Robust Optimal Frequency Response Enhancement Using Energy Storage-Based Grid-Forming Converters

Abstract

To enhance frequency and active power control performance, this research proposes a decentralized robust optimal tuning approach for power grid frequency regulation support using energy storage systems (ESSs) as the primary source of grid-forming (GFM) converters. The proposed approach employs the robust Kharitonov theory to find a family of stabilizing sets of a proportional-integral (PI)-based supplementary controller, which is used in the outer control layer of the GFM control system. A family of stabilizing parameter sets is found in the presence of system uncertainties and disturbances that are common in power grid operation. Then, using a developed Bayesian optimization algorithm, an optimal set of parameters is determined among the mentioned family member sets. The proposed sophisticated combination of a robust control theorem and an optimization algorithm provides a promising solution for the robust and optimal tuning of control system parameters in ESS-based GFM converters. The efficacy of the proposed method is demonstrated via simulation and laboratory real-time experiment results for a given detailed case study.

1. Introduction

A modern power grid consists of distributed generations (DGs) such as renewable energy sources (RESs). It is characterized by decentralized and distributed control and operation systems. There are several options available to customers when considering marketing and deregulation policies. In this type of power grid, the use of remote and automated monitoring/control is common, including a wide range of generation and demand. Modern power grids are known for several new features, such as full generation/load control, bidirectional flows of power and data, the integration of numerous DGs and RESs, a wide network of monitoring units such as phasor measurement units (PMUs) and intelligent electronic devices, while the complexity in these systems is increased [1].

A modern power grid has sophisticated metering technology that enables remote reading and monitoring as well as remote connection and disconnection. This infrastructure enhances consumer visibility and provides a pathway for home and grid automation. However, the primary issues affecting the dynamic performance and stability of a modern power grid arise from a decrease in the rotational inertia of the system, which occurs when grid-connected converters (GCCs) associated with DGs and RESs progressively replace synchronous generators (SGs).

Decreasing the rotational inertia in a power grid may hurt the system reaction and could affect the control capability and performance [2]. This may result in significant fluctuations in grid frequency, angle, and voltage, and it may potentially cause system instability [1]. Therefore, the increasing penetration of GCCs leads to a reduction in power grid inertia, which poses significant challenges for load–power balance and frequency management. The inherent nature of renewable power production further intensifies this problem [2]. Consequently, many studies have been conducted in this area, and several techniques have been introduced using grid-forming (GFM) converters’ capabilities to address this challenge.

GFM converters are now commonly acknowledged as a viable option for enhancing modern electrical grids. These systems can operate in both grid-connected and islanded operating modes [1]. The integration of GFM converters can reduce frequency nadir in a hybrid diesel microgrid. Researchers in [3] investigate the transient dynamics of a real-world hybrid diesel microgrid. It proposes a transition from a diesel spinning reserve to a battery energy storage system (BESS) operating reserve scheme. The proposed transition improves power quality in terms of voltage deviation and frequency nadir when the microgrid is subjected to a loss of wind generation scenario. Additionally, Ref. [4] explores the use of hybrid energy storage systems with GFM conversion in low-inertia systems. It highlights the importance of these systems in addressing reduced system inertia and ensuring grid stability. The study highlights the operational advantages of hybrid systems over traditional BESS systems, emphasizing the need for prioritizing their deployment and optimization for sustainable energy.

In a DC microgrid, photovoltaic systems (PVSs) usually function in the maximum power point tracking (MPPT) mode, while the DC bus voltage is mainly regulated by the energy storage system (ESS). Thus, to ensure the stability of the bus voltage in DC microgrids, it is often necessary to have large-scale ESSs, which leads to higher investments and maintenance expenses. Ref. [5] introduces an adaptive power point tracking (APPT) control approach for PVSs to run in the GFM mode and actively regulate the voltage for a DC microgrid. Additionally, a virtual inertia control mechanism is suggested to enable the ESS to function in the GFM mode and mimic the dynamic properties of an SG. Furthermore, the suggested method introduces adaptive non-singular terminal sliding mode control loops for the voltage and current control loops of the GFM converters. This approach successfully enhances the voltage and frequency regulation performance.

BESSs can rapidly modify their power output beyond the capabilities of traditional power production. This makes them valuable assets for restoring and maintaining appropriate levels of frequency control capacity. BESSs are often linked to the electrical grid using a power converter that can function in GFM modes. Ref. [6] provides a quantitative analysis of the influence of large-scale BESSs on the ability of a low-inertia power grid to regulate frequency. It also compares the effectiveness of GFM converters’ management mode in this regard. Additionally, Ref. [7] presents a new approach called virtual inertia control that uses a feedforward decoupling strategy to solve the problem of low inertia in GFM converter-interfaced microgrids. The feedforward control strategy is used to mitigate the interdependence between active and reactive power resulting from line impedance. An active power–voltage droop is used for the battery converter in hybrid energy storage systems (HESSs). This updated voltage input control approach has been developed to enhance the inertia of the microgrid by improving the supercapacitor converter in a HESS. An in-depth analysis was performed to create a precise small-signal model of the supercapacitor converter using the suggested virtual inertia control. This resulted in the derivation of a transfer function model. Additionally, virtual inertia and virtual damping were investigated using the pole–zero map technique to determine their parameters.

With the widespread integration of GFM-based sources into power grids, there is a growing interest in distributed energy systems. Nevertheless, reduced inertia adversely affects the stability of voltage or frequency and the capacity of systems to withstand disturbances, making power quality susceptible to the intermittent and unpredictable nature of solar or wind plants. Hence, virtual inertia control is suggested to sustain system stability. Ref. [8] presents a virtual adaptive inertia control technique. The dual extended Kalman filter is used to predict the online states of energy storage battery packs (ESBPs). The virtual inertia and droop parameters are determined using fuzzy logic and virtual battery algorithms, which consider the battery states and bus voltage changes. The objective is to allocate inertia and power during dynamic and steady periods, respectively. It can mitigate voltage fluctuations, enhance system stability, and accomplish decentralized and coordinated control throughout the whole operation of the microgrid. Furthermore, the implementation of a microgrid with a significant proportion of RESs is now essential. ESSs offer sustainable and uninterrupted active power assistance to the power grid, while also being cost-effective and environmentally friendly. Nevertheless, as mentioned above, the lack of inertia is a drawback of RES-based systems, as it has a detrimental impact on the dynamic performance of the microgrid and may result in serious repercussions such as a loss of synchronization and instability. Various strategies have been suggested to simulate virtual inertia to solve the problem of inertia deficit. Out of these options, the derivative control technique (DCT) is widely favored because of its straightforwardness and efficiency. The abovementioned study employed ESSs to provide virtual inertia assistance via DCT during load imbalance. However, this approach fails to consider the potential for harnessing virtual inertia support derived from ESSs. Ref. [9] presents an innovative approach to harnessing the virtual inertia capability of ESS by enhancing the derivative control mechanism, hence enhancing the dynamic performance of the system.

In DC microgrids, the size of the DC connection capacitor is minimized to ensure the presence of inherent inertial properties. Consequently, significant voltage variations arise when there are changes in the load or unpredictable fluctuations in the power supply from unreliable sources. This results in a deterioration in voltage quality. To address the issue of low inertia, Ref. [10] suggests the use of a high-speed ESS technology, such as a supercapacitor, which can imitate inertial reactions by using a specific control algorithm. A bidirectional DC-DC converter is used to connect a supercapacitor ESS to a DC microgrid. The control method consists of a virtual capacitor and a virtual conductance. The implementation is carried out inside the inner loop controls, namely the current loop control, to provide a high-speed emulation of the concepts of inertia and damping.

Recent research has focused on the incorporation of GFM converters with energy storage devices, since they have the potential to significantly improve grid stability and performance. Nevertheless, there is a significant deficiency in the existing body of literature about the use of ESS-based GFM converters to effectively enhance frequency performance robustly. The present work aims to overcome this limitation by proposing an efficient and reliable frequency control technique that combines Kharitonov’s theorem with a Bayesian optimization approach. Kharitonov’s theorem gives a framework for studying the robustness of closed-loop control system while faced with unknown parameters. Additionally, Bayesian optimization provides a systematic approach for optimizing the obtained robust results to reach the most acceptable performance. The proposed control strategy seeks to improve the frequency and active power control by integrating the two techniques mentioned above. This would eventually result in a more robust and dependable electrical grid that can effectively handle the growing use of RESs. The proposed control strategy is successfully examined in a power grid case study using a supercapacitor-based GFM converter. The results are verified in both simulation and laboratory real-time experimental environments.

Optimizing a complex system involves selecting suitable optimization algorithms and tuning their parameters for efficiency. Genetic algorithms (GAs) are a widely recognized optimization method that aims to identify the optimal solution within a wide range of feasible options. This method finds the optimal solution in complex control systems as presented in [11]. In addition, control schemes for the coordination of multiple microgrid generators are presented in [12], especially with voltage source converter type interfaces, for both grid-connected and autonomous modes. To maintain required control performance and power quality during severe conditions, an effective optimal control parameter-tuning method using the particle swarm optimization (PSO) algorithm is proposed.

The novelty of this work lies in the application of a robust stability method utilizing the Kharitonov theorem, combined with a systematic D-stability approach through Bayesian optimization for a GCC powered by ESSs. Additionally, this study demonstrates the capability of frequency regulation support achieved with a simple optimally adjusted PI controller, ensuring robust performance.

This paper is organized as follows: Section 2 provides the theoretical background essential for understanding the discussed concepts. Section 3 introduces the microgrid case study that serves as the basis for the control analysis and synthesis. Section 4 focuses on power–frequency response modeling and analysis, detailing the methodologies employed. In Section 5, the proposed robust optimal secondary controller design is presented, highlighting the key design principles. Section 6 discusses the simulation and laboratory real-time results, showcasing the practical implications of the proposed approach. Finally, Section 7 concludes the paper, summarizing the findings and their significance.

2. Theoretical Background

2.1. Kharitonov’s Theorem

Kharitonov’s theorem is a widely recognized and reliable technique in the field of control. However, its use in modern power grid control is not very common. This theorem offers a straightforward method for examining the robust stability of interval polynomials [13]. Kharitonov’s theorem states that a family of interval polynomials, denoted as $P$, with a certain order, is considered robustly stable if and only if all four of its Kharitonov polynomials are stable. This technique provides a strong framework for constructing robust controllers that can efficiently manage the parametric uncertainties and disturbances inherent in modern power grids.

Kharitonov’s theorem offers a significant benefit in ensuring robust stability by examining the stability of a limited number of polynomials, hence simplifying the complexity of stability analysis. This characteristic makes it as a promising method for power grid control applications, where the stability and performance of systems in the face of parameter variations are crucial.

Theorem 1. 

(Kharitonov’s theorem): A polynomial with variable coefficients, such as K(s):

$$K\left(s\right)=c_0+c_{1}s+c_2{s}^2+c_3{s}^3+c_4{s}^4+\cdots$$

(1)

is considered Hurwitz stable if and only if the following four extreme polynomials, which are obtained from the Kharitonov polynomials, are likewise Hurwitz stable [13]:

$$\begin{gathered} K_1\left(s\right)=c_0^{+}+c_1^{+}s+c_2^{-}{s}^2+c_3^{-}{s}^3+\cdots \\ {K}_2\left(s\right)=c_0^{-}+c_1^{-}s+c_2^{+}{s}^2+c_3^{+}{s}^3+\cdots \\ {K}_3\left(s\right)=c_0^{-}+c_1^{+}s+c_2^{+}{s}^2+c_3^{-}{s}^3+\cdots \\ {K}_4\left(s\right)=c_0^{+}+c_1^{-}s+c_2^{-}{s}^2+c_3^{+}{s}^3+\cdots \end{gathered}$$

(2)

The symbols ‘−’ and ‘+’ represent the lower and upper limits of the polynomial coefficients, respectively. In the context of Kharitonov’s theorem, the polynomial $K\left(s\right)$ denotes the characteristic polynomial of the closed-loop system.

The proof is given in [13].

2.2. Bayesian Optimization Technique

Bayesian optimization is an advanced method designed to optimize global black-box functions, which are not transparent to the observer and require inputs and outputs. It offers advantages over conventional optimization techniques like network search or random search, such as superior performance in managing expensive and noisy function evaluations, incorporating previous knowledge or perspectives, and maintaining a balance between exploration. Bayesian optimization involves multiple sequential processes, starting with random sample points and initializing the surrogate model. The acquisition function optimizes the balance between exploration, and the process is updated using observed data until a convergence threshold is reached. The process systematically explores the search space to find the best possible solution [14,15].

Figure 1 illustrates a basic Bayesian network with variables A, B, and C. The variables B and C are statistically contingent upon the variable A. Consequently, an edge from A to B in Figure 1 signifies that knowledge of A can facilitate the prediction of B. This may or may not signify a causal link, wherein A directly or indirectly influences B. Bayesian networks can manage partial datasets. For instance, consider a scenario involving two highly anti-correlated input variables. Due to the inability to express the connection among input variables, effective predictions using most existing models become impossible when one of the inputs is unobservable. Bayesian networks provide an inherent method to represent these dependencies.

Bayesian networks facilitate the understanding of causal links among many variables. It facilitates forecasts among interventions. Bayesian networks, together with Bayesian statistical methods, enable the integration of domain knowledge and data. Their causal semantics facilitate the straightforward embedding of causal prior information. Bayesian approaches, in combination with Bayesian networks and other models, provide an effective and principled strategy to prevent data overfitting, eliminating the necessity to reserve a portion of the available data for testing purposes. In other words, the Bayesian technique allows for the smoothing of study models, enabling the utilization of all available data for training purposes [15].

3. Microgrid Case Study

3.1. Overal Framework

The power network diagram shown in Figure 2 is considered as a case study in the present work. The power grid includes two DGs. DG 1 uses a GFM converter which uses a set of supercapacitors as the primary source to inject the required power into the network. This power source is mainly used for grid frequency regulation support. DG 2 uses a grid-following converter as the main supplier of the grid load. In addition to the main load (grid load), a local load is also available to represent the domestic demand. The parameters of two DGs, lines, and loads are the same as the given parameters for voltage source converters 1 and 2 (VSC 1 and VSC 2) in [16].

3.2. Energy Storage-Based GFM Converter

Figure 3 depicts the simplest topology for connecting a GFM-based DG to an AC grid. The structure includes a GFM converter that is linked to an AC grid via an RL circuit and a DC voltage source $u_{dc}$. Both the inner and outer control layers are shown. However, here, the focus is on the outer control layer, which is realized by a PF-based supplementary controller for grid frequency regulation support. The modulated voltage $v_{m_{abc}}$ is assumed to be driven directly by a set of three-phase reference signals in the control system. Additionally, the GFM converter is assumed to be connected to the AC grid through a simple R-L filter. For high-voltage power transmission, it is assumed that the resistive part is considerably less than the impedance $L_c{\omega }_g\gg R_c$. This is often referred to as the leakage impedance of the transformer. It is also assumed that the AC grid is balanced. The model uses a Thevenin equivalent AC voltage source in series with its equivalent impedance to provide an AC voltage $v_{g_{abc}}={\left[\begin{array}{ccc}{v}_{g_a}& v_{g_b}& v_{g_c}\end{array}\right]}^T$. Additionally, the grid currents are indicated by $i_{g_{abc}}={\left[\begin{array}{ccc}{i}_{g_a}& i_{g_b}& i_{g_c}\end{array}\right]}^T$. This model is used for introducing synchronization and the power exchange principle between the GFM converter and the AC grid.

To simplify the analysis, the resistor has not been included in the line impedance. The active power at point of common coupling (PCC) can be expressed as

$$P=\mathfrak{R}e\left({\displaystyle \frac{\overline{{ V}_g}(\overline{{ V}_m}-\overline{{ V}_g)}}{j{X}_c}} \right)={\displaystyle \frac{V_m{V}_g}{X_c}}sin\left(\psi \right)$$

(3)

where $X_c$ and $\psi$ are the converter side impedance and the angle between $\overline{{ V}_g}$ and $\overline{{ V}_m}$, and $V_m=V_g=1 pu$.

From Equation (3), it is obvious that the active power is linked with the modulated angle ${\delta }_m$ and the root mean square (RMS) value of the modulated voltage $V_m$. In practice, the active power will be controlled using ${\delta }_m$ (via the $\psi$) because it is not acceptable to apply a significant variation in $V_m$. Thus, the control must produce a reference (${\delta }_m^{*}$) for this angle as depicted in Figure 4. The generation of ${\delta }_m^{*}$ only necessitates information on the active power, so the control does not need any information on the grid voltage. This control ensures that $V_m$ stays within a specified $V_m^{*}$ setpoint. The time domain-modulated voltage angle ${\theta }_m$ can be deduced from the phasor modulated voltage angle ${\delta }_m$ (${\theta }_m={\omega }_{m}t+{\delta }_m$).

4. Power–Frequency Response Modeling and Analysis

The objective of this section is to analyze the dynamic characteristics of the GFM converter and its compatibility with an actual AC system. In addition to ensuring synchronization and consistent power injection, the converter must also be capable of providing grid control support. Therefore, a GFM control is employed to provide an inertial effect, which aids in restricting the rate of change of frequency (RoCoF) following a serious load-generation disturbance.

The GFM converter connected with an AC system which is shown in Figure 3 is redrawn in Figure 5. In this figure, the AC power grid is replaced by its frequency response model, which is widely recognized for power system frequency and active power response analysis and relevant control synthesis issues. Frequency regulation support can be provided using the GFM converter by implementing an outer droop control loop. The aggregated turbine dynamics are simplified with a swing equation and an associated lead–lag filter [17,18]. To support the grid frequency, a governor has been added. The ratio between the rated powers of AC sources as given in Equation (4) is the basis for achieving a realistic situation. Its aim is to demonstrate how the GFM converter supports the AC grid and affects the system frequency response.

$$PR=100{\displaystyle \frac{S_{b1}}{S_{b1}+S_{b2}}}=100{\displaystyle \frac{S_{b1}}{S_T}} [\%]$$

(4)

PR, $S_{b1}$, and $S_{b2}$, are the power ratio, converter side base power, and grid side base power, respectively.

In this research, the GFM converter and AC grid deliver a total of 1000 MW of power ($S_T=1000 \mathrm{M}\mathrm{W}$). It is noteworthy that, here in the reduced-order frequency response model, the nonlinearities and fast dynamics of both the grid and GFM converter are neglected. Moreover, it is assumed that the system is already equipped with other required protection and control units such as a synchronizing system [19,20].

4.1. A Simplified Power–Frequency Response Model

The use of simplified models for power system analysis is a well-established practice, particularly while the focus is on studying system frequency response and active power characteristics. Similarly, in the context of modern power grids including GFM-based DGs connected to the grid, this modeling approach is acceptable [1]. In this subsection, the detailed non-linear model of the system under investigation for the sake of frequency response analysis and supplementary control design is reduced to the second-order model. Simulation findings also demonstrate that, for frequency and active power studies, the reduced second-order model can be employed as a substitute for the detailed dynamics model. This model simplification is specifically validated for supplementary control system analysis and synthesis, as its dynamics are slower than the dynamics of primary control layer.

Like Figure 4, a quasi-static model is utilized for the grid-connected GFM converter in Figure 6. According to Figure 6, a differential equation for small signal frequency and active power analysis can be written as follows:

$${\mathsf{\Delta }\delta }_m=\left[\left({\mathsf{\Delta }p}^{*}-{\mathsf{\Delta }p}_{GFM}\right){\displaystyle \frac{k_i}{{\omega }_b}}-\widetilde{{∆\omega }_g}\right]\left({\displaystyle \frac{{\omega }_c}{{\omega }_c+s}}\right){\displaystyle \frac{{\omega }_b}{s}}$$

(5)

Here, $k_i, {\omega }_{c }, \mathrm{and} {\omega }_{b }$ are controller gain, cut-off frequency, and base frequency, respectively. The active power flow between the GFM converter and the AC grid is calculated as follows:

$${\mathsf{\Delta }p}_{GFM}=K_C({\mathsf{\Delta }\delta }_m-{\mathsf{\Delta }\delta }_g)$$

(6)

where ${ K}_C$ is the synchronization coefficient of the converter. By combining (5) and (6), the following equation is obtained:

$${\mathsf{\Delta }p}_{GFM}={ K}_C\left[\left(\left({\mathsf{\Delta }p}^{*}-{\mathsf{\Delta }p}_{GFM}\right){\displaystyle \frac{k_i}{{\omega }_b}}-\widetilde{{∆\omega }_g}\right)\left({\displaystyle \frac{{\omega }_c}{{\omega }_c+s}}\right){\displaystyle \frac{{\omega }_b}{s}}-{\displaystyle \frac{{\mathsf{\Delta }\omega }_g{\omega }_b}{s}}\right]$$

(7)

Then, ${\mathsf{\Delta }p}_{GFM}$ can be expressed in terms of ${\mathsf{\Delta }p}^{*},$ $\widetilde{{∆\omega }_g},$ and ${\mathsf{\Delta }\omega }_g$:

$${\mathsf{\Delta }p}_{GFM}={\displaystyle \frac{1}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}{\mathsf{\Delta }p}^{*}+{\displaystyle \frac{\frac{{\omega }_b}{k_i}}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}\widetilde{{∆\omega }_g}-{\displaystyle \frac{\frac{{\omega }_b}{k_i}\left(1+\frac{s}{{\omega }_c}\right)}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}{\mathsf{\Delta }\omega }_g$$

(8)

Neglecting the frequency estimation error, $\widetilde{{(∆\omega }_g}={\mathsf{\Delta }\omega }_g)$, (8) can be rewritten as follows:

$${\mathsf{\Delta }p}_{GFM}={\displaystyle \frac{1}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}{\mathsf{\Delta }p}^{*}-{\displaystyle \frac{\frac{{\omega }_b}{k_i{\omega }_c}}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}{\mathsf{\Delta }\omega }_g$$

(9)

Ignoring the power losses in Figure 5, the power flowing from $v_{mabc}$ to $v_{gabc}$ is the same as the power flowing from $v_{gabc}$ to $v_{eabc}$. A similar relation as in Equation (6) can be derived by using ${\mathsf{\Delta }\delta }_g$ and ${\mathsf{\Delta }\delta }_e$. ${\delta }_e$ is the equivalent angle obtained from the grid frequency response model using the concept of the center of inertia frequency (angle) for a control area [1].

$${\mathsf{\Delta }p}_{GFM}=K_g({\mathsf{\Delta }\delta }_g-{\mathsf{\Delta }\delta }_e)$$

(10)

where $K_g$ is a synchronizing coefficient of the grid (with respect to its aggregated model) ($K_g\approx {\displaystyle \frac{V_{in}{V}_e}{X_c+X_g}}$). Using ${\mathsf{\Delta }\omega }_g$ and ${\mathsf{\Delta }\omega }_e$ results in the following:

$${\mathsf{\Delta }p}_{GFM}=K_g{\displaystyle \frac{({\mathsf{\Delta }\omega }_g-{\mathsf{\Delta }\omega }_e)}{s}}{\omega }_b$$

(11)

Assuming constant frequency in the equivalent frequency response model (${\mathsf{\Delta }\omega }_e=0$), the relation between ${\mathsf{\Delta }\omega }_g$ and ${\mathsf{\Delta }p}_{GFM}$ can be derived as follows:

$${\mathsf{\Delta }\omega }_g={\displaystyle \frac{s{\mathsf{\Delta }p}_{GFM}}{{\omega }_b}}\left({\displaystyle \frac{1}{K_g}}\right)$$

(12)

Finally, by replacing ${\mathsf{\Delta }\omega }_g$ from Equation (12) to Equation (9), a second-order model is obtained:

$${\mathsf{\Delta }p}_{GFM}={\displaystyle \frac{1}{1+\frac{X_c}{k_i{V}_m{V}_e}s+\frac{X_c+X_g}{k_i{V}_m{V}_e{\omega }_c}{s}^2}}{\mathsf{\Delta }p}^{*}$$

(13)

The reduced frequency response model given by (13) is compared to the full dynamic linear model and detailed non-linear model to demonstrate its accuracy. It is noteworthy that the detailed nonlinear model is implemented in the Sim-power toolbox of the MATLAB (R2024b) invironment. The full dynamic linearized model can be provided by Equations (14) and (15).

$$∆x=A∆x+B∆u$$

(14)

$$∆x=\left[\underset{grid current}{\underbrace{∆i_{cd}∆i_{cq}∆i_{gd}∆i_{gq}}}\underset{PCC voltage}{\underbrace{∆v_{gd}∆v_{gq}}}\underset{TVR}{\underbrace{∆{\zeta }_{TVRd}∆{\zeta }_{TVRq}}}\underset{coupling dynamics}{\underbrace{{\mathsf{\Delta }\delta }_m{\mathsf{\Delta }\omega }_m}}\underset{PLL}{\underbrace{∆{\zeta }_{PLL}∆{\delta }_{PLL}}}\right]$$

(15)

where TVR represents transient virtual resistance. The obtained results indicate that all models behave in a similar manner, which confirms the validity of the proposed simplified model for frequency response analysis. The active power and frequency dynamics for various control and system parameters can be analyzed using this model in the next subsection [16]. The linearized model represented by Equation (15) is a 12-order dynamic model.

4.2. An Updated Simplified Model

The behavior of a GFM converter under variable frequency of the equivalent grid control area is investigated in this subsection. Indeed, with the fixed frequency of the equivalent grid configuration (i.e., ${\mathsf{\Delta }\omega }_e=0$), the active power dynamics were impacted only by the grid impedance as proved by Equation (13). The variable frequency has a significant impact on the system dynamics because ${\mathsf{\Delta }\omega }_e$ is no longer null.

Equation (9) is employed to examine the dynamic of active power. In Equation (9), ${\mathsf{\Delta }\omega }_g$ must be expressed with respect to the converter and the AC grid parameters.

$${\mathsf{\Delta }\omega }_g={\displaystyle \frac{S_{b1}}{S_{b2}}}{\displaystyle \frac{({\mathsf{\Delta }p}_{GFM}-{\mathsf{\Delta }P}_{load})s}{K_g{\omega }_b}}+{\mathsf{\Delta }\omega }_e$$

(16)

${\mathsf{\Delta }P}_{load}$ is the power consumed by the domestic load. By inserting Equation (15) into Equation (9), the following expression is obtained:

$${\mathsf{\Delta }p}_{GFM}={\displaystyle \frac{1}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}{\mathsf{\Delta }p}^{*}+{\displaystyle \frac{\frac{{\omega }_b}{k_i{\omega }_c}}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}{\mathsf{\Delta }\omega }_e-{\displaystyle \frac{\frac{S_{b1}}{S_{b2}}\frac{1}{k_i{K}_g{\omega }_c}s}{1+\frac{1}{k_i{K}_C}s+\frac{1}{k_i{K}_C{\omega }_c}{s}^2}}({\mathsf{\Delta }p}_{GFM}-{\mathsf{\Delta }P}_{load})$$

(17)

Using the grid frequency response model in Figure 5, ${\mathsf{\Delta }\omega }_e$ can be written as follows [21]:

$$\begin{gathered} {\mathsf{\Delta }\omega }_{e=}{\displaystyle \frac{R_{SG}\left(1+T_{D}s\right)}{\left(1+\left(T_N+2H_{eq}{R}_{SG}\right)s+2H_{eq}{R}_{SG}{T}_D{s}^2\right)}}{\mathsf{\Delta }P}_g \\ {\mathsf{\Delta }P}_g\cong {\displaystyle \frac{S_{b1}}{S_{b2}}}({\mathsf{\Delta }p}_{GFM}-{\mathsf{\Delta }P}_{load}) \end{gathered}$$

(18)

$T_N, T_D, R_{SG}, and H_{eq}$ are the parameters of the grid frequency response model. By replacing ${\mathsf{\Delta }\omega }_e$ in Equation (17) with Equation (18), the transfer functions of the active power with respect to the setpoint ${\mathsf{\Delta }p}^{*}$ and ${\mathsf{\Delta }P}_{load}$ are obtained:

$${\mathsf{\Delta }p}_{GFM}={\displaystyle \frac{1+\left(T_N+2H_{eq}{R}_{SG}\right)s+2H_{eq}{R}_{SG}{T}_D{s}^2}{1+a{s+bs}^2+c{s}^3+d{s}^4}}{\mathsf{\Delta }p}^{*}+{\displaystyle \frac{1}{S_{b2}}}{\displaystyle \frac{\frac{R_{SG}{\omega }_b\left(1+T_{D}s\right)s}{k_i{\omega }_c}}{1+a{s+bs}^2+c{s}^3+d{s}^4}}{\mathsf{\Delta }P}_{load}$$

(19)

where $a$, $b$, $c$, and $d$ are given in Appendix A. The obtained transfer function is of the fourth order.

The control structure depicted in Figure 6 provides the potential for grid active power (frequency) tracking because of the incorporation of ${\omega }_g$. Since the present work is mainly focused on a grid with frequency support, Figure 6 must be updated to find a clear representation between the GFM converter and the grid frequency response models, i.e., a relationship between ${\mathsf{\Delta }p}_{GFM}, {\mathsf{\Delta }p}^{*}, {\mathsf{\Delta }\omega }_e$, and the given frequency set point (${\omega }_{set}$). The updated model is shown in Figure 7. In the updated control scheme, the output frequency${ \omega }_m$ is obtained as follows:

$${\displaystyle \frac{{\omega }_b}{{k_i\omega }_c}}{\displaystyle \frac{d{∆\omega }_m}{dt}}={∆p}^{*}-{∆p}_{GFM}-{\displaystyle \frac{{\omega }_b}{k_i}}(∆{\omega }_m-{∆\omega }_{set})$$

(20)

According to Figure 7, the differential equation presented in Equation (5) can be written as follows:

$${\mathsf{\Delta }\delta }_m=\left[\left({\mathsf{\Delta }p}^{*}-{\mathsf{\Delta }p}_{GFM}\right){\displaystyle \frac{k_i}{{\omega }_b}}+{\mathsf{\Delta }\omega }_{set}\right]\left({\displaystyle \frac{{\omega }_c}{{\omega }_c+s}}\right){\displaystyle \frac{{\omega }_b}{s}}$$

(21)

Then, Equation (6) can be updated as follows:

$${\mathsf{\Delta }p}_{GFM}={K_C}^{\prime }({\mathsf{\Delta }\delta }_m-{\mathsf{\Delta }\delta }_e)$$

(22)

where ${K_C}^{\prime }={\displaystyle \frac{V_m{V}_e}{X_c+X_g}}$ [18]. Then, the closed-loop transfer function is determined by putting Equation (22) in Equation (23) as follows:

$${\mathsf{\Delta }p}_{GFM}={\displaystyle \frac{1}{1+\frac{1}{k_i{K_C}^{\prime }}s+\frac{1}{k_i{K_C}^{\prime }{\omega }_c}{s}^2}}{\mathsf{\Delta }p}^{*}-{\displaystyle \frac{\frac{{\omega }_b}{k_i}(1+\frac{s}{{\omega }_c})}{1+\frac{1}{k_i{K_C}^{\prime }}s+\frac{1}{k_i{K_C}^{\prime }{\omega }_c}{s}^2}}({\mathsf{\Delta }\omega }_{set}-{\mathsf{\Delta }\omega }_e)$$

(23)

The steady-state error (ε) of the transfer function of Equation (23) can be expressed as follows:

$$\epsilon =\underset{s\to 0}{\lim }s\left[{\mathsf{\Delta }p}^{*}-{\mathsf{\Delta }p}_{GFM}\right]=-{\displaystyle \frac{{\omega }_b}{k_i}}({\mathsf{\Delta }\omega }_{set}-{\mathsf{\Delta }\omega }_e)$$

(24)

Considering Equation (24), in a steady state, if the grid frequency changes, the active power will be adjusted with respect to the following equation:

$${\mathsf{\Delta }p}^{*}-{\mathsf{\Delta }p}_{GFM}={\displaystyle \frac{{\omega }_b}{k_i}}\left({\mathsf{\Delta }\omega }_{set}-{\mathsf{\Delta }\omega }_e\right)$$

(25)

The above equation is equivalent to the representation of the primary frequency control, often known as droop control. However, because of the power mismatch, the frequency is derived, and this function is alternatively referred to as “Inverse droop control” [22,23]. Hence, the ratio of the feedback gain $k_i$ to the input frequency ${\omega }_b$ is referred to as the droop gain. It affects the distribution of the total load among several sources. Based on the findings in Equations (20) and (25), it can be inferred that the addressed control structure acts as a hybrid control that serves as both a primary frequency control and an emulator of inertial effects.

The polynomial function that represents the characteristic equation of the closed-loop system is obtained from Equation (23) as follows:

$$F\left(s\right)=1+{\displaystyle \frac{1}{k_i{K_C}^{\prime }}}s+{\displaystyle \frac{1}{k_i{K_C}^{\prime }}}{s}^2$$

(26)

The associated damping ratio $\zeta$ and the natural frequency${ \omega }_n$ are obtained as follows:

$$\zeta ={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{{\omega }_c}{k_i{K_C}^{\prime }}}}, {\omega }_n=\sqrt{k_i{K_C}^{\prime }{\omega }_c}$$

(27)

The simulation results indicate that the error between the simplified model and the full dynamic model can be ignored. Therefore, it may be first inferred that the separation between the inner and outer control loops is assured. Furthermore, the reduced model will suffice for quasi-static analysis. Using Equation (23), the updated simplified model for GFM power control with respect to the relevant power is a second-order system:

$${\displaystyle \frac{{\mathsf{\Delta }p}_{GFM}}{{\mathsf{\Delta }p}^{*}}}={\displaystyle \frac{1}{1+\frac{X_c}{k_i{V}_m{V}_e}s+\frac{X_c+X_g}{k_i{V}_m{V}_e{\omega }_c}{s}^2}}$$

(28)

Remark 1. 

In this study, the primary focus is on the dynamics of frequency and active power responses. Consequently, other dynamics are not addressed, and the analysis is limited to the perspectives of active power and frequency.

5. Proposed Robust Optimal Secondary Controller Design

5.1. Finding All Robust Stabilizing PI Controllers

By combining the PI-based supplementary controller and the inner power control loop given by Equation (28), the polynomial characteristic equation of the closed-loop system can be obtained. This equation enables the utilization of Kharitonov’s theorem to analyze the system robust stability. The closed-loop system, considering the PI controller ($K\left(s\right)=k_{sp}+{\displaystyle \frac{k_{si}}{s}}$), is defined as follows:

$$T\left(s\right)={\displaystyle \frac{k_i{V}_m{V}_e{\omega }_c{k}_{sp}s+k_i{V}_m{V}_e{\omega }_c{k}_{si}}{\left(X_c+X_g\right)s^3+X_c{\omega }_c{s}^2+\left(k_i{V}_m{V}_e{\omega }_c+k_i{V}_m{V}_e{\omega }_c{k}_{sp}\right)s+k_i{V}_m{V}_e{\omega }_c{k}_{si}}}$$

(29)

Considering the incorporation of a PI controller as the supplementary control layer for the case study presented in Figure 3, the characteristic equation of the closed-loop system is derived as follows:

$$∆\left(s\right)=\left(X_c+X_g\right)s^3+X_c{\omega }_c{s}^2+\left(k_i{V}_m{V}_e{\omega }_c+k_i{V}_m{V}_e{\omega }_c{k}_{sp}\right)s+k_i{V}_m{V}_e{\omega }_c{k}_{si}$$

(30)

Considering 50% variation in $X_c$ and $X_g$ (max $X_c=X_g=0.225 \mathsf{\Omega }$and min $X_c=X_g=0.075 \mathsf{\Omega })$as uncertain parameters, $c_i^{-}$ and $c_i^{+}$ in Equation (2) can be obtained, and Kharitonov’s polynomials can be calculated (i denotes the number of coefficients). The line parameters $X_c$ and $X_g$ are identified as uncertainties for two primary reasons: (i) The line parameters are derived from estimation methods, which inherently introduce uncertainties. This indicates that these parameters are not fixed values but are subject to variation. (ii) Alterations in grid topology can result in modifications to the line parameters. These changes can be modeled as uncertainties, reflecting the dynamic nature of the system. As given in Equation (30), the order of the closed-loop system is three, and as mentioned earlier, for applying Kharitonov’s theorem on a third-order characteristic equation, the Hurwitz criterion just needs to be tested for the third polynomial of Equation (2):

$$K_3\left(s\right)=c_0^{-}+c_1^{+}s+c_2^{+}{s}^2+c_3^{-}{s}^3$$

(31)

where

$${c_0^{-}=\mathrm{m}\mathrm{i}\mathrm{n}(k}_i{V}_m{V}_e{\omega }_c{k}_{si})$$

$$c_1^{+}=\max \left(k_i{V}_m{V}_e{\omega }_c+k_i{V}_m{V}_e{\omega }_c{k}_{sp}\right)$$

$$c_2^{+}=\max \left(X_c{\omega }_c\right)$$

$$c_3^{-}=\mathrm{m}\mathrm{i}\mathrm{n}(X_c+X_g)$$

To ensure the stability of $K_3\left(s\right)$, using a Routh–Hurwitz table, the following inequality must be derived, which provides the necessary and sufficient condition:

$$c_1^{+}{c}_2^{+}>c_3^{-}{c}_0^{-}$$

(32)

Considering the above inequality, it is possible to find the allowed values of $k_{sp}$ and $k_{si}$ to satisfy the robust stability. In the next section, the results of this stage are shown in detail.

In fact, the Kharitonov polynomial $K_3\left(s\right)$ leads to the determination of a family of PI parameters ($k_{sp-i}, k_{si-i}$) to keep stabilizing the closed-loop system in the presence of the perturbation of specified uncertain parameters. Figure 8 illustrates all members of the solution family for the PI parameters.

The zero-exclusion condition [13] is a fundamental concept for assessing system stability. In this study, the zero-exclusion condition is analyzed within the frequency range of $0<\omega <50$ kHz, and the resulting geometric representation is presented in Figure 9. According to the zero-exclusion condition, the considered closed-loop system is robustly stable because the rectangle plots do not encompass the origin of the plane. This condition is crucial in ensuring the robust stability of the system for specific parameter perturbation and disturbances.

5.2. Finding Optimal Robust Solutions

To achieve robust stability for the selected case study, a family of solutions for the proportional gain ($k_{sp-i}$) and integral gain ${(k}_{si-i})$ parameters has been obtained. This solution family ensures the stability of the closed-loop system under various operating conditions regarding parametric uncertainty. However, to determine the optimal values for $k_{sp-i}$ and $k_{si-i}$ to achieve the desired performance, a Bayesian optimization method has been employed.

To enhance robust optimal performance using Bayesian optimization, the system performance metrics are considered as the step response maximum overshoot and settling time to perform the required objective functions. The objective was to find the best values of $k_{sp-i}$ and $k_{si-i}$ that minimize these performance metrics.

Through the Bayesian optimization process, the best values for $k_{sp-i}$ and $k_{si-i}$ among the Kharitonov’s solutions family were identified. These values were selected based on their ability to minimize the maximum overshoot and settling time, indicating improved system performance. By using this optimization method, an optimal PI controller is obtained, ensuring both robust stability and optimal performance for the given case study.

To evaluate the performance of each PI parameter combination ($k_{sp-i}, k_{si-i}$), a cost function is defined. The cost function includes the performance metric obtained via the step response of the closed-loop system for each pair of $k_{sp-i}$ and $k_{si-i}$ values within the mentioned solution family. The step response is analyzed to determine the relevant settling time ($t_{s-i}$) and overshoot ($M_{p-i}$).

The objective function is calculated as a linear combination of the settling time and overshoot, where the settling time is multiplied by a weight of 10 to prioritize its minimization. The weight of 10 assigned to the settling time in the objective function was chosen based on the relative importance of minimizing the settling time compared to the overshoot. Since the settling time is a critical performance metric that directly impacts the system’s responsiveness and performance, we decided to prioritize its minimization by weighting it more heavily than the overshoot in the overall cost function. This approach reflects the fact that excessive settling time can be detrimental to system performance, even if the overshoot is relatively low.

The specific weight of 10 was determined through accurate testing and sensitivity analysis, as it provided a good balance between minimizing the settling time while also considering the overshoot. However, the choice of the weight factor is ultimately dependent on the specific requirements and priorities of the system under study. In a different application, the relative weights assigned to the settling time and overshoot in the objective function may need to be adjusted accordingly. The optimization process includes four main steps:

Step 1: Determine the step response for the given $k_{sp-i}$ and $k_{si-i}$.

Step 2: Calculate the corresponding settling time ($t_{s-i}$) and overshoot ($M_{p-i}$).

For a step response vector, $t_{s-i}$ and $M_{p-i}$ can be calculated as follows:

$${t=t}_{s-i}; if \left|{y(t}_i)-1\right|<0.01$$

(33)

$$M_{p-i}=\underset{i}{\max }(y_i-1)$$

(34)

Step 3: Calculate the corresponding cost function (CF) as follows:

$$CF=10\times t_{s-i}+M_{p-i}$$

(35)

Step 4: If the optimization criteria are satisfied, stop; otherwise, $i=i+1$ and go back to step 1.

The developed algorithm flowchart is shown in Figure 10. The key aspect here is that the Bayesian optimization algorithm adjusts the values of $k_{sp-i}$ and $k_{si-i}$ in an iterative manner, guided by the internal model’s understanding of the relationship between the parameter values and the CF. This allows the algorithm to efficiently explore the parameter space and converge to the optimal values.

By iteratively investigating different combinations of $k_{sp-i}$ and $k_{si-i}$ values and evaluating their performance using the defined CF, the Bayesian optimization algorithm determines the optimal values that minimize the CF, providing an efficient and data-driven approach to tuning the controller parameters for improved system performance as desired.

6. Simulation and Laboratory Real-Time Results

6.1. Simulation Results

In this section, the performance evaluation of the proposed supplementary control is presented. All simulations were conducted using the Julia programming language version 1.10.2 [24], on a desktop machine with a MacBook Pro M2 system. The following Julia packages are used to implement the proposed control methodology: ControlSystem.jl, CSV.jl, DataFrame.jl, Plot.jl, and PlotlyJS.jl [24,25,26].

To evaluate the effectiveness of the proposed controller, a step input is applied to the nonlinear model of the system equipped with a robust PI controller, where $k_{sp}$ and $k_{si}$ are randomly selected from the Kharitonov solution family (Figure 9). As illustrated in Figure 11, the closed-loop system is robust and stable in the face of parameter perturbation for all pairs of $k_{sp}$ and $k_{si}$ selected from the Kharitonov solution family.

As a second test scenario, the system behavior is investigated and compared in two stages. In the first stage, the PI parameters are randomly selected from the family of all stabilizing Kharitonov-based solutions (${[k}_{si-i}, k_{si-i}]=[25, 16]$). The step response for the closed-loop system is illustrated in Figure 12, in which the system exhibited sufficient stability when subjected to a 50% variation in the $X_c$ and $X_g$ parameters. In the second stage, as shown in the same figure, the system response is significantly improved when employing the optimal PI gains obtained by the applied Bayesian optimization algorithm ([$k_{si.opt}, k_{sp.opt}]=[1.001, 1.087]$). The green curve in the same figure illustrates the system response for the same scenario, but without considering the support control layer for frequency. In this case, the controller was tuned using a trial-and-error method.

Thus, the results of this study demonstrate that the proposed control approach not only ensures robust stability but also enables the achievement of optimal performance for the system in the presence of 50% variation in the $X_c$ and $X_g$ parameters.

6.2. Laboratory Real-Time Results

Figure 13 presents laboratory real-time results obtained using Typhoon HIL to evaluate the performance of the proposed method. Figure 13a illustrates the active power and frequency response during a load increase (active power—red; frequency—blue). Initially, only DG1 supplies the load. However, at t = 0.2 s, the load demand rises, causing an increase in active power. This rise in demand is related to a decrease in frequency. Notably, at t = 0.3 s, the GFM converter, supported by supercapacitors, activates, effectively restoring the frequency to a stable value (50 Hz). In Figure 13b, the properties of the supercapacitors are shown, and the supportive GFM converter is indicated to be “on” by a green light. The over-voltage protection mechanism remains inactive during this study, while the supercapacitors operate in discharging mode to provide active power. Key metrics such as the charge–discharge current set-point (defaulted to 0.5 pu), output voltage ($u_{dc}$), and state of charge (SoC), which reaches 98.95% after contributing to frequency control, are also highlighted. Finally, Figure 13c shows the experimental setup employed at Nagoya University, providing context for the implementation of these results.

7. Conclusions

This paper introduces an optimal supplementary frequency and active power regulation approach for an AC grid with a GFM converter in the presence of parameter perturbation. The GFM converter provides regulation power support from an energy storage system. The proposed strategy involves the application of a robust PI controller designed by the Kharitonov method. The PI controller is tuned to address uncertainties and disturbances commonly encountered in power grid operations, resulting in improved system stability and performance.

Using Kharitonov’s theorem, the PI controller demonstrates robust stability, effectively mitigating system uncertainties and contributing to enhanced system stability and performance through frequency regulation. Moreover, based on the assigned performance targets, Bayesian optimization is employed to select optimal parameters from the Kharitonov solution family. The integration of Bayesian optimization ensures optimal parameter selection, further enhancing the effectiveness of the proposed approach.