{matrix}

(47)

in the above equation, $ω_{c}$ represents the cut-off frequency of a low-pass filter utilized in primary power controller.

Figure 3.

Block diagram of DMPC-based secondary voltage control.

DMPC-based frequency restoration

This portion implements a linear DMPC to synchronize frequency $ω_{i}$ of individual DGs with reference frequency $ω_{ref}$ . Droop control acts as primary controller as a result of which frequency deviates from nominal value under variation of load as per (1). Now, for restore frequency, the secondary control aims to choose appropriate value of nominal frequency $ω_{ni}$ , which is subsequently used as a reference by the power control block. Even though the original dynamics of the system is nonlinear, for application of DMPC in secondary frequency control, a linear relationship is established between the frequency $ω_{i}$ and auxiliary control input $v_{ω_{i}}$ using IOFL. For the IOFL approach, the frequency droop characteristic (1) is differentiated to obtain the required linearization as:

\begin{matrix} {\overset{\cdot}{ω}}_{i} = {\overset{\cdot}{ω}}_{ni} - m_{i} \overset{\cdot}{P_{i}} = v_{ω_{i}} \end{matrix}

(48)

where $v_{ω_{i}}$ represents the auxiliary control input. Transformed dynamics of the DG are represented as a linear state-space system as:

\begin{matrix} \overset{\cdot}{z_{i}} & = A_{i} z_{i} + B_{i} v_{ω_{i}} \\ y_{i} & = C_{i} z_{i} \end{matrix}

(49)

where $z_{i} = [ω_{i}]$ , $A_{i} = [0]$ , $B_{i} = [1]$ , and $C_{i} = [1]$ are the state and system matrices. An assumption can be made that DGs interact with each other via the designated digraph G in order for successful synchronization. Only a few DGs can function as leader nodes in the digraph have access of reference frequency $ω_{ref}$ ; the objective of DMPC in this section is to determine $v_{ω_{i}}$ such that $y_{i} \to ω_{ref} \forall i$ .

To apply DMPC approach for frequency control, a discrete-time model is developed using the forward Euler method of discretization of derivative. Thus, the linear state-space system presented in (50) can be stated as:

\begin{matrix} z_{i} (k + 1) & = A_{i} z_{i} (k) + B_{i} v_{ω_{i}} \\ y_{i} (k) & = C_{i} z_{i} (k) \end{matrix}

(50)

The linear DMPC approach centers on minimizing error between predicted output and given reference signal. Tracking synchronization in an MAS aims to eliminate local neighborhood error and offer convergence to reference given to the leader node. The subsequent receding horizon optimization index for frequency control of the MG is given as follows (Bordons et al., 2020; Lou et al., 2018):

\begin{matrix} \min J_{i} & = ∥ \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k) - Y_{i} (k + 1, T_{P} | k) ∥_{Q_{i}}^{2} \\ + ∥ Y_{r} (k) - Y_{i} (k + 1, T_{P} | k) ∥_{W_{i}}^{2} + ∥ Δ V_{ω_{i}} (k, T_{U} | k) ∥_{R_{i}}^{2} \end{matrix}

(51)

where $Y_{i}$ is formed in a similar manner as was in (40). The neighboring DGs of the $i^{th}$ DG are represented by $N_{i}$ , $T_{P}$ , and $T_{U}$ denote the prediction and control horizons, respectively, for frequency control using DMPC. The reference signal provided to the leader node is $ω_{ref}$ . The future incremental control action at sampling instant k in vector form can be given as $Δ V_{ωi} (k, T_{U} | k) = {[Δ v_{ωi} (k), Δ v_{ωi} (k + 1), \dots, Δ v_{ωi} (k + T_{U} - 1)]}^{T}$ with $v_{ωi} (k + h) = v_{ωi} (k - 1) + \sum_{r = 0}^{h} Δ v_{ωi} (k + r)$ . The incremental control can be written as:

\begin{matrix} Y_{i} (k + 1, T_{p} | k) = N_{i} Δ V_{ωi} (k, T_{U} | k) + M_{i} (k) \end{matrix}

(52)

where $N_{i}$ and $M_{i}$ are similar to as given in section “DMPC-based voltage restoration.” Substituting (52) into (51), the cost function formulated into a quadratic program problem in terms of $Δ V_{ωi}$ and should be solved in each sampling instant:

\begin{matrix} \min_{Δ V_{ωi} (k, T_{U} | k)} J_{i} = \\ Δ V_{ωi}^{T} (k, T_{U} | k) H_{i} Δ V_{ωi} (k, T_{U} | k) - L_{i}^{T} (k) Δ V_{ωi} (k, T_{U} | k) \end{matrix}

(53)

where $H_{i}$ and $L_{i}$ have the form same as in section “DMPC-based voltage restoration.” Explicit solution that minimizes the cost function is given as $\partial J_{i} ∕ \partial Δ V_{ωi} (k, T_{U} | k) = 0$ . The solution is obtained as:

\begin{matrix} Δ V_{ωi} (k, T_{U} k) = (1 ∕ 2) H_{i}^{- 1} L_{i} (k) \end{matrix}

(54)

First component of the optimal control sequence applied to the system allowing control action to be represented as $v_{ωi} (k) = v_{ωi} (k - 1) + τ_{i} Δ V_{ωi} (k, T_{U} | k)$ , where $τ_{i} = [1 0 \dots 0]$ . Final control action is given as:

\begin{matrix} v_{ωi} (k) & = v_{ωi} (k - 1) + τ_{i} H_{i}^{- 1} N_{i}^{T} [W_{i} (M_{i} (k) - Y_{r} (k)) + Q_{i} (M_{i} (k) \\ - \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k))] \end{matrix}

(55)

As per (48), the input to the primary controller from the DMPC for frequency control can be derived as follows:

\begin{matrix} ω_{ni} = \int (v_{ω_{i}} + m_{i} \overset{\cdot}{P_{i}}) dt \end{matrix}

(56)

After execution of the secondary control, output powers of DGs need to be in proportion with their ratings. Thus:

\begin{matrix} m_{i} P_{i} = m_{j} P_{j} \dots = m_{n} P_{N} \end{matrix}

(57)

An additional DMPC strategy to meet this requirement can be formulated by regulator synchronization for a linear first-order MAS:

\begin{matrix} m_{i} \overset{\cdot}{P_{i}} & = v_{p_{i}} \end{matrix}

(58)

\begin{matrix} y_{i} & = P_{i} \end{matrix}

(59)

where $v_{p_{i}}$ is the auxiliary control for proportional active power sharing, and $y_{i} = P_{i}$ is the output active power for $i_{th}$ DG. A receding horizon optimization index is formulated for sharing active power according to DG’s ratings as (Bordons et al., 2020; Lou et al., 2018):

\begin{matrix} \min J_{i} = ∥ \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k) \\ - Y_{i} (k + 1, T_{P} | k) ∥_{Q_{i}}^{2} + ∥ Δ V_{ω_{i}} (k, T_{U} | k) ∥_{R_{i}}^{2} \end{matrix}

(60)

and no reference value is given here as it becomes a cooperative regulator synchronization problem. The future incremental control action for sampling instant k in vector form can be given as $Δ V_{pi} (k, T_{U} | k) = {[Δ v_{pi} (k), Δ v_{pi} (k + 1), \dots, Δ v_{pi} (k + T_{U} - 1)]}^{T}$ and $Δ v_{pi} (k + h) = v_{pi} (k + h) - v_{pi} (k + h - 1)$ , $h = 0, \dots, T_{U} - 1$ . The positive definite symmetric weighting matrix $W_{i} = 0$ in this case as there is no reference present. The control effort is now calculated as $v_{pi} (k + h) = v_{pi} (k - 1) + \sum_{r = 0}^{h} Δ v_{pi} (k + r)$ . The incremental control can be written as:

\begin{matrix} Y_{i} (k + 1, T_{p} | k) = N_{i} Δ V_{pi} (k, T_{U} | k) + M_{i} (k) \end{matrix}

(61)

where $N_{i}$ and $M_{i}$ are similar to as given in section “DMPC-based voltage restoration.” Substituting (61) into (60), the cost function is formulated as a QP problem in terms of $Δ V_{pi}$ and should be solved in each sampling instant:

\begin{matrix} \min_{Δ V_{pi} (k, T_{U} | k)} J_{i} = \\ Δ V_{pi}^{T} (k, T_{U} | k) H_{i} Δ V_{pi} (k, T_{U} | k) - L_{i}^{T} (k) Δ V_{pi} (k, T_{U} | k) \end{matrix}

(62)

where $H_{i}$ is the positive definite matrix, given as $H_{i} = N_{i}^{T} Q_{i} N_{i} + R_{i}$ :

\begin{matrix} L_{i} (k) = 2 N_{i}^{T} [Q_{i} (M_{i} (k) - \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k))] \end{matrix}

Explicit solution that minimizes the cost function is given as $\partial J_{i} ∕ \partial Δ V_{pi} (k, T_{U} | k) = 0$ . The solution is obtained as:

\begin{matrix} Δ V_{pi} (k, T_{U} k) = (1 ∕ 2) H_{i}^{- 1} L_{i} (k) \end{matrix}

(63)

First component of optimal sequence applied to the procedure allowing that control action to be represented as $v_{pi} (k) = v_{pi} (k - 1) + τ_{i} Δ V_{pi} (k, T_{U} | k)$ , where $τ_{i} = [1 0 \dots 0]$ . Final control action is given as:

\begin{matrix} v_{pi} (k) & = v_{pi} (k - 1) + τ_{i} H_{i}^{- 1} N_{i}^{T} [Q_{i} (M_{i} (k) \\ - \frac{1}{| N_{i} |} \sum_{j \in N_{i}} Y_{j} (k + 1, T_{P} | k))] \end{matrix}

(64)

Figure 4 shows the block diagram for secondary DMPC-based frequency control and control input is obtained as follows:

\begin{matrix} ω_{ni} = \int (v_{ω_{i}} + v_{p_{i}}) dt \end{matrix}

(65)

Figure 4.

Block diagram of DMPC-based secondary frequency control.

This control input ensures both the frequency restoration of the MG to nominal value as well as proportional power sharing among the DGs.

The deployment workflow includes system linearization, model prediction and optimization, and distributed coordination of DGs. A local MPC problem is formulated for each DG unit, at each sampling instant. The problem is formulated using the state evaluation of DG and the neighboring DGs. This problem is solved to minimize a quadratic cost function on a finite prediction horizon, capturing voltage or frequency tracking objectives and control effort minimization.

Case studies

To verify efficacy of MPC-based secondary voltage and frequency control, an MG test system simulated in MATLAB R2024a/Simulink as practical system is illustrated in Figure 5. MG test system comprises four DGs, each connected to individual loads. These DGs are linked via four transmission lines modeled as RL branches with series connection. The specifications for DGs, loads, and transmission lines briefly listed in Table 1. Each DG coordinate with its neighboring DGs through a cyber topology given in Figure 5. DG1 is designated as the leader node in this configuration and has access to voltage and frequency references. The MAS operated in MATLAB/Simulink with the secondary control scheme implemented through MPC. In this section, four scenarios are discussed, focusing on the evaluation of controller’s performance for voltage and frequency control and its robustness for load variations.

Figure 5.

Microgrid test system and communication topology.

Table 1.

Parameters of DGs for microgrid test system.

Parameter	Value	Parameter	Value
Parameters of all DGs rating 35 KVA each
m	$5 \times 1 0^{- 5}$	n	$3 \times 1 0^{- 3}$
$R_{c}$	$0.05 Ω$	$L_{c}$	0.35 mH
$R_{f}$	$0.1 Ω$	$L_{f}$	3.5 mH
$C_{f}$	$50 μ$ F
$K_{PV}$	0.2859	$K_{IV}$	595
$K_{PC}$	54.98	$K_{IC}$	1570.9
Line specifications
$R_{L}$	$0.15 Ω$	$L_{L}$	0.35 mH
Load ratings
$P_{L 1}$	6.2	$Q_{L 1}$	4
$P_{L 2}$	6.2	$Q_{L 2}$	4
$P_{L 3}$	6.2	$Q_{L 3}$	4
$P_{L 4}$	6.2	$Q_{L 4}$	4
Secondary voltage control
$Q ∕ W ∕ R$	$10 ∕ 5 ∕ 1$	$T_{s}$	0.1
Secondary frequency control
$Q ∕ W ∕ R$	$10 ∕ 10 ∕ 1$	$T_{s}$	0.01

Case 1: Simulation results for various reference voltage tracking

This scenario illustrates transient behavior of proposed DMPC-based voltage controller. At $t = 0$ seconds, MG is detached from main grid, with primary control active from start and secondary control deliberately activated at $t = 0.3$ seconds. As DG 1 is designated the leading node, Figure 6(a)–(c) depicted simulations for various reference voltages 320 V, 350 V, and 380 V provided to DG 1 and it is seen that while DG terminal voltages are stable with primary control, all DG terminal voltages successfully achieve references after application of secondary control that acts like supervisory control. An additional validation of voltage restoration is studied in a 10 DG MG system, and here, reference voltage provided is 340 V. Secondary control is applied at $t = 0.3$ seconds as shown in Figure 6(e), and all DGs in the system track the reference voltage.

Figure 6.

DG terminal voltages for Case 1 and Case 2: (a) when $v_{ref} = 320$ V, (b) when $v_{ref} = 350$ V, (c) when $v_{ref} = 380$ V, (d) load switching, and (e) when $v_{ref} = 340$ V for 10 DGs.

Case 2: Simulation results for secondary voltage control during load switching

The performance of the MG is analyzed during load switching in this case as depicted in Figure 6(d), where additional loads are connected and disconnected to observe the system’s response. Additional RL loads are connected at bus 2 with rating $S_{1} = 8 KW + 1 KV Ar$ , and at bus 4 with rating $S_{2} = 7 KW + 1 KV Ar$ , at $t = 1.5$ seconds, these are detached at $t = 2.5$ seconds. Voltage controller when given a reference 350 V demonstrates good robustness during the load switching process despite slight transients, effectively maintaining stable voltage levels despite the changes in load.

Case 3: Simulation results for frequency restoration and proportional power sharing

Performance of the MG under DMPC-based frequency control is demonstrated in this case. As depicted in Figure 7(a), during droop control, operating frequency initially drops from nominal value. However, once secondary control applied at $t = 0.3$ seconds, DG frequency begins to rise and is fully restored to $50 Hz$ within few seconds. To evaluate controller’s performance under load switching, additional loads are applied on bus 2 and bus 4, with ratings of $S_{1} = 4 KW + 0.2 KV Ar$ and $S_{2} = 4 KW + 0.2 KV Ar$ respectively, at $t = 2$ seconds and removed at $t = 4$ seconds. During this period, frequency control demonstrates robust performance, as shown in Figure 7(b)–(c). An additional validation of frequency restoration and active power sharing is studied in a 10 DG MG system. Secondary control is applied at $t = 0.3$ seconds, as shown in Figure 7(e)–(f). All DGs in the system share equal active power and track the reference frequency.

Figure 7.

Frequency results for Case 3: (a) active power without load switching, (b) frequency restoration with DMPC, (c) active power during load switching, (d) frequency restoration during load switching, (e) active power of 10 DG system, and (f) frequency restoration for 10 DG system.

Case 4: Plug and play capability

In this study, performance and robustness of proposed controller is evaluated under plug and play condition. The active power results are shown in Figure 8(a), DG 4 is disconnected from the MG system at $t = 2$ seconds, and then it shares no active power with other DGs and DG 4 reconnected to the system at $t = 4$ seconds and the active power results illustrates that after re-connection of DG 4 with MG system it shares the appropriate active power. The voltages of all DGs remain at their provided reference values 350 V after some small transients during the plug and play of the DGs, as shown in Figure 8(b). The frequencies of all DGs are remains stable to their reference value during the plug and play operation with some small transients during removal and re-connection of DG 4, as shown in Figure 8(c). The results illustrated that the proposed control scheme provides the plug and play capabilities of the DGs.

Figure 8.

Plug and play capability results for Case 4: (a) active power of DGs, (b) output voltages of DGs, and (c) frequency of DGs.

Case 5: Comparison of proposed controller with Sun et al. (2022) and satya nagasri Dangeti and Marimuthu (2025)

Output terminal voltage of DGs proposed in Sun et al. (2022) is illustrated in Figure 9(a). The output DG voltages of the proposed work are presented in Figure 9(b). Control methods presented in both works are applied in same MG test, and secondary control is applied in $t = 0.3$ seconds, as illustrated in Figure 9. The proposed control approach shown a good convergence speed, as at $t = 1$ seconds, voltage is regulated form 300 V to 350 V; however, in Sun et al. (2022), voltage regulated from 307 V to 311 V at $t = 1.1$ seconds. The frequency control results presented in satya nagasri Dangeti and Marimuthu (2025) are shown in Figure 9(c). The results of proposed frequency control are shown in Figure 9(d), when the secondary control applied at $t = 0.3$ seconds, the proposed frequency control shows faster convergence speed as the frequency converges in proposed work at $t = 1$ seconds, that is faster as compare to $t = 2$ seconds presented in satya nagasri Dangeti and Marimuthu (2025).

Figure 9.

Comparison between the voltage and frequency control approach for Case 5: (a) voltage restoration results in Sun et al. (2022), (b) voltage restoration results in this paper, (c) frequency restoration in satya nagasri Dangeti and Marimuthu (2025), and (d) frequency restoration in this paper.

Conclusion

In this work, a distributed MPC-based secondary control scheme for the regulation of voltage and frequency in an islanded MG is proposed. Although the original dynamics of the inverter-interfaced DG is highly nonlinear, IOFL is used to define a linear model between control input and output variables, that is, voltage and frequency. Following this transformation, a secondary voltage regulation strategy, which is based on linear DMPC, is implemented. This distributed control works on exchange of information between local and neighboring DGs and obviates the need of a centralized controller, enhancing reliability and improving overall system performance. Moreover, the proposed approach is equipped with the plug and play feature that allows to easily connect or disconnect the DG without affecting rest of the MG. DMPC-based secondary control also facilitates frequency restoration when a droop in frequency occurs during droop control and guarantees the proportional power sharing between the DGs. The effectiveness of the proposed control scheme is demonstrated through simulation results across various test scenarios. The simulation results validate the control strategy by restoring the frequency and output voltage of the MG to their reference values in the presence of uncertainties such as load variations, while maintaining accuracy in active power sharing.

Footnotes

Acknowledgements

The authors would like to acknowledge the support of Dr B R Ambedkar National Institute of Technology Jalandhar, India.

Author contributions

A.C. and V.S. designed the controller, and D.S. made the conclusions and edited them.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Arun Chantola

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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