Dual-Mode Laguerre MPC and Its Application in Inertia-Frequency Regulation of Power Systems

Abstract

This paper studies the collaborative inertia-frequency regulation strategies for the high renewable energy penetrated low inertia power system. Firstly, a systematic investigation is conducted to reveal the dominant dynamic characteristics and the possible challenges for such systems, and then proved the effectiveness of virtual inertia. Subsequently, a novel Laguerre-based model predictive control strategy is accordingly pro-posed, which ensures a better system states convergence ability and a reduced computational burden. The controller takes into account the system’s dual-mode feature to ensure timely response for both the inertia and the frequency support. Then, the regulation quality, operational burden and the cost are mathematically defined. The control trajectory is determined by the rolling optimization. The Gravity Searching Algorithm is utilized to determine the optimal control parameters. Finally, the proposed control strategy is validated through five case studies, demonstrating enhanced robustness, superior dynamic performance and cost-effective operation. This study provides new insights for the analysis and control strategies of the high RE penetrated low inertia systems.

1. Introduction

The fourth industrial revolution is marked by the integration of carbon-independent renewable energy (RE), posing significant challenges to the power system. Researches have extensively examined RE characteristics and their impacts on the grid, highlighting the dilemma between RE penetration and weak grid stability [1,2]. Ref. [3] noted that most REs depend on power electronics at the grid interface and are strongly influenced by climate variability. They exhibit three main undesirable traits: output power fluctuations, variations, and low inertia contribution. Consequently, high RE penetration weakens frequency regulation in three ways: (1) existing controllers lack robustness against atmospheric volatility-induced disturbances [4]; (2) inertia reduction causes rapid frequency change (ROCOF) and larger frequency nadir [5]; (3) more frequent regulation actions and challenges to Energy Storage System (ESS) duration are required [6].

As the trend towards decarbonization continues, RE integration in modern power systems is increasing rapidly. Photovoltaic power generation is growing at an annual rate of 15%, while wind power generation in-creases by 10% annually [7]. In the near future, RE’s impact on frequency stability could lead to significant frequency nadirs and major events like blackouts [8]. Future frequency regulation strategies must prevent such disasters while ensuring stability, reliability, and cost-effectiveness. Existing literature divides studies into two categories for large power systems: (1) traditional primary and secondary frequency control [9]; (2) inertia control providing virtual inertia (VI) [10].

Research hotspots for traditional frequency control in high RE, low inertia systems include: (1) ESS application strategies [11], like pumped storage plants [12]; (2) advanced collaborative control methods [13] for multiple energies and ESS; (3) multi-energy compensation methods, including hydro-solar, wind-solar, and wind-wave power. Ref. [14] indicated that the Load Frequency Control (LFC) system is a common low-order linear model for frequency analysis. Primary and secondary frequency control research is relatively mature. Ref. [15] suggested that advanced controllers, such as intelligence algorithm-based control [16], adaptive control, and receding horizon control, should be more widely used to enhance robustness and dynamic performance under various conditions. They proposed a GPC-based adaptive control for high RE penetrated power systems. However, most literature overlooks that in the initial seconds, only the kinetic energy (inertia) in the rotating mass of the remaining generators responds to frequency changes [5]. This can trigger blackouts before conventional controllers activate [7], referred to as ‘the time gap’ in this paper.

Inertia control is a novel control technology de-signed for the low inertia power system. It employs ESS to provide virtual inertia through the droop control of the RE or ROCOF control of the system. Theoretically, inertia control mostly depends on the electromagnetic components such as Battery Energy Storage Systems (BESS). It can respond more timely than the primary and secondary frequency controllers. In Ref. [17], it is defined as ‘the controlled response from a generating unit to mimic the exchange of rotational energy from a synchronous machine with the power system’. Inertia control aims to eliminate the fast fluctuations and the huge nadir of the frequency caused by the low system inertia [18]. For research on inertia control, the main focuses include: (1) system identification algorithm based inertia monitoring and estimation techniques; (2) artificial intelligence algorithm-based adaptive control strategies for inertia control strategies [10]. The extensively adopted numerical models for inertia related frequency regulation are built within the frame-work of LFC [19,20,21]. According to Ref. [19], incorporating both conventional load frequency control and the virtual inertia creates a blue ocean opportunity. Refs. [20,21,22] introduced a novel model method for the integrated inertia-LFC system, the control strategies are still designed without considering the coupling characteristic of the system of different control loops.

From a modeling perspective, grid-following and grid-forming controls represent two different approaches to inverter operation. Grid-following inverters are typically modeled as current-controlled devices that synchronize with an existing grid, providing stability in systems where grid inertia is still present [10]. This method is well-suited for traditional power grids with high RE penetration, where the grid’s frequency and voltage are relatively stable. The main advantage of grid-following control lies in its simplicity and maturity, which allows for reliable operation with established modeling techniques and fewer computational demands. However, grid-following inverters depend on the grid for synchronization and may face limitations in weak or isolated grids. On the other hand, grid-forming control is designed to establish their own reference for voltage and frequency, thus enabling stable operation in grids with high renewable penetration or even islanded conditions [19,23]. The complexity of their modeling and control introduces challenges in implementation, particularly in real-time system dynamics. Given the current context of our re-search—focused on the frequency control strategies and integrating renewable energy into a large, traditional power grid. grid-following control offers a more practical solution, balancing reliability with simplicity and aligning with the system’s existing architecture.

This paper establishes a theoretical foundation for future grid frequency regulation system modifications (Figure 1). The proposed strategy minimizes costs through a three-layer collaborative architecture: (1) Modeling & Analysis layer: Integrating Virtual Inertia Control, Primary and Secondary Frequency Regulation with inertia/damping coefficient analysis in time/frequency domains; (2) Dual-Mode MIMO-MPC layer: Implementing Rolling Optimization with coordinated SISO-MPC (fast response) and MIMO-MPC (steady-state precision) stages; (3) Validation layer: Quantifying Reduced Operation Burden and Robustness via dynamic trajectory comparisons (L-MPC with/without J₂).

By retaining conventional LFC loops while adding an inertia control loop, the conflict between low-inertia dynamics and control responsiveness is resolved. Specifically: analyze virtual inertia’s transient compensation effect (Section 2.2.2); propose multi-time-scale Laguerre-MPC for frequency regulation; verify less operation burden reduction via ACE trajectories.

The contributions of this paper are:

(1) Construction of an inertia control loop integrated LFC model for small disturbance cases, rarely addressed in existing literature.

(2) Introduction of operational burden for multi-time scale optimization.

(3) Modification of ROCOF’s mathematical ex-pression into a calculable form, avoiding compromises on its original physical meaning as seen in Refs. [19,23].

(4) Adoption of Laguerre-based MPC for MIMO systems, facilitating cooperative control of different loops.

The paper is structured as follows: A cooperative mathematical model of inertia, primary, and secondary frequency control is built. Section 2 provides a theoretical analysis of the low inertia system’s dynamic characteristics and the effects of inertia and damping control. Section 3 designs the L-MIMO controller, ensuring robustness for variable conditions. Section 4 presents case studies verifying the proposed strategy’s effectiveness. Section 5 concludes the paper.

2. The Theoretical Analysis and the Modeling Method for a Virtual Inertia Involved Single Area Power System

The increasing penetration of renewable energy makes the system frequency more sensitive to load disturbances, affecting the oscillation mode due to reduced inertia [20,21]. To address the challenges of frequency regulation in low-inertia power systems, mathematical models need to be established, basic characteristics analyzed, and feasible solutions pro-posed. A concept of inertia response interval and time gap is defined, and an Energy Storage System (ESS) based inertial control loop is added to the traditional Load Frequency Control (LFC) model to provide inertia assistance. Simulations are conducted to verify the analysis, demonstrating how virtual inertia improves system stability. Limitations of basic frequency control methods and challenges in applying virtual inertia are discussed, highlighting the need for the proposed con-troller in system frequency regulation.

2.1. The Modeling of the Frequency Regulation System

The concept of time gap and inertia response must be addressed. The traditional LFC system includes a primary and secondary control loop: primary control maintains balance between generation and load, while secondary control restores pre-disturbance frequency. In practice, rotor inertia responds to power imbalance within the first few seconds before these controllers activate, necessitating the inclusion of an inertia control loop for enhanced overall inertia [24].

The mathematical representation of the frequency regulation model encompasses three control loops as demonstrated in Figure 2. The frequency deviation Δf is measured at the Point of Common Coupling (PCC) through distinct methods based on device types. For synchronous generators, Δf is derived from rotor speed measured by Keyphasor sensors. For power-electronics-interfaced devices (e.g., ESS), Δf is obtained from grid frequency transducers that directly sample system frequency at the PCC.

The traditional inertia controller is depicted in block diagram (a), which is indicated by the blue dotted area on the right. This block diagram (a) can be expressed as u₂ = K_VI(dΔf/dt) + K_pΔf, where K_VI and K_p represent the control gains for rate-of-change-of-frequency (ROCOF) and Δf. K_VI(dΔf/dt) represents the ROCOF control. And K_pΔf stands for the damping control. The proposed L-MPC concentrates on the ROCOF control and makes a sacrifice on the damping control. Equations (1)–(6) have been previously derived in Ref. [25], and thus their detailed derivation is omitted here for brevity.

$${\dot{X}}_g=-{\displaystyle \frac{K_{gov}}{R\cdot T_{gov}}}\Delta f-{\displaystyle \frac{1}{T_{gov}}}{X}_g+{\displaystyle \frac{K_{gov}}{T_{gov}}}{u}_1$$

(1)

$${\dot{P}}_m={\displaystyle \frac{K_{tur}}{T_{tur}}}{X}_g-{\displaystyle \frac{1}{T_{tur}}}{P}_m$$

(2)

$${\dot{P}}_V={\displaystyle \frac{K_{ESS}}{T_{ESS}}}{u}_2-{\displaystyle \frac{1}{T_{ESS}}}{P}_V$$

(3)

$$\Delta \dot{f}=-{\displaystyle \frac{D}{2H}}\Delta f+{\displaystyle \frac{K_{PS}}{2H}}{P}_m+{\displaystyle \frac{K_{PS}}{2H}}{P}_V+{\displaystyle \frac{K_{PS}}{2H}}{P}_L$$

(4)

$$\begin{array}{l}RO\dot{C}OF={({\displaystyle \frac{D}{2H}})}^2\Delta f-({\displaystyle \frac{D{K}_{PS}}{4H^2}}+{\displaystyle \frac{K_{PS}}{2H{T}_{tur}}})P_m+{\dot{P}}_L\\ -({\displaystyle \frac{D{K}_{PS}}{4H^2}}+{\displaystyle \frac{K_{PS}}{2H{T}_{ESS}}})P_V-{\displaystyle \frac{D{K}_{PS}}{4H^2}}{P}_L+{\displaystyle \frac{K_{PS}{K}_{tur}}{2H{T}_{tur}}}Xg+{\displaystyle \frac{K_{PS}{K}_{ESS}}{2H{T}_{ESS}}}{u}_2\end{array}$$

(5)

$$ACE=\beta f+P_{tie}$$

(6)

where X = [X_gij, P_m, P_v, Δf, ROCOF]^T is chosen as the state variable, U = [u₁, u₂] is the system input and Y = [ACE, ROCOF]^T is the system output. The load disturbance and the intermittent energy are integrated into one term P_L.

However, if the proposed MPC controller is to be implemented, the term ${\dot{P}}_L$ in Equation (5) has to be transformed into a calculable form. Therefore, this paper proposes a method to approximate it, as shown in Equation (7).

$$ROCOF=s\Delta f\approx {\displaystyle \frac{s}{ps+1}}\Delta f(p\to 0)$$

(7)

Theoretically, if p is small enough, the value of the approximated ROCOF should be about the real ROCOF *. To testify this theory, a simulation is conducted respectively using PI and PD controllers for the secondary and inertia control. The approximated ROCOF is compared with ROCOF * which is calculated by ROCOF *(k) = (Δf(k) − Δf(k − 1))/Δt.

The interpretations of the corresponding variables in Equations (1)–(4), (6) and (7) are listed in the Appendix A. The values adopted according to Ref. [26].

The continuous-time dynamics defined by Equations (1)–(6) were solved using a fixed-step fourth-order Runge-Kutta (RK4) method with a step size of h = 0.01 s. This solver was selected for its balance between computational efficiency and accuracy in capturing high-ROCOF dynamics (>0.5 Hz/s).

2.2. The Influence of H, D and the Virtual Inertia on the Frequency Stability

Based on the model presented in Section 2.1, a few topics need to be discussed to show the necessity of introducing the proposed controller into the system. Simulations are conducted to discuss: (1) the influence of the low inertia on the frequency stability, which is the decreasing H and D reflected in the constructed model; (2) the compensation that the virtual inertia can make on the small H and D system; (3) the conflict of the smaller nadir and longer settling time of the system frequency with different inertia constant value; (4) the conclusions and limitations of conventional control methods for such system.

This part conduct two groups of comparisons: (1) A comparison of the SISO LFC system without inertia control under different values of H and D. (2) A comparison of the SISO LFC system without inertia control with a MIMO LFC system including inertia control under varying values of H and D. Given that ACE and u₁ represent the control objective and input for load frequency control in conventional power systems. And both the models in the first comparison and the reference group in the second are SISO systems, ACE-u1 is chosen as an appropriate example for the frequency-domain analysis.

2.2.1. System Stability Analysis with Smaller H and D

Zero-pole maps of the system model with varying inertia and damping coefficients are shown in Figure 3. When D is constant at 1.5 s and H decreases from 25 s to 20 s to 15 s, the system’s poles and zeros are illustrated in Figure 3a, with different working conditions marked in red, green, and blue. Observing the six poles on the real axis, three poles near the origin exhibit minimal positional changes with varying H. However, the other three poles, located farther from the imaginary axis at around 3.5 on the real axis, have minimal impact on the system’s dynamic performance.

The orange dashed boxes highlight three sets of poles that significantly affect the system’s frequency regulation quality. As H decreases, these poles gradually approach the imaginary axis, leading to degraded system dynamics. The red markers (H = 25 s) exhibit the smallest imaginary component, indicating the longest frequency oscillation period and optimal stability under disturbances. In contrast, the blue markers (H = 15 s) have a larger imaginary axis coordinate, making them more susceptible to high-frequency oscillations and weaker stability.

The time- domain responses of the power system under a P_L of −0.2% pu are depicted in Figure 4. Figure 4a,b illustrate the dynamic responses of ACE and ROCOF for D = 1.5 s while H respectively adopt 25 s, 20 s, and 15 s. As H decreases, the peak value of ACE increases, reaching 0.024 pu, 0.027 pu, and 0.032 pu, respectively. Additionally, the frequency oscillation counts increase. 2 oscillations are observed for the first two cases and 3 for the last case. The stabilization time also decreases, with respective values of 33.2 s, 33.1 s, and 29.9 s. ROCOF exhibits a similar trend as well. A smaller H results in a faster rate of frequency change, leading to a shorter time to reach the first significant overshoot of frequency. Figure 4c,d present the dynamic responses of the system when H is a constant of 15 s and D adopt varying values of 1.5 s, 1.2 s, and 1 s. As D decreases, the oscillation period of ACE undergoes minimal changes, but the overshoot obviously decreases, and the stabilization time increases. However, ROCOF is relatively insensitive to changes in D and can be considered negligible. Figure 4 indicate that the power system’s dynamic performance can be significantly influenced by both the inertia coefficient (H) and the damping coefficient (D). Appropriately adjusting H and D can effectively offer a trade-off solution between the system’s stability and transient response characteristics.

2.2.2. System Stability Analysis with Virtual Inertia

As discussed above, a deterioration in the dynamic performance can be expected in the studied systems because of its low inertia and poor damping capability. When facing disturbances, the frequency of the systems would suffer from larger deviations, increased ROCOF values, low nadir values and more oscillations. By applying energy storage devices based virtual inertia can effectively enhance the system frequency regulation capability. Section 2.2.2. analyzes the impact of the inertia control loop. A comparison is made between the low-inertia power system integrated with virtual inertia and the actual high-mechanical-inertia power system, demonstrating the importance of inertia control.

Figure 5 illustrates the time-domain response of the power system under a P_L = −0.2% pu disturbance. The red, green, and blue icons represent the low-inertia power system (with H = 15 s and D = 1 s), the conventional power system (with H = 25 s and D = 1.8 s), and the low-inertia power system integrated with an inertia control loop (with H = 15 s and D = 1 s), respectively. As shown in Figure 5a, the integration of the inertia control loop results in a significant leftward shift of the three sets of poles that have a major impact on the system. It leads to a notable improvement in system stability, which quality is similar comparing to the conventional power system with higher acutal mechanical inertia. Figure 5b,c further corroborate the conclusion drawn from Figure 5a. It can be observed that after the integration of the inertia control loop, the ACE overshoot decreases from 0.013 pu to 0.009 pu, but the settling time increases from 33.6 s to 46.2 s. Additionally, the ROCOF stabilizes to −1.28 × 10⁻⁵ at 16.3 s, which is a reduction of 4.9 s compared to the 21.2 s of the low-inertia power system.

The integration of the inertia control loop can rapidly mitigate the ROCOF value in the low-inertia power system, effectively reduce the frequency and ACE overshoot, and improve the system’s stability and frequency regulation performance.

Notably, the proposed VI-based approach fundamentally differs from Virtual Synchronous Generator (VSG) methods:

VI provides targeted inertia emulation without mimicking full synchronous machine dynamics, reducing model complexity and oscillation risks.

VSG requires solving electro-mechanical equations to simulate rotor behavior, increasing computational burden and parameter sensitivity [27].

The proposed L-MPC further enhances VI by adaptively optimizing control parameters in multi-time-scale scenarios.

3. Laguerre-Based Dual-Mode MPC Strategy

To maintain low construction and operational costs for inertia control while ensuring excellent frequency regulation capabilities in the power system under large and random disturbances, Section 3 proposes a Laguerre function based multi-time-scale multi-input multi-output model predictive control algorithm that considers operational burden.

‘Time gap’ is adopted in this paper to demonstrate ‘the time difference of the activation time of different controllers’, which usually ranges from 0 to 10 s in different systems. This paper takes 4 s as an example. Additionally, this chapter introduces the Gravitational Search Algorithm (GSA) for the parameter optimization.

3.1. The Discrete-Time LAGUERRE Function

According to the introduction of the Laguerre basis functions in Appendix B. At sample k, the control trajectory Δu(k|k), Δu(k + 1|k), Δu(k + 2|k)… can be reformulated as:

$$\Delta u(k+m|k)={\displaystyle \sum _{i=1}^N{l}_i{c}_i=L^{\mathrm{T}}(m)\cdot {[\begin{array}{cccc}{c}_1& c_2& \cdots & c_N\end{array}]}^{\mathrm{T}}}=L^{\mathrm{T}}(m)\cdot \eta$$

(8)

where Δu(k + i|k) represents the control increments calculated at the kth step and predicted for the next ith step; η = [c₁, c₂, …, c_N] are the decision coefficients. Additionally, N determines the dimensions of η instead of N_p. Consequently, the prediction horizon can be lengthened without adding to the computation burden.

3.2. The SISO-MPC Stage

The mathematical expressions of the discrete-time system are stated in Appendix C. According to [4,28], the predicted values of the state variables at sample k can be calculated as follows:

$$\begin{array}{lll}x(k+m|k)& =& A^{m}x(k)+{\displaystyle \sum _{j=0}^{m-1}{A}^{m-j-1}B\Delta u_2}(k+j|k)+\\ & & {\displaystyle \sum _{j=0}^{m-1}}{A}^{m-j-1}F\Delta P_L(k+j|k-1)\\ & =& A^{m}x(k)+\phi (m)L(j)\eta (k)+f_d\end{array}$$

(9)

where $\phi ={\displaystyle \sum _{j=0}^{m-1}{A}^{m-j-1}{B}_i};f_d={\displaystyle \sum _{j=0}^{m-1}{A}^{m-j-1}F\Delta P_L}(k+j|k-1)$.

Basing on Equation (9), the cost function for the rolling optimization can be gained as Equation (10). And Equation (10) turns the cost function into a quadratic programming problem (QP). And three types of constraints are considered: (1) control input constraint; (2) generation rate constraint (GRC); (3) system frequency fluctuation constraint, which respectively can be explained as Equations (11)–(13).

$$\begin{array}{l}J(\mathrm{k})={\displaystyle \sum _{m=1}^{Np}x{(k+m|k)}^{\mathrm{T}}Q}x(k+m|k)\\ +{\displaystyle \sum _{m=1}^{Np}\Delta u_2{(k+m-1|k)}^{\mathrm{T}}R}{u}_2(k+m-1|k)\\ =\eta {(k)}^{\mathrm{T}}[{\displaystyle \sum _{m=1}^{Np}{\phi }^{\mathrm{T}}(m)Q}\phi (m)+R_L]\eta (k)\\ +2\eta {(k)}^{\mathrm{T}}[{\displaystyle \sum _{m=1}^{Np}{\phi }^{\mathrm{T}}(m)Q}(A^{m}x(k)+f_d)]\\ +{\displaystyle \sum _{m=1}^{Np}{(A^{m}x(k)+f_d)}^{\mathrm{T}}Q}(A^{m}x(k)+f_d)\end{array}$$

(10)

$$u_{i\_\min }-u(k-1)\le {\displaystyle \sum _{j=0}^{i}L(j)\eta (k)}\le u_{i\_\max }-u(k-1)$$

(11)

$$\begin{array}{l}\Delta P_{mi\_\min }-\nu \le {\displaystyle \sum _{j=1}^m{S}_{\Delta P_{mi}}{A}_i^{m-j-1}{B}_i}L(j-1)\eta (k)\\ \le \Delta P_{mi\_\max }-\nu \end{array}$$

(12)

$$\begin{array}{l}\Delta f_{mi\_\min }-\kappa \le {\displaystyle \sum _{j=1}^m{\displaystyle \sum _{r=1}^j{S}_{\Delta \_\Delta f_{mi}}{A}_i^{m-r}{B}_i}L(r-1)\eta (k)}\\ \le \Delta f_{mi\_\max }-\kappa \end{array}$$

(13)

where Q_i and R_i are the weight matrix, and R_L is calculated as $R_L={\displaystyle \sum _{m=1}^{Np}{L}^{\mathrm{T}}}(m-1)RL(m-1)$, u_{i_min} and u_{i_max} are respectively the lower and upper limits of the accepted control variable, S_Δpm = [0, 1, 0, 0, 0, 0], ΔP_{mi_min} and ΔP_{mi_max} respectively are the lower and upper limits of GRC, η is the decision variable, $\nu =S_{\Delta P_{mi}}{A}_i^m{x}_i(k)+{\displaystyle \sum _{j=1}^m{S}_{\Delta P_{mi}}{A}_i^{m-j}{F}_i\Delta P_L}(k+j-1|k-1)$, $\kappa =\Delta f_i(k)+{\displaystyle \sum _{j=1}^m{S}_{\Delta \_\Delta f_{mi}}{A}_i^m{x}_i(k)}+{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{r=1}^j{S}_{\Delta \_\Delta f_{mi}}{A}_i^{m-r}{F}_i\Delta P_L}(k+r-1|k-1).}$

3.3. The MIMO-MPC Stage

Equations (9)–(13) can be extended to MIMO systems. For the MIMO-MPC controller proposed in this paper, the control variable increments are as follows:

$$\Delta u(k)={[\Delta u_1(k),\Delta u_2(k)]}^{\mathrm{T}}$$

(14)

The input and output matrix can be partitioned to:

$$B={[B_1,B_2]}^{\mathrm{T}}C={[C_1,C_2]}^{\mathrm{T}}$$

(15)

The choice of a and N parameters is fully flexible for each Δu_i(k), as shown in Equation (16):

$$\Delta u_i(k)=L_i^{\mathrm{T}}(k)\cdot {\eta }_i$$

(16)

where L_i^T(k) and η_i are the Laguerre network description of the ith control: ${\mathbf{L}}_{\mathbf{i}}^{\mathrm{T}}(k)=[l_1^i\left(k),l_2^i\right(k\left)\right]$.

The predicted state variables at sample k can be expressed as follows:

$$\begin{array}{l}{x}_i(k+m|k)={\displaystyle \sum _{j=0}^{m-1}{A}_i^{m-j-1}[B_1{L}_1^{\mathrm{T}}(j),B_1{L}_2^{\mathrm{T}}(j)]\eta (k)}\\ +{\displaystyle \sum _{j=0}^{m-1}{A}_i^{m-j-1}{F}_i\Delta P_L}(k+j|k-1)+A_i^m{x}_i(k)\\ =A_i^m{x}_i(k)+{\phi }_i(m)[L_1^{\mathrm{T}}(j),O_m;O_m,L_2^{\mathrm{T}}(j)]\eta (k)+f_{di}\end{array}$$

(17)

where ${\phi }_i={\displaystyle \sum _{j=0}^{m-1}{A}_i^{m-j-1}[B_1},B_2];f_{di}={\displaystyle \sum _{j=0}^{m-1}{A}_i^{m-j-1}{F}_i\Delta P_L}(k+j|k-1).$

Similarly, the cost function can be defined as follows:

$$\begin{array}{l}J(\mathrm{k})={\displaystyle \sum _{m=1}^{Np}x{(k+m|k)}^{\mathrm{T}}Q}x(k+m|k)\\ +{\displaystyle \sum _{m=1}^{Np}\Delta u{(k+m-1|k)}^{\mathrm{T}}R}\Delta u(k+m-1|k)\\ =\eta {(k)}^{\mathrm{T}}[{\displaystyle \sum _{m=1}^{Np}{\phi }^{\mathrm{T}}(m)Q}\phi (m)+R_L]\eta (k)\\ +2\eta {(k)}^{\mathrm{T}}[{\displaystyle \sum _{m=1}^{Np}{\phi }^{\mathrm{T}}(m)Q}(A^{m}x(k)+f_d)]\\ +{\displaystyle \sum _{m=1}^{Np}{(A^{m}x(k)+f_d)}^{\mathrm{T}}Q}(A^{m}x(k)+f_d)\end{array}$$

(18)

where Q_i and R_i are the weight matrix, and R_L can be interpreted as:

$$R_L={\displaystyle \sum _{m=1}^{Np}{[L_1^{\mathrm{T}}(m-1),O_m;O_m,L_2^{\mathrm{T}}(m-1)]}^{\mathrm{T}}R[L_1^{\mathrm{T}}(m-1),O_m;O_m,L_2^{\mathrm{T}}(m-1)]}.$$

The treatment of the constraints is similar to SISO systems by adopting new L, which can be calculated as L = [L₁^T(m − 1), Om; Om, L₂^T(m − 1)]. Therefore, by applying QP, the optimal solution of L can be obtained, and the optimized Δu₁(k) and Δu₂(k) can be implemented.

3.4. The Tow Stage Parameter Optimization

As analyzed in Section 2, the virtual inertia contributes to the dynamic responses of the system just as the real inertia. Furthermore, the new inertial constant of the system can be regarded as the sum of two parts, (1) H_eq: the inertia from the SGs; (2) H_vir: the virtual inertia:

$$H=H_{eq}+H_{vir}$$

(19)

For better frequency regulation performances, the ESS is responsible for providing the inertia power until the critical value of frequency is reached. The power reference of inverter-based DERs should be altered as follows [28]:

$$P_{new}=P_{old}+2H_{vir}\cdot {\displaystyle \frac{df}{dt}}$$

(20)

where P_new and P_old are the power reference with VI and only of the DERs respectively. H_vir is the value of virtual inertia.

H_vir is directly linked to the capacity and the maximum power of ESS, which typically have a significant impact on construction costs during the capacity planning phase. By minimizing H_vir, it is possible to achieve a lower costs.

Therefore, H_vir is adopted as an economic index in this paper, which is also the first optimization objective:

$$J_1=H$$

(21)

The second economic index considered in this paper, is the operational burden [11] of ESS. J₂ can be obtained as Equation (22):

$$J_2={\displaystyle \sum _{k=2}^{\mathrm{end}}|P_v(\mathrm{k})-P_v(\mathrm{k}-1)|}$$

(22)

J₂ can be considered as the regulation milage index which can limit the ESS output variation. Therefore the ESS operational burden can be reduced, and lifespan can be extended.

Additionally, the closed-loop performance should be addressed as well, which is different for SISO system and MIMO system. The frequency regulation performance indexes for the SISO-MPC stage and MIMO-MPC stage can be respectively explained as Equations (23) and (24):

$$J_3={\displaystyle {\int }_0^{\infty }({\omega }_1|e(t)|+{\omega }_2{u}_2^2(t)+{\omega }_3\Delta u_2^2(t))}dt$$

(23)

$$\begin{array}{l}{J}_4={\displaystyle {\int }_0^{\infty }({\omega }_1|e_1(t)|+{\omega }_2{u}_2^2(t)+{\omega }_3\Delta u_2^2(t)}\\ +{\omega }_4|e_2(t)|+{\omega }_5{u}_1^2(t)+{\omega }_6\Delta u_1^2(t))\mathrm{dt}\end{array}$$

(24)

where ω₁, ω₂, ω₃, ω₄, ω₅ and ω₆ are weights, which is roughly decided as 0.4, 0.05, 0.05, 0.4, 0.05 and 0.05, respectively. e₁(t) = ROCOF − 0, e₂(t) = Δf − 0 which not only guarantee a good overall performance but also prevent the drastic energy changes when transitioning between SISO and MIMO modes.

The Gravitational Search Algorithm (GSA) optimizes control parameters by simulating mass agents in a gravitational field. Each agent’s position represents a candidate solution (e.g., H_vir and MPC weights), with its ‘mass’ determined by the cost function (Equations (21)–(24)). Agents move toward solutions with higher fitness (lower J) under virtual gravitational forces.

Parameter settings: Population size = 50, maximum iterations = 200, initial gravitational constant G₀ = 100. The algorithm terminates when solutions converge or reach iteration limits.

The framework (Figure 6) initiates with Data Acquisition from synchronous generators (Keyphaser Sensors) and ESS grid transducers. Δf Calculation modules process inputs to compute frequency deviations. State Estimation constructs dynamic vector X(k), feeding Dual-MPC Stages:

SISO-MPC rapidly minimizes ROCOF under constraints, generating state adjustments Δx(k);

MIMO-MPC optimizes steady-state performance, outputting control increments Δu₁(k), Δu₂(k).

Integrated commands u₁(k), u₂(k) actuate Governor Valves (conventional plants) and PCS Setpoints (ESS). New measurements (Δf(k + 1), ROCOF(k + 1)) close the loop.

4. Case Study

This section aims to compare the dynamic performance of the system under varying working conditions and inertia constants before integrating primary and secondary frequency regulation. Various controllers are evaluated to demonstrate the theoretical advantages of the proposed controller. The analysis focuses on four main aspects: (1) performance degradation due to time gaps and improvements by the proposed controller; (2) the interaction between secondary and inertia control principles; (3) the impact of changing working conditions on dynamic performance; (4) the economic and dynamic effects of different virtual inertia compared to optimized virtual inertia and the improvements from optimization.

The controller for both secondary frequency regulation and the inertia control adopts PI. The control parameters are optimized in ideal condition which doesn’t need to take time gap into consideration (K_p = 0.001, K_i = 0.067 for the secondary control loop, K_p = 0.764, K_i = 0.232 for the inertia control loop). In Figure 7, a power disturbance of 0.5% pu happens at the 5th second, the red lines display the dynamic responses of the ideal system which simplifies the time gap as 0 s. The blue lines are the responses of real system with time gap of 4 s when control parameters stays equal as the red line.

Validating the time-gap theory in Section 2.1, Figure 7a compares the primary control objective of the LFC system, ACE. Ideally, the system exhibits no overshoot and an adjusting time of 15 s without considering time delays. However, in practice, a significant overshoot of 0.011 pu (83% of the nadir) occurs due to unexpected time delays. The adjusting time is also extended by 2.6 s (32.5% longer). With a bias factor of 2, the nadirs are 49.7 Hz and 49.67 Hz for the two systems, respectively. Figure 7b illustrates the second control objective, ROCOF, showing larger oscillations and a more variable frequency under practical conditions, indicating degraded transient behavior. Figure 7c,d show the output powers of the conventional generators and the ESS, which together match the load disturbance. Oscillations and a prolonged transient response are evident when time delays are considered.

Further evidence supporting the adoption of the MIMO controller is presented below. Many studies isolate the inertia control loop to examine its impact on system frequency regulation, leading to limitations when applying theoretical findings to practice due to the neglect of interactions between different control loops. While some studies incorporate the inertia control loop into the LFC model and use predictive controllers like GPC, they still treat the system as SISO, preventing optimal adjustment of the two control signals. Consequently, existing research lacks a study on the coordination principles of different control loops.

This part provides three groups of simulations with increasing drastic disturbances to test the robustness of the system with the proposed controller.

Figure 8 presents the dynamic performance comparisons between the MIMO-MPC, two-GPC, and two-PI controllers under load disturbances of −0.5%, −1%, and −1.2% pu. The comparison includes the system’s ACE, ROCOF, ESS adjustment mileage (P_v), and traditional generator speed governor mileage (P_t), as shown in panels (a–d).

As illustrated in Figure 8a, the ACE nadir values for the two-PI controllers are significantly higher: −0.0153 pu, −0.0306 pu, and −0.0367 pu under the three load disturbances, indicating poor inertia compensation and higher system inertia. Consistent with the dual-mode coordination principle in Section 3.3, MIMO-MPC controller demonstrates substantial improvements, with the ACE nadir reduced by 47.7%, 67.9%, and 68.7% compared to the PI controllers, achieving values of −0.0080 pu, −0.0098 pu, and −0.0115 pu, respectively. This reduction in ACE values highlights the superior inertia control and adaptability of MIMO-MPC under varying disturbances. The two-GPC controllers show improvements over the PI controllers but still perform worse than MIMO-MPC, with ACE values of −0.0087 pu, −0.0133 pu, and −0.0159 pu, which are 8.5%, 26.4%, and 11.2% higher than MIMO-MPC’s corresponding values.

In terms of ROCOF, the two-PI controllers show larger fluctuations and slower damping, especially at higher disturbances. At a 1.2% pu disturbance, the maximum ROCOF for MIMO-MPC is reduced by approximately 50% compared to the two-PI controllers, with faster recovery and lower peak values. The MIMO-MPC controller also outperforms the GPC controllers, which exhibit slightly higher oscillation frequencies and less optimal synchronization between the inertia control loop and LFC, confirming the enhanced damping and stability of MIMO-MPC.

The ESS adjustment mileage data in Figure 8c reveals that MIMO-MPC generally shows larger values than the two-PI and two-GPC controllers. For example, at a 0.5% pu disturbance, the ESS adjustment mileage for MIMO-MPC is 0.94 pu, while for the two-PI and two-GPC controllers, the values are 0.545 pu and −0.7 pu, respectively. A larger ESS adjustment mileage for MIMO-MPC indicates better handling of dynamic load fluctuations, ensuring stability and effective regulation quality. Minimizing ESS adjustment mileage may compromise the system’s regulation accuracy and dynamic response.

The Pt adjustment mileage (Figure 8d) shows less variation between controllers. At a 0.5% pu disturbance, the Pt adjustment mileage for MIMO-MPC is 0.33 pu, compared to −0.45 pu and −0.5 pu for the two-PI and two-GPC controllers. This suggests that the two-PI and two-GPC controllers regulate P_t more conservatively, potentially sacrificing faster recovery or more precise regulation compared to MIMO-MPC.

In summary, the MIMO-MPC controller outperforms both the two-PI and two-GPC controllers across all performance metrics. It significantly improves inertia control, dynamic response, and system stability, especially under higher load disturbances. The larger ESS adjustment mileage of MIMO-MPC indicates better adaptability and more stable regulation, while its superior performance in ROCOF and ACE highlights its overall advantage in dynamic load management. Although the Pt adjustment mileage for MIMO-MPC is slightly larger, this trade-off is beneficial for faster recovery and enhanced regulation accuracy.

The economic index and the performance index adopted in the optimization process are mutually exclusive. Theoretically, smaller H and less ESS regulation mileage can directly lead to a performance deterioration. A group of simulations are needed to quantitatively analyze the performance degradation and the operational burden suppression effect and cost-cutting effect brought by the economic index.

Figure 9 displays the comparison of the dynamic responses of systems respectively optimized with and without J₂. Similarly, Figure 9a,b shows that a slightly larger overshoot can be observed when the system considers the regulation mileage index J₂, but the difference is insignificant. In terms of the regulation mileage, the J₂ optimized controller achieved a mileage of 0.94 pu. However without J₂ the mileage can reach 0.98 pu, representing an increase of 4.37% in 20 s. Therefore, the effectiveness of the mileage index has been confirmed. And it could be concluded that with a small sacrifice on the performance, both the working life of ESS and the economic benefits of the grid can be ensured.

As for the mplementation possibility, The L-MPC controller was deployed on a standard PC (Intel i7-12700H, 32GB RAM) running MATLAB R2023b. For a prediction horizon N_p = 20 and control horizon N_c = 5, the average computation time per control cycle (10 ms step) was 8.2 ms– well below the 10 ms threshold. The 10 ms control cycle aligns with standard grid-following inverter switching frequencies (1–10 kHz). GSA optimization (offline) required 200 iterations in 42 s for parameter tuning, which is feasible for day-ahead planning.

5. Conclusions

This paper proposes an adaptive L-MPC controller to provide a collaborative control strategy for the inertia, primary and secondary controller. The controller is aimed at the frequency regulation of a single area power system because the concrete applications would be different for multi area systems. Then a series of researches are conducted to testify its superiority.

(1) It is necessary to consider the inertia response interval (time gap) for the design and optimization of the controller. Otherwise, a degradation in the frequency performance is expected. The ignorance of time gap could cause a 17% higher ROCOF for PI dominated systems and a 1.3% higher ROCOF for MIMO MPC dominated systems;

(2) The coordination of conventional LFC and virtual inertia should adopt MIMO controllers. The cooperative adaptive control can be guaranteed with decrease the frequency nadir by 50% when considering the system as a MIMO system;

(3) The proposed MIMO-MPC controller has been testified with enhanced self-adaptability and robustness when dealing with a variety of operating conditions in low-inertia power systems. Additionally, it minimizes the total adjustment mileage and decreases equipment losses, which contributes to the economical operation of the system.

(4) Unlike VSG methods that emulate full synchronous machine dynamics at high computational cost, L-MPC focuses on economical inertia support via adaptive gain optimization, reducing ESS burden while avoiding possible VSG-induced oscillations [26].

It is denoted that the focus of this study is the frequency stability of power system under small perturbations. Therefore, only mechanical dynamics is emphasized in the system modelling. The system plant is a linearized system only applicable to small disturbance scenarios around the equilibrium. The study is at the numerical simulation stage, we will try to extend it to the physical lab prototype or practical application in the future.