Contents Introduction
Microgrids are more vulnerable to becoming unstable due to faults and large disturbances than larger and interconnected power grids because microgrids are small-scale and localized power systems with low or no inertia distributed energy resources (DERs). Evaluating the robustness of microgrids in the face of significant disturbances, the process referred to as large-signal stability assessment, helps to understand the dynamic behavior of DERs, ensures stable operation during islanded mode and generation shortages, addresses nonlinearities and control interactions, and ensures transient and steady-state stability.
The importance and distinction of large-signal stability analysis for microgrids can be attributed to their unique characteristics and operational requirements—DERs in microgrids exhibit fast dynamic responses due to their electronic converters and power electronics interfaces [1], [2], [3]. Their response to large disturbances, such as sudden load changes or faults, can significantly impact the overall system stability. Moreover, microgrids can operate in an islanded mode when they are disconnected from the main utility grid. In this mode, the microgrid must maintain stable frequency and voltage without external support [4]. Additionally, microgrids involve complex interactions between various control loops and power electronics converters. Nonlinearities, such as voltage and current limits and control interactions, can affect the system’s stability. Furthermore, the high penetration of power electronics-based DERs in microgrids can introduce harmonic distortion and resonance phenomena [5], [6], [7], [8]. Harmonics can affect system stability and lead to instability if not properly addressed.
Large-signal stability analysis considers both transient and steady-state stability aspects [5], [7]. It assesses the system’s response to large disturbances, such as faults or sudden changes in generation or load, and examines whether the system can recover and reach a stable operating point. This analysis ensures the robustness and reliability of microgrid operation under different operating conditions.
Several studies evaluate the stability of inverter-based DERs using various stability analysis methods and dynamic models of inverter-based resources (IBR). However, there are only a few comprehensive review studies that focus on the large-signal stability assessment of microgrids and summarize existing methods and approaches in a single study. Current review papers do not provide a comparative and analytical assessment of existing methods and approaches that are needed to examine the accuracy and authenticity of the large-signal stability analysis of microgrids.
In [9], authors provide a classification of the stability analysis based on the disturbance and duration, list the important features of microgrid and inverter-interfaced networks that must be considered in stability analysis, and process the factors that impact the large-signal stability of a converter-based microgrid. Moreover, the paper has tabulated the stability analysis tool and techniques. The work in [10] investigates the existing valuable studies about large-signal stability analysis on the distribution systems and IBRs as well as applications of Lyapunov-based stability assessment in detail.
The work in [11] presents one of the well-known reviews in the area of large-signal stability assessment. The authors have investigated the large-signal stability of the DC power system that includes motor drives and constant power load. Different direct methods for large-signal stability analysis of DC power systems are processed, including Takagi-Sugeno (T-S) multi-modeling, Brayton-Moser’s mixed potential, block diagonalized quadratic Lyapunov function, and reverse trajectory tracking. The application of the T-S fuzzy modeling as one of the direct methods for stability analysis of a system with a DC-link connection is also proposed. Furthermore, the work in [11] proves the performance of the proposed method by providing numerical analysis and experimental results. The case study is an electric motor connected to a DC power source via an inadequately damped LC filter that is examined under the listed tools.
In [12], a comprehensive study on all aspects of stability challenges in microgrids is provided. The paper focuses on techniques for small-signal stability assessment and discusses the important contents of the stability assessment of microgrids under both small and large disturbances. Review studies [13] and [14] provide a review of the voltage stability of microgrids and categorize the voltage stability based on the operation modes (grid-connected or islanded), disturbance intensity, and controlling system. In [15], the stability of the microgrid is assorted based on the fault type, disturbance duration, and different energy resources. Also, the stability challenges of microgrids for both grid-connected and islanded modes are discussed. The work in [16] classifies the models and approaches of the stability analysis based on the large- and small-signal analysis. Moreover, the authors review the advantages and disadvantages of the models and approaches.
Although the aforementioned studies provide valuable reviews of existing research on the large-signal stability assessment of microgrids and IBRs, none of them analytically provides comparative scrutiny with numerical assessment and proofs in the large-signal assessment of AC microgrids. In other words, a critical review needs to compare and analyze different stability analysis methods and approaches to identify the accuracy and integrity of microgrid stability analysis. A numerical, analytical study and review of large-signal stability assessment of microgrids can help to determine the advantages and disadvantages of existing methods and approaches as well as introduce the best possible applications for each large-signal stability technique.
The compelling benefits, stability concerns associated with microgrids, and the research gap for a comparative investigation of different stability assessment techniques, as outlined previously, have inspired our concentration on their stability assessments. This involves a thorough exploration of pertinent definitions and a comprehensive, analytical scrutiny of the existing scholarly literature on the topic.
In this study, a comparative perusal of the large-signal stability assessment of AC microgrids and IBRs is provided such that the system’s large-signal stability analysis is numerically scrutinized based on the different dynamic models and different mathematical methods for stability analysis. Based on the obtained results, the advantages and disadvantages of each method and dynamic model are discussed. This paper not only reviews and investigates the exciting techniques and methods for the microgrid stability assessment but also presents a numerical, comparative assessment of the large-signal stability analysis of IBRs, which has not been addressed in the other large-signal stability assessment surveys [9], [10], [13], [14]. In this context, our work is focused on assessing the large-signal stability of the system without the implementation of any stabilization techniques. The study objective is to identify crucial parameters from a system stability standpoint. The contributions of this study are summarized as follows:
Provide a comprehensive literature review on the large-signal stability assessment of AC microgrids and classify existing stability analyses based on applied methods, microgrid dynamic models, and system configurations and characteristics.
Propose a detailed mathematical dynamic state-space model and modify stability assessment methods for Lyapunov-based stability analysis accordingly.
Conduct numerical assessments and comparative analytical review using the Lyapunov-based stability assessment including Krasovskii’s, Popov-Lure, and sum of squares(SOS) method.
Evaluate microgrid dynamic models including time-domain differential equations and synchronverter models in terms of accuracy, scalability, and computational time.
The rest of the paper is organized as follows (classified in Figure 1). Section II presents the challenges of stability analysis in a microgrid. In this Section, classifications of small- and large-signal stability assessment are expanded and discussed, and the concept of the domain of attraction (DOA) and critical clearing time (CCT) is elaborated in detail. Section III represents the existing dynamic model of inverter-based DERs. In Section IV, the techniques and methods for stability analysis are developed and discussed. Section V develops a case study and numerically expatiates the approaches of existing stability analysis. The numerical analysis includes modeling, simulation, analysis, and comparison of the results. In Section VI, a discussion on the advantages and disadvantages of the simulated approaches is provided and the conclusion is presented.
Stability Challenges in Microgrids
The stability analysis of microgrids is generally classified into two main categories (small- and large-signal stability analyses), which are explained in the following sections in detail. The small-signal stability is defined as steady-state stability for small disturbances, and the large-signal stability is defined as transient stability for large disturbances. Also, the large-signal stability assessment can present the microgrids’ dynamic behavior during small disturbances. Although the main goal of this paper is to discuss and analyze the stability of AC microgrids to large disturbances, we briefly present the small-signal stability analysis for the sake of comparison and emphasize the importance of large-signal stability analysis for microgrids.
A. Steady-state (Small-signal) Stability Study
Small-signal stability analysis of a microgrid involves studying the stability of the system under small disturbances and deviations from its steady-state operating conditions. The small-signal stability studies of inverter-based microgrids have been mainly conducted based on eigenvalue analysis and impedance-based approach [17], [18].
Using the eigenvalue technique, a state-space model of the system is linearized around the steady-state operating point to obtain the eigenvalues. The eigenvalues are then used as a stability criterion. For instance, when all eigenvalues of a continuous state-space model have negative real parts, the system is called stable. However, having at least one eigenvalue with a positive real part results in system instability. For instance, the eigenvalue assessment is applied to a system dynamic model to analyze the impact of changing controller parameters in inverters [19], filters, line, and load specifications on the system stability.
In addition, the sensitivity analysis of eigenvalues with respect to parameter changes holds significant importance within this field. Thus, the parameters that may affect the stability are realized and can be used as a viable solution for designing optimal controllers in the system. In [20], a novel Laplacian matrix eigenvalue is introduced to build a certified stability region. The computation burden of the proposed method is almost independent of the inverters’ penetration level in the distribution grid. In [21], a small-signal stability criterion based on the extended Gershgorin Theorem is formulated by reshaping the range of eigenvalue estimates. The test results suggest that the proposed method possesses the capability to evaluate the stability of high-order systems with minimal computational complexity. However, the method is applied for DC microgrids, not AC grids.
In [22], the stability analysis model of an islanded microgrid based on a linearized state-space model is broadened to encompass the dynamics of a smart load alongside converter-interfaced distributed generators.
Another technique used in small-signal studies is impedance-based analysis [23], [24], where the load and sources are represented by their input and output (I/O) impedance. This method evaluates the whole system instead of details of inner subsystems. It applies the Nyquist criterion to the ratio of output and input impedance to assess stability. The average linearization model is usually adopted for this analysis.
In [25], an efficient technique based on the perturbation signal quadratic residue binary (QRB) sequence is proposed for impedance measurement and stability improvement of microgrids with an adaptive control algorithm. The study given in [26] proposes a numerical-based technique to obtain impedance models of power-electronic dominated grids. The proposed technique using the finite-difference method shows accurate results compared with perturbation and analytical methods. In [27], authors address the small-signal assessment of AC islanded microgrid and show that active filtering function remarkably affects current-controlled converters (CCC), phase-locked loop (PLL), and the dynamic behavior of microgrid. They also offer useful guidelines for designing the PLL controller for multi-functional inverters.
Microgrid modeling has also been an important topic in small-signal analysis. In [28], the limitation of different reduced-order models with virtual impedance is studied by evaluating the computation time, the parametric stability boundary, eigenvalues, and the relative error of the natural frequency and damping factor. In [29], the reduced-order model of the microgrid’s distributed generator is developed, including virtual impedance and internal control. The authors also demonstrate that using quasi-stationary virtual impedance led to a larger stability region compared to the transient virtual impedances.
Small-signal stability assessment scrutinizes the stability of microgrids under small changes, which makes the results valid only around system operating points. Since microgrids are prone to both large and small disturbances, the results of small-signal analysis cannot be generalized. In contrast, large-signal stability assessment uses nonlinear models, enabling it to comprehensively evaluate system stability under both small and large disturbances. However, this can add computational complexity to the stability studies. Critically discussing the advantages and disadvantages of existing large-signal stability analysis methods and their associated computational challenges using theoretical and numerical analyses is the main motivation of this paper.
B. Transient (Large-signal) Stability Analysis
Large-signal stability analysis for transmission power systems has been extensively studied in the literature [30], [31], and several methods have been commonly adopted by power companies. On the other hand, large-signal stability analysis for microgrids is still evolving especially with the high penetration of IBRs, where each method is applicable under certain system conditions. Applying the methods proposed for the stability assessment of the conventional power grids cannot present comprehensive and authentic results for AC microgrids that have complicated structures such as IBRs. The main differences between microgrids and conventional power grids are itemized as follows:
The microgrid includes inverters and other power electronics devices, making their dynamic response faster than that in transmission power systems equipped with mechanical components used in the generators [32].
The complexity and nonlinearity of the dynamic model of the microgrids are elevated by the utilization of power electronic devices, rendering them arduous to investigate the stability [33].
The IBRs in microgrids have lower inertia (or no inertia) compared to the synchronous generators and electrical machines in transmission grids [34], decreasing the system’s stability and increasing its susceptibility to disturbances.
The
R/X Microgrids are considered small energy systems, and any changes and disturbances in microgrids can have an extreme effect on the whole system’s operation.
Microgrids are susceptible to experiencing unbalanced loads and power flow, leading to adding more complexity and nonlinearity to dynamic model [35], [36].
Consequently, the large-signal stability analysis for the inverter-based AC microgrid is inevitably complicated and needs close and extensive investigation. As the large-signal stability analysis of microgrids based on the time-domain simulation is an extremely time-consuming process, recently, research studies have focused on the mathematical and theoretical methods to appraise the large-signal stability in a computationally tractable manner.
Generally, the large-signal stability analysis can be classified based on the system dynamic model (discussed in Section III) and stability assessment method (discussed in Section IV), as shown in Figure 2.
In the realm of dynamic model-based classification for inverter-based microgrids, two main categories exist: the full- or reduced-order dynamic model and the synchronverter (second-order) model. The full- or reduced-order dynamic model is defined based on the time-domain differential equations of IBRs and circuit laws, including both controlling systems and circuits (discussed in Section V-B). For the synchronverter (second-order) model proposed by [37] and [38], it has been assumed that an IBR mimics the dynamic behavior of a synchronous generator (SG).
Moreover, for the classification of large-signal stability analysis of IBRs based on the stability analysis tools, there are a variety of methods performed based on the Lyapunov-based stability theory. Popov-Lure (discussed in Section IV-B) [39], [40], [41], Krasovskii’s method (discussed in Section IV-A) [42], [43], [44], linear matrix inequality (LMI)-based method (e.g., Takagi-Sugeno (T-S) fuzzy) (described in Section IV-D) [4], [10], [11], [45], [46], [47], [48], [49], [50], and the sum of squares methods (discussed in Section IV-E) [51], [52], [53] are among the variety of methods used.
Choosing an appropriate method for stability analysis as well as an accurate dynamic model for large-signal stability analysis of inverter-based microgrids are important factors, which vary depending on the application, microgrid configuration, and case study. For instance, in online applications where both fast and accurate responses are required, it is essential to utilize a dynamic model that is simple with reasonable approximations. Additionally, the selected stability analysis method must be fast and computationally tractable to meet real-time requirements.
On the other hand, for conducting a post-fault analysis to adjust the protection system and relay settings, the time-consuming computational process imposed by using an accurate and detailed dynamic model and the stability analysis method is not considered as an obstacle.
There are a few research studies addressing the complexity of the large-signal stability assessment of microgrids tabulated in Table 1. The applied method and investigated dynamic model are highlighted for each study. Moreover, there are a few survey studies on large-signal stability assessment of microgrids listed in Table 2. Generally, the large-signal stability assessment for microgrids has been conducted for a single IBR connected to an infinite bus. However, there are a few perusals that address the challenges of the transient stability of microgrids that are constructed with more than one IBR, such as [43], [50], and [54] (detailed in Table 1). It is worth noting that the main focus of this paper is on the theoretical and analytical large-signal stability analysis of AC microgrids rather than DC microgrids. However, there are a variety of studies on DC microgrids and the challenges of large-signal stability of the DC network [52], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68].
C. Critical Clearing Time and Domain of Attraction
The stability analysis has been carried out for two main purposes: determining critical clearing time (CCT) for temporary disturbances and the feasibility assessment of stable transitions for permanent disturbances [79], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94]. The CCT refers to the time duration following the occurrence of a contingency or disturbance in the power system. It is necessary to clear the contingency and restore the system to its pre-fault state within this duration in order to maintain stable operation. In other words, CCT represents the maximum duration during which a power system can tolerate a disturbance and remain stable without experiencing a system-wide collapse [95], [96], [97].
The CCT is an important factor in power system planning and operation since it serves as a crucial parameter by providing system operators with a time limit to take necessary corrective actions during events. It is also used as a criterion for designing protective relays and determining the necessary backup protection actions [54], [98], [99], [100], [101]. This time is calculated based on the system parameters and network topology considering the fault location and type. Typically, precise CCTs for each contingency can be determined by conducting a suitable stability analysis of the grid, utilizing either time-domain simulation or theoretical evaluation.
In the case of temporary disturbances, the main objective of the stability analysis is finding the possible maximum time to clear the fault or contingencies without any blackouts or losing system operation—i.e., the fault or disturbance must be cleared before the determined clearing time duration to guarantee a stable operation. To determine the CCT using large-signal stability assessment [102], [103], [104], [105], [106], the evaluation and determination of the system parameters’ critical boundaries and limits are undertaken to ensure system stability. The driven boundaries help in setting the protection system in a more realistic and less conservative manner to guarantee a stable operation and prevent unnecessary trips. As a consequential result, this approach allows for optimal utilization of the majority of the system’s capabilities during transient faults and contingencies.
The other application for applying the large-signal stability assessment is for permanent and large disturbances or changes in the system. The assessment of large-signal stability during permanent disturbance is to determine whether the system can reach its new steady-state operational/equilibrium point (EP) after the system changes.
To ensure a stable transition regardless of permanent or temporary events, the disturbance trajectory must be inside the stability region of the system around the EP, representing the domain of attraction (DOA)—i.e., the domain of attraction, alternatively identified as the stability region, encompasses the set of all initial states that eventually converge to a specific equilibrium point of a system. Essentially, it demarcates the range of initial conditions that give rise to a stable state within the system. Each operational point or equilibrium point is confined to a specific domain of attraction [107], [108]. For each DOA, a critical energy boundary is defined as determining the unstable boundary for any initial point and disturbance trajectories such that if the disturbance trajectory passes the critical energy boundary, the system cannot return to its original steady-state operation point or equilibrium point and experiences an unstable scenario or blackout [95], [96].
For a temporary disturbance, the critical energy boundary is exploited to identify CCT [109]. For a permanent disturbance, if the pre-disturbance equilibrium point is not inside of the DOA of the new equilibrium point, the system experiences instability during the transient response. In other words, the previous operating point works as an initial point for the new equilibrium point to the steady-state operating point. According to the DOA definition [107], [108], if this initial point (previous steady-state operating point) is not inside of the DOA of the new steady-state operating point, trajectories will not converge to the new equilibrium point of the system, and cannot achieve a stable state within the system.
In this regard, for a specific transient stability assessment for permanent disturbances and changes in the system, the previous equilibrium point can be considered as a set of initial points and new trajectories for the new equilibrium point that is cleared at the time of starting permanent change or contingency. If the previous steady-state operation point is inside of the DOA of the new equilibrium point, it means that the energy of the system at clearing time is not more than the critical energy boundary of the new equilibrium point, and the system will successfully transfer to the new steady-state operational point.
For more clarification, consider a new equilibrium point in the middle of the light-green area shown in Figure 3; since the pre-disturbance equilibrium points
Generally, there are two ways to obtain the domain of attraction of a specific equilibrium point in a system, including time-domain simulation and direct method (e.g., Lyapunov-based methods) [46]. As mentioned before, although the utilization of time-domain simulation is meticulous and authentic, it poses challenges in terms of computational burden and process complexity. On the other hand, direct methods offer not only time and cost efficiency but also achieve acceptable accuracy levels when compared to the accuracy of time-domain simulations [10], [110].
This study focuses on the large-signal stability analysis of inverter-based microgrids based on the direct methods and Lyapunov-based stability approaches. In the following sections, the challenges of large-signal stability assessment of IBRs are analytically discussed, and the studies in this area are numerically reviewed in detail. The numerical analysis and comparative study are presented for the existing techniques and methods.
Microgrid Component Dynamics and Modeling
The primary challenges associated with utilizing dynamic models for IBRs lie in achieving a balance between acceptable model accuracy and maintaining low computational costs during the stability analysis process. This is particularly important for online applications where the system needs to respond quickly to changes in system conditions. To address this problem, the reduced-order dynamic models of IBRs are introduced. They are designed to be computationally efficient and have reasonable approximations. This can be achieved by simplifying the model and reducing the number of states in the state-space model of the system [111], [112].
The reduced-order dynamic models of inverter-based DERs can provide a reasonable approximation of the system’s dynamic behavior, satisfying the accuracy of the model. It is worth noting that the accuracy of the reduce-order dynamic models depends on the level of detail included in these models and the complexity of the system being analyzed. Therefore, to ensure the high accuracy of the dynamic model, it may be necessary to include more states and details in the state-space model of the system dynamics [46], [113].
The full-order dynamic model can be modified to a reduced-order model based on the approximation and desired outputs and achieved different numbers of order, including 11th, 9th, 7th, 5th, 3rd, and 1st order. The details of the reduced-order models can be found in [46]. The full-order model is described and discussed in Section V-B.
The virtual synchronous generator (VSG) model can be deemed as a reduced-order dynamic model of IBRs. According to the VSG model, IBR’s dynamics emulate the dynamic behavior of a synchronous generator in a power system. In fact, the VSG model is a second-order dynamic model of IBRs and is simple to apply for any online control application. The details of the VSG model are discussed in Section V-A.
Depending on the application and intended results, an appropriate dynamic model for inverter-based microgrids must be chosen. Nevertheless, the dynamic model is not the sole crucial element in the large-signal stability analysis of microgrids. The methodology for acquiring the system’s stability region and defining stability criteria is another vital aspect of large-signal stability analysis for microgrids. In the subsequent section, various stability analysis methods are elucidated and formulated.
Techniques for Constructing DOA With the Lyapunov-Based Method
The Lyapunov-based method is a powerful tool used in control theory for analyzing the stability and domain of attraction of dynamic systems. The Lyapunov function search technique involves searching for a Lyapunov function that satisfies certain conditions, such as being positive definite and having a negative definite derivative [114], [115], [116]. Once a Lyapunov function of a system is found, the DOA of the system around a specific equilibrium point can be estimated as the largest set of initial conditions for which the function is positive, and the derivative of the function is negative.
There are several techniques for constructing the DOA of a system around a specific equilibrium point using this method (e.g., LMI-based method, Krasovskii’s, Popov-Lure, SOS-based). These techniques provide powerful tools for constructing the domain of attraction using the Lyapunov-based method. The choice of a technique depends on the specific system being analyzed and the complexity of the Lyapunov function needed to characterize the system’s dynamic behavior. The existing well-known methods for stability analysis can be classified as follows.
A. Krasovskii’s Method
One of the simplest methods to construct the Lyapunov function for a nonlinear system is Krasovskii’s method. In particular, Krasovskii’s method involves constructing a Lyapunov function that satisfies certain conditions, such as being positive definite and having a negative definite derivative along the system trajectories. This method is often used in the analysis and design of nonlinear control systems, and it has been shown to be effective for a wide range of applications [42], [117].
Mathematically, for constructing a Lyapunov function of a nonlinear system based on Krasovskii’s method, the state-space model of the system must be defined as follows:
\begin{equation*} \dot {\mathbf {x}}=f(\mathbf {x}),~~\text {and }~~f(0)=0. \tag{1}\end{equation*}
The Lyapunov function for this system can be defined as follows:\begin{equation*} V(\mathbf {x})=f^{T}(\mathbf {x})\;\mathbf {P}\;f(\mathbf {x}), \tag{2}\end{equation*}
\begin{equation*} \dot {V}(\mathbf {x})=\dot {f}^{T}(\mathbf {x})\;\mathbf {P}\;f(\mathbf {x})+f^{T}(\mathbf {x})\; \mathbf {P}\;\dot {f}(\mathbf {x}), \tag{3}\end{equation*}
\begin{align*} &\dot {f}(\mathbf {x})= \frac {\partial {d}f(\mathbf {x})}{\partial {d}\mathbf {x}} \frac {\mathrm {d}\mathbf {x}}{\mathrm {d}t}= \frac {\partial {d}f(\mathbf {x})}{\partial {d}\mathbf {x}} \dot {\mathbf {x}}= \frac {\partial {d}f(\mathbf {x})}{\partial {d}\mathbf {x}} f(\mathbf {x}), \tag{4a}\\ &\text {Jacobean matrix: } J_{f}(\mathbf {x})= \frac {\partial {d}f(\mathbf {x})}{\partial {d}\mathbf {x}}. \tag{4b}\end{align*}
Therefore, the derivative of the Lyapunov function can be modified as follows:\begin{align*} \dot {V}(\mathbf {x})&=f^{T}(\mathbf {x})J_{f}^{T}(\mathbf {x})\;\mathbf {P}\;f(\mathbf {x})+f^{T}(\mathbf {x})\; \mathbf {P}\;J_{f}(\mathbf {x}) \,f(\mathbf {x}), \\ \dot {V}(\mathbf {x})&=f^{T}(\mathbf {x})\biggl (J_{f}^{T}(\mathbf {x})\;\mathbf {P}+ \mathbf {P}\;J_{f}(\mathbf {x})\biggr )\;f(\mathbf {x}), \\ ~~ \text {such that }~~\mathbf {Q}&=-\biggl (J_{f}^{T}(\mathbf {x})\;\mathbf {P}+ \mathbf {P}\;J_{f}(\mathbf {x})\biggr ), \tag{5}\end{align*}
B. Popov-Lure
If the state-space model of a system is defined as follows:\begin{align*} \begin{cases} \displaystyle \dot { {\mathbf {x}}}=\mathbf {A} {\mathbf {x}} +\mathbf {b}\,f(\tau )\\ \displaystyle \tau =\mathbf {c}^{T} {\mathbf {x}} \end{cases}~, \tag{6}\end{align*}
\begin{equation*} G(s)=-\mathbf {c}^{T}\bigl [\,s\mathbf {I}-\mathbf {A }\bigr ]^{-1}\,\mathbf {b}~. \tag{7}\end{equation*}
Lure theory and Popov criteria ([118] and [119]) can provide conditions for stability analysis. According to Kalman [120], the state-space model of the system can be transferred to a reduced-order model as follows:\begin{align*} \begin{cases} \displaystyle \dot { \tilde {\mathbf {x}}}=\mathbf {F} \tilde {\mathbf {x}} -\mathbf {g}\,f(\tau )\\ \displaystyle \\ \displaystyle \dot {\mu }=-f(\tau )\\ \displaystyle \\ \displaystyle \tau =\mathbf {h}^{T} \tilde {\mathbf {x}}+\alpha \,\mu \end{cases}~, \tag{8}\end{align*}
\begin{equation*} \tau \,f(\tau ) >0 \;,\;\forall \,\tau \not =0 \;, \text {and} \quad f(0)=0~. \tag{9}\end{equation*}
The aforementioned system has two parts, linear and nonlinear parts determined by \begin{equation*} G(s)=\biggl (\mathbf {h}^{T}\,\bigl [\,s\mathbf {I}-\mathbf {F }\bigr ]^{-1}\,\mathbf {g}\biggr )+\frac {\alpha }{s}~. \tag{10}\end{equation*}
According to the Lure-Lyapunov lemma [120], to construct the Lyapunov function, the two parameters in the following equation, real number \begin{equation*} V( \tilde {\mathbf {x}},\mu ,\tau )= \tilde {\mathbf {x}} ^{T} {\mathbf {K}}\, \tilde {\mathbf {x}} +\frac {1}{2}\,\alpha \,\mu ^{2}+\,\gamma \int _{0}^{\tau } f(\tau ) d\tau ~. \tag{11}\end{equation*}
To find the required parameters in (11), the next steps listed below need to be followed:\begin{align*} \begin{cases} \displaystyle {\scriptstyle \bullet } \quad & W(\omega )=\Re \biggl [(1+\gamma j\omega )\,G(j\omega )\, \Gamma (j\omega )\,\Gamma (-j\omega )\biggr ]\\ \displaystyle \\ \displaystyle \quad & \quad \text {where} \begin{cases} \displaystyle \Gamma (s)=\text {det} \bigl [\,s\mathbf {I}-\mathbf {F }\bigr] \\ \displaystyle \gamma \geq \text {0 and is a real number such that: } \\ \displaystyle \Re \biggl [(1+\gamma j\omega )\,G(j\omega )\biggr ]\geq 0 \quad ,\quad \forall \omega >0 \end{cases}\\ \displaystyle {\scriptstyle \bullet } \quad & \text {Find } \phi (j\omega )\, \in \; W(\omega )= \phi (j\omega )\,\phi (-j\omega )\\ \displaystyle {\scriptstyle \bullet } \quad &\text {Consider a real vector} \mathbf {q} \text {such that: } \mathbf {q}= a\, \varrho (z)\\ \displaystyle \quad & \quad \text {where} \begin{cases} \displaystyle a \text {can be any real numbers.}\\ \displaystyle \,\varrho (z)= \sqrt {r}\,\Gamma (z)-\phi (z)\;,\quad \forall \;j\omega =z \\ \displaystyle r=\gamma \,(\alpha +\mathbf {h}^{T}\,\mathbf {g}) \end{cases}\\ \displaystyle {\scriptstyle \bullet } \quad & \text {Find } {\mathbf {K}} \in \; \begin{cases} \displaystyle {\mathbf {F}}^{T} {\mathbf {K}}+ {\mathbf {K}} {\mathbf {F}}= -\mathbf {q}\mathbf {q}^{T},\\ \displaystyle \text {where } {\mathbf {K}} \text {is positive definite.} \end{cases} \end{cases} \tag{12}\end{align*}
By following the aforementioned steps in (12), the Lure-Lyapunov function in (11) can be obtained. To determine the region of stability, two following boundaries obtained by finding the local maximum of (11), are defined as follows:\begin{align*} \begin{cases} \displaystyle R_{1}=\displaystyle \frac {l_{1}^{2}}{\mathbf {c}^{T}\mathbf {K}^{-1}\mathbf {c}} +\,\gamma \int _{0}^{l_{1}} f(\tau ) d\tau \\ \displaystyle \\ \displaystyle R_{2}=\displaystyle \frac {l_{2}^{2}}{\mathbf {c}^{T}\mathbf {K}^{-1}\mathbf {c}} +\,\gamma \int _{0}^{l_{2}} f(\tau ) d\tau \\ \displaystyle \\ \displaystyle \text {Critical energy boundary:} \quad R=\text {min} \,\{R_{1}, R_{2}\} \\ \displaystyle \qquad \qquad \qquad \qquad \quad \Rightarrow \quad V(\tilde {\mathbf {x}},\mu ,\tau ) < R \end{cases} \tag{13}\end{align*}
C. Linear Matrix Inequalities (LMI)
The use of LMIs makes the Lyapunov function construction process tractable and computationally efficient, even for complex nonlinear systems [121], [122]. Constructing a Lyapunov function for a nonlinear system utilizing an LMI-based method involves a rigorous set of steps to ensure stability analysis of the system with profound implications. The procedure can be summarized as follows:
Define the system dynamics: First, the system dynamics must be expressed in the form of a differential equation or a set of differential equations defined. Let us consider a nonlinear system described by the differential equation:
\begin{equation*} \dot {\mathbf {x}} = f(\mathbf {x})~, \tag{14}\end{equation*} View Source\begin{equation*} \dot {\mathbf {x}} = f(\mathbf {x})~, \tag{14}\end{equation*}
\mathbf {x} f(\mathbf {x}) Choose a candidate Lyapunov function: The next step is to choose a candidate Lyapunov function that satisfies certain properties. The Lyapunov function is a scalar function of the state variables that can help to determine the stability of the system. Choose a quadratic function of the form as follows:
\begin{equation*} V(\mathbf {x}) = \mathbf {x}^{T} \mathbf {P} \mathbf {x}~, \tag{15}\end{equation*} View Source\begin{equation*} V(\mathbf {x}) = \mathbf {x}^{T} \mathbf {P} \mathbf {x}~, \tag{15}\end{equation*}
\mathbf {P} Compute the derivative of the Lyapunov function: Compute the derivative of the Lyapunov function with respect to time. Using the chain rule,
\dot {V}(\mathbf {x}) \begin{equation*} \dot {V}(\mathbf {x}) = \mathbf {x}^{T} {\mathbf {P}} \dot {\mathbf {x}} + \dot {\mathbf {x}}^{T}\mathbf {P} \mathbf {x}~. \tag{16}\end{equation*} View Source\begin{equation*} \dot {V}(\mathbf {x}) = \mathbf {x}^{T} {\mathbf {P}} \dot {\mathbf {x}} + \dot {\mathbf {x}}^{T}\mathbf {P} \mathbf {x}~. \tag{16}\end{equation*}
Substituting
\dot {\mathbf {x}} f(\mathbf {x}) \begin{equation*} \dot {V}(\mathbf {x}) = \mathbf {x}^{T} {\mathbf {P}} f({\mathbf {x}}) + f({\mathbf {x}})^{T}\mathbf {P} \mathbf {x}~. \tag{17}\end{equation*} View Source\begin{equation*} \dot {V}(\mathbf {x}) = \mathbf {x}^{T} {\mathbf {P}} f({\mathbf {x}}) + f({\mathbf {x}})^{T}\mathbf {P} \mathbf {x}~. \tag{17}\end{equation*}
Choose a Lyapunov candidate: To guarantee the stability of the system, a positive definite matrix
\mathbf {P} \begin{equation*} \mathbf {A}^{T} \mathbf {P} + \mathbf {P} \mathbf {A} + \mathbf {Q} < 0~, \tag{18}\end{equation*} View Source\begin{equation*} \mathbf {A}^{T} \mathbf {P} + \mathbf {P} \mathbf {A} + \mathbf {Q} < 0~, \tag{18}\end{equation*}
\mathbf {A} f(\mathbf {x}) \mathbf {Q} Check the stability of the system: Once a positive definite matrix
\mathbf {P} V(\mathbf {x}) = \mathbf {x}^{T} \mathbf {P} \mathbf {x} \dot {V}(\mathbf {x}) < 0 \mathbf {x} \dot {V}(\mathbf {x}) > 0 \mathbf {x}
D. Takagi-Sugeno Fuzzy
The T-S (Takagi-Sugeno) fuzzy modeling technique is a popular approach for modeling nonlinear systems using a set of local linear models. This technique can also be used to construct Lyapunov functions for analyzing the stability of the system [121], [123].
The T-S fuzzy modeling technique involves partitioning the state space into a set of fuzzy regions, each of which is associated with a local linear model [124]. The fuzzy membership function determines the degree of membership of a given state vector to each of the fuzzy regions. The T-S fuzzy model can be written as:\begin{equation*} \text {If $\mathbf {x}$ is $A_{i}$ then $f(\mathbf {x}) = g_{i}(\mathbf {x})$}~, \tag{19}\end{equation*}
To construct a Lyapunov function for the T-S fuzzy model, we can use the following approach:
Define the system dynamics: Define a common quadratic Lyapunov function for each of the local models:
\begin{equation*} V_{i}(x) = \mathbf {x}^{T} \mathbf {P}_{i} \mathbf {x}~, \tag{20}\end{equation*} View Source\begin{equation*} V_{i}(x) = \mathbf {x}^{T} \mathbf {P}_{i} \mathbf {x}~, \tag{20}\end{equation*}
\mathbf {P}_{i} i^{th} Choose a candidate Lyapunov function: Combine the local Lyapunov functions into a global Lyapunov function that satisfies the stability conditions for the entire system. This can be done using a weighted sum of the local Lyapunov functions as follows:
\begin{equation*} V(\mathbf {x}) = \sum _{i=1}^{K} w_{i} V_{i}(\mathbf {x})~, \tag{21}\end{equation*} View Source\begin{equation*} V(\mathbf {x}) = \sum _{i=1}^{K} w_{i} V_{i}(\mathbf {x})~, \tag{21}\end{equation*}
w_{i} i^{th} K Compute the derivative of the Lyapunov function and check the stability of the system: Choose the weights
w_{i}
E. Sum of Squares
In Parrilo’s thesis, the sum of squares method was first presented [125]. In this method, numerous problems in system analysis that were previously challenging to answer have been addressed. One of the addressed challenges is the algorithmic stability analysis of nonlinear systems using Lyapunov techniques. In this paper, the application of the SOS-based Lyapunov function is suggested for stability analysis of the inverter-based microgrid to find a region of attraction with lower conservativeness. The SOSTOOL is applied to find the polynomial Lyapunov function. The sum of squares polynomial optimization programs can be created and solved using the free MATLAB toolbox SOSTOOLS [126].
Constructing a Lyapunov function for a nonlinear system using the sum of squares techniques involves the following steps [127], [128], [129]:
Define the system: Consider the following nonlinear system:
\begin{equation*} \dot {\mathbf {x}} = f(\mathbf {x})~, \tag{22}\end{equation*} View Source\begin{equation*} \dot {\mathbf {x}} = f(\mathbf {x})~, \tag{22}\end{equation*}
\mathbf {x} f(\mathbf {x}) Choose a candidate Lyapunov function: The next step is to choose a candidate Lyapunov function that satisfies certain properties. The Lyapunov function is a scalar function of the state variables that can help to determine the stability of the system. For the SOS-based Lyapunov function, choose a polynomial function of the form as follows:
\begin{equation*} V(\mathbf {x}) =\sum _{i_{1}=0}^{d}\sum _{i_{2}=0}^{d} \cdots \sum _{i_{n}=0}^{d} p_{k} \cdot \biggl (x_{1}^{i_{1}} x_{1}^{i_{1}} \cdots x_{n}^{i_{n}}\biggr )~, \tag{23}\end{equation*} View Source\begin{equation*} V(\mathbf {x}) =\sum _{i_{1}=0}^{d}\sum _{i_{2}=0}^{d} \cdots \sum _{i_{n}=0}^{d} p_{k} \cdot \biggl (x_{1}^{i_{1}} x_{1}^{i_{1}} \cdots x_{n}^{i_{n}}\biggr )~, \tag{23}\end{equation*}
p_{k} d x_{n} n \begin{align*} V(x)= p_{0}+ p_{1} x_{1}+p_{2} x_{2} + p_{3} x_{1} x_{2}^{2} +\cdots + p_{24}\; x_{1}^{4} x_{2}^{4}. \\{}\tag{24}\end{align*} View Source\begin{align*} V(x)= p_{0}+ p_{1} x_{1}+p_{2} x_{2} + p_{3} x_{1} x_{2}^{2} +\cdots + p_{24}\; x_{1}^{4} x_{2}^{4}. \\{}\tag{24}\end{align*}
Compute the derivative of the Lyapunov function: The derivative of the Lyapunov function with respect to time is calculated. Using the chain rule:
\begin{equation*} \dot {V}(\mathbf {x}) =\nabla V(\mathbf {x}) f(\mathbf {x})~, \tag{25}\end{equation*} View Source\begin{equation*} \dot {V}(\mathbf {x}) =\nabla V(\mathbf {x}) f(\mathbf {x})~, \tag{25}\end{equation*}
\nabla V(\mathbf {x}) f(\mathbf {x}) \begin{align*} \dot {V}(\mathbf {x}) = \left({ \frac {\partial {d}V}{\partial {d}x_{1}}}\right) f_{1}(\mathbf {x}) + \left({ \frac {\partial {d}V}{\partial {d}x_{2}}}\right) f_{2}(\mathbf {x}) +\cdots + \left({ \frac {\partial {d}V}{\partial {d}x_{n}}}\right) f_{n}(\mathbf {x}). \\{}\tag{26}\end{align*} View Source\begin{align*} \dot {V}(\mathbf {x}) = \left({ \frac {\partial {d}V}{\partial {d}x_{1}}}\right) f_{1}(\mathbf {x}) + \left({ \frac {\partial {d}V}{\partial {d}x_{2}}}\right) f_{2}(\mathbf {x}) +\cdots + \left({ \frac {\partial {d}V}{\partial {d}x_{n}}}\right) f_{n}(\mathbf {x}). \\{}\tag{26}\end{align*}
Choose a Lyapunov candidate: To guarantee the stability of the system, a Lyapunov candidate must be obtained that satisfies the following SOS constraint:
\begin{equation*} \dot {V}(\mathbf {x}) + \alpha V(\mathbf {x}) < 0, \tag{27}\end{equation*} View Source\begin{equation*} \dot {V}(\mathbf {x}) + \alpha V(\mathbf {x}) < 0, \tag{27}\end{equation*}
\alpha Check the stability of the system: Once we find a Lyapunov candidate that satisfies the SOS constraint, we can use the Lyapunov function
V(\mathbf {x}) \dot {V}(\mathbf {x}) + \alpha V(\mathbf {x}) < 0 \mathbf {x} \dot {V}(\mathbf {x}) + \alpha V(\mathbf {x}) > 0 \mathbf {x}
Constructing a Lyapunov function using SOS techniques involves choosing a candidate function, computing its derivative, and finding a Lyapunov candidate that satisfies an SOS constraint. Some numerical examples for simple systems can be found in [128] and [130]. By analyzing the derivative of the Lyapunov function, the stability of the system and its domain of attraction can be determined.
The studies in other applications [90], [131] show that the SOS-based Lyapunov function can provide a larger and less conservative domain of attraction for a certain system compared to the other existing methods (Krososkii’s, Popov-Lure, and T-S fuzzy methods), guaranteeing the accurate and less conservative stability assessment of the system. However, by increasing the order number of the state-space model of the system, the computational process gets extremely time-consuming [131].
In the following sections, a numerical analysis of the large-signal stability of a microgrid with a single inverter-based DER is developed. A comparative study is presented by using different dynamic models and methods of Lyapunov function construction.
Case Studies and Simulation Setups
In the following sections, a numerical analysis of the existing methods and dynamic models is developed and assessed. The analytical, numerical assessment is provided for the second-order (VSG) and full-order dynamic models. The comparative assessment of the microgrid’s stability is obtained based on the outcomes of the three Lyapunov-based stability methods named Krasoskii’s, Popov-Lure, and SOS methods.
A. Methods for Constructing DOA for the VSG Dynamic Model
The dynamic model employed for a system must be sufficiently accurate to predict the system’s behavior reliably. However, using an excessively complex model can lead to computationally demanding processes. The dynamic model described in this section for inverter-based DERs is the second-order or virtual synchronous generator (VSG) model. This model treats the dynamic behavior of IBRs such as synchronous machines, resulting in defining an equivalent damping factor and inertia values. Consequently, the VSG model simplifies the modeling process while still capturing the essential dynamics of inverter-based DERs.
1) The Mathematical Model for Inverter-Based Microgrids Based on Second-Order Model-VSG
The active power can be effectively modeled based on the droop control mechanism employed by the inverters [132], [133], [134]. The model can be expressed as follows:\begin{align*} \omega _{d,ctrl}&=\omega _{out}-m_{p}\;(P_{d,ctrl}-P_{out})~, \tag{28a}\\ P_{d,ctrl}&=\frac {\omega _{f}}{s+\omega _{f}}P_{e}~, \tag{28b}\\ P_{e} &=\,v_{oq}\,i_{oq}+\,v_{od}\,i_{od}, \tag{28c}\end{align*}
\begin{align*} &\omega _{d,ctrl}=\omega _{out}-m_{p}\left({\frac {\omega _{f}}{s+\omega _{f}}P_{e}-P_{out}}\right), \\ &P_{e}=\left({\frac {s}{\omega _{f}}+1}\right)\left({\frac {1}{m_{p}}(\omega _{d,ctrl}-\omega _{out})+P_{out}}\right). \tag{29}\end{align*}
Comparison of DOAs constructed by SOS method and Popov-Lure method for VSG model of IBRs.
According to (28a) and (28b) and transposing related terms, the aforementioned equation can be written as follows:\begin{align*} &\frac {1}{\omega _{f}m_{p}}s \,\omega _{d,ctrl}=P_{out}-P_{e}-\frac {1}{m_{p}}\cancelto {\Delta \omega }{(\omega _{d,ctrl}-\omega _{out})}, \\ &\frac {1}{\omega _{f}m_{p}}\dot {\omega }_{d,ctrl}+\frac {1}{m_{p}}(\triangle \omega )=P_{out}-P_{e}. \tag{30}\end{align*}
To determine the virtual inertia and damping factor of IBRs, the equation obtained can be compared with the standard definition of the swing equation used for synchronous generators. This allows for the formulation of the following expressions:\begin{equation*} M= \frac {1}{\omega _{f}m_{p}} \quad \text {and}\quad D=\frac {1}{m_{p}}. \tag{31}\end{equation*}
Numerous studies have confirmed that IBRs exhibit synchronous machine-like characteristics, with various definitions of inertia and damping factor proposed in [37], [38], and [135]. Despite the differences in these proposed definitions, the resulting values are notably close to one another. For instance, the model developed in [38] treats the inverter-based source as an electrostatic machine and calculates the virtual inertia (M) and damping factor (D) using the inverter-based equations given below:\begin{equation*} M=\frac {1+2\,H\,m_{p}\, \omega _{f}}{m_{p}\, \omega _{f}}\approx \frac {1}{m_{p}\, \omega _{f}} \;{~\text {and}}\quad D=\frac {1}{m_{p}}, \tag{32}\end{equation*}
In the next step, the state-space dynamic model of the network is defined. The network includes an IBR connected to an infinite bus and the voltage at the point of common coupling (PCC) is assumed to remain constant. Consequently, the dynamic variables are limited to the angle and frequency. However, when applying this model to the islanded mode of operation, additional complexities arise due to voltage fluctuations. In such scenarios, the assumption of a constant voltage at PCC no longer holds true.
According to the obtained VSG model of IBR and the assumed network, the state-space dynamic model of the network can be formulated as follows:\begin{align*} \begin{cases} \displaystyle M\dot {\omega }=-D\,\triangle \omega +P_{out}-P_{e},\\ \displaystyle \displaystyle \frac {\mathrm {d}\delta }{\mathrm {d}t}=\triangle \omega ,\\ \displaystyle P_{e}=A\, \sin (\delta +\theta _{Y})+B\quad (*), \end{cases} \tag{33}\end{align*}
\begin{align*} \begin{cases} \displaystyle \dot {\mathbf {x}}= \begin{bmatrix} 0 & 1 \\ 0 & -D/M \end{bmatrix}\mathbf {x}+ \begin{bmatrix} 0\\ -1 \end{bmatrix}f(\tau ),\\ \displaystyle \tau =[ \;1\quad 0\; ]\;\mathbf {x},\\ \displaystyle f(\tau )=-\displaystyle \frac {\bigl (P_{out}-P_{e}\bigr )}{M}\\ \displaystyle \,\qquad =\displaystyle \frac {A}{M}\, \biggl (\sin (\tau +\delta _{s}+\theta _{Y})-\, \sin (\delta _{s}+\theta _{Y})\biggl ), \end{cases} \tag{34}\end{align*}
\begin{align*} \mathbf {x}= \begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}= \begin{bmatrix} \delta -\delta _{s}\\ \triangle \omega \end{bmatrix}. \tag{35}\end{align*}
\begin{align*} \begin{cases} \displaystyle \dot {\tilde {x}}=-\displaystyle \frac {D}{M}\tilde {x}-f(\tau )~,\\ \displaystyle \dot {\mu }=\,-f(\tau )~,\\ \displaystyle \tau =-\displaystyle \frac {M}{D}\tilde {x}+\displaystyle \frac {M}{D}\mu ~,\\ \displaystyle \text {where }\;\;\tilde {x}=[\;0 \quad 1\;]\,\mathbf {x}=x_{2}=\triangle \omega ~, \end{cases} \tag{36}\end{align*}
\begin{align*} \begin{cases} \displaystyle {\scriptstyle \bullet } \quad G(s)=-\mathbf {c}^{T}\bigl [\,s\mathbf {I}-\mathbf {A }\bigr ]^{-1}\,\mathbf {b}=\displaystyle \frac {1}{s\left({s+\frac {D}{M}}\right)}~,\\ \displaystyle \begin{cases} \displaystyle \text {Applying Popov Criterion: } \displaystyle \frac {\gamma \displaystyle \frac {D}{M}-1}{\omega ^{2}+\left({\displaystyle \frac {D}{M}}\right)^{2}}\geq 0~,\\ \displaystyle \text {Here}~, {\gamma}~ {{\mathrm {is~ considered~ such~ that}}: }\gamma =\displaystyle \frac {nM}{D}\;,\; \forall \, n\geq 1~,\\ \displaystyle \;W(\omega )=\Re \biggl [(1+\gamma j\omega )\,G(j\omega )\, \Gamma (j\omega )\,\Gamma (-j\omega )\biggr ]\hspace {-1ex}= \displaystyle \frac {D^{3}}{M^{3}}~,\\ \displaystyle \;\Gamma (s)=det \bigl [\,s\mathbf {I}-\mathbf {F }]=s+\displaystyle \frac {D}{M}~. \end{cases}\\ \displaystyle {\scriptstyle \bullet } \quad \text {Find } \phi (j\omega )\,\Rightarrow W(\omega )= \phi (j\omega )\,\phi (-j\omega )=-\sqrt {\displaystyle \frac {D^{3}}{M^{3}}}~.\\ \displaystyle {\scriptstyle \bullet } \quad \text {The real vector}~{\mathbf {q}}~{{\mathrm {is~ considered~ such~ that:}} }\\ \displaystyle \begin{cases} \displaystyle \mathbf {q}= a\, \varrho (z)=\sqrt {\displaystyle \frac {D^{3}}{M^{3}}}~,\\ \displaystyle r=\gamma \,(\alpha +\mathbf {h}^{T}\,\mathbf {g})=\gamma \,\left({\displaystyle \frac {M}{D}-\displaystyle \frac {M}{D}\times 1}\right)=0~,\\ \displaystyle \varrho (z)= \sqrt {r}\,\Gamma (z)-\phi (z)=-\phi (z)=\sqrt {\displaystyle \frac {D^{3}}{M^{3}}}\;,\; \forall \;j\omega =z~,\\ \displaystyle a=1~. \end{cases}\\ \displaystyle {\scriptstyle \bullet } \quad \text {Find }{\mathbf {K}} \in \; {\mathbf {F}}^{T} {\mathbf {K}}+ {\mathbf {K}} {\mathbf {F}}= -\mathbf {q}\mathbf {q}^{T} \; \Rightarrow \; \mathbf {K}=\displaystyle \frac {D^{2}}{2M^{2}}.\end{cases} \\{}\tag{37}\end{align*}
Then, the Lur’e-type Lyapunov function is expressed in the following form:\begin{align*} V( \tilde {\mathbf {x}},\mu ,\tau )&= \tilde {\mathbf {x}} ^{T} {\mathbf {K}}\, \tilde {\mathbf {x}} +\frac {1}{2}\,\alpha \,\mu ^{2}+\,\gamma \int _{0}^{\tau } f(\tau ) d\tau \\ &=\displaystyle \frac {D^{2}}{2M^{2}} {\triangle \omega }^{2} +\frac {1}{2}\,\displaystyle \frac {M}{D}\,\mu ^{2}+\,\displaystyle \frac {nM}{D}\int _{0}^{\tau } f(\tau ) d\tau \\ &=\displaystyle \frac {D^{2}}{2M^{2}} {\triangle \omega }^{2} +\frac {1}{2}\,\displaystyle \frac {M}{D}\,\mu ^{2}+ \displaystyle \frac {nA}{D}\, \biggl (\cos (\delta _{s}+\theta _{Y}) \\ &\quad -\,\cos (\tau +\delta _{s}+\theta _{Y})-\, \tau \sin (\delta _{s}+\theta _{Y})\biggl ). \tag{38}\end{align*}
By replacing \begin{align*} &V( {\mathbf {x}},\mu ,\tau )=\displaystyle \frac {D^{2}}{2M^{2}} x_{2}^{2} +\frac {1}{2}\,\displaystyle \frac {M}{D}\,\left({\displaystyle \frac {D}{M} x_{1}+x_{2}}\right)^{2}+ \\ &\displaystyle \frac {nA}{D}\biggl (\cos (\delta _{s}+\theta _{Y}) -\cos (x_{1} +\delta _{s}+\theta _{Y})-x_{1} \,\sin (\delta _{s}+\theta _{Y})\biggl ) \\ &=\left({\displaystyle \frac {D^{2}}{2M^{2}}+\displaystyle \frac {M}{2D}}\right) x_{2}^{2} +x_{1} x_{2}+\displaystyle \frac {D}{2M} x_{1}^{2}+\displaystyle \frac {nA}{D} \biggl (\cos (\delta _{s}+\theta _{Y}) \\ &\quad -\,\cos (x_{1} +\delta _{s}+\theta _{Y})-\, x_{1} \,\sin (\delta _{s}+\theta _{Y})\biggl ). \tag{39}\end{align*}
Using Lagrange multipliers, the minimum non-zero boundary indicated by \begin{align*} \begin{cases} \displaystyle R=\displaystyle \frac {l^{2}}{\mathbf {c}^{T}\mathbf {K}^{-1}\mathbf {c}} +\,\gamma \int _{0}^{l} f(\tau ) d\tau \\ \displaystyle \quad \;\,= \displaystyle \frac {(\pi -2(\delta _{s}+\theta _{Y}))^{2}}{2M^{2}/D^{2}} +\,\displaystyle \frac {nM}{D}\int _{0}^{\pi -2(\delta _{s}+\theta _{Y})} f(\tau ) d\tau \\ \displaystyle \quad \;\,= \displaystyle \frac {(\pi -2(\delta _{s}+\theta _{Y}))^{2}}{2M^{2}/D^{2}} +\displaystyle \frac {nA}{D}\, \biggl (\cos (\delta _{s}+\theta _{Y})- \\ \displaystyle \;\cos (\pi -(\delta _{s}+\theta _{Y}))-\, \bigl (\pi -2(\delta _{s}+\theta _{Y})\bigr ) \,\sin (\delta _{s}+\theta _{Y})\biggl ). \end{cases} \tag{40}\end{align*}
2) Numerical Analysis for VSG Model
The state-space representation based on the VSG model is constructed for a grid-connected IBR detailed in Section V-A1. The Popov-Lur’e and sum of squares (SOS) methods, detailed in Sections IV-B and IV-E, respectively, are utilized to construct the Lyapunov function and the domain of attraction. Additionally, Table 3 provides the parameters for the IBR.
3) Results and Discussion for VSG Model
The stability analysis results, which are based on the VSG model and SOS-based Lyapunov function, are presented in Figure 4 in a counter plot. Figure 4a and 4b illustrate the SOS-based Lyapunov function and its derivative, respectively. According to the Lyapunov stability theory, the intersection between the positive Lyapunov function and the negative derivative of the Lyapunov function represents the region of stability. By satisfying these two conditions, the domain of attraction is determined, as shown in Figure 4b.
Figures 6a and 6b illustrate the 3- and 2-dimension presentations of the Lyapunov function constructed based on the Popov-Lure theory considering the VSG model of IBRs, respectively. As described in Section IV-B, the DOA is defined by the positive Lyapunov function and the negative derivative of the Lyapunov function around the original equilibrium point. The critical energy boundary is determined by Lagrange multipliers (
The comparison presented in Figure 7 between DOAs constructed using the SOS-based and Popov-Lyapunov methods reveals that the Popov-based DOA is overly conservative and smaller than the SOS-based DOA. This conservatism leads to unnecessary constraints on system parameters, which can have negative effects in practical applications such as protection systems and relay settings. Strict limitations may result in unnecessary triggering of the protection system, causing it to miss out on utilizing the full potential of the system. Therefore, an inaccurate and conservative DOA can impede the system’s efficiency and effectiveness.
B. Methods of Constructing DOA for the Full-order Dynamic Model
1) The Mathematical Model of Inverter-based Microgrids based on Full-order Differential Equations
To create a further reliable dynamic model of the inverter-based microgrid, it is crucial to model all existing components, including DGs, loads, converters, and batteries, using appropriate approximations.
This section focuses on the IBR, its network, and control systems, as depicted in Figure 8. The network includes an IBR connected to the main grid and two loads. The dynamic model used for the IBR in this study is based on [50] and [137]. The subsequent sections explain the theoretical model of the IBR, which covers power control, voltage control, current control, LCL filter, lines, connection, and loads. The state-space model of the microgrid is developed by combining and modeling all components according to the network configuration using the network mapping model. In this study, the DC-side components are excluded from the model to simplify the computation process. Additional information about the DC-side dynamic model can be found in [50].
a: Power Controller
The power controller comprises several components, including the low-pass filter (\begin{align*} P_{d,ctrl}&=\frac {\omega _{f}}{s+\omega _{f}}P_{e}~,\quad Q_{d,ctrl}= \frac {\omega _{f}}{s+\omega _{f}} Q_{e}~, \tag{41a}\\ P_{e} &=\,v_{oq}\,i_{oq}+\,v_{od}\,i_{od}~,\quad Q_{e} =v_{od}\,i_{oq}-\,v_{oq}\,i_{od}. \\{}\tag{41b}\end{align*}
In the rest of the paper, for simplification, \begin{align*} m_{p}\, P&=\omega _{n}-\omega \;,\quad n_{q}\,Q=V_{n}-v^{\ast }_{od}\;,\quad v^{\ast }_{oq}=0, \tag{42a}\\ \delta _{n}&=\int (\omega -\omega _{ref}) \quad , \quad \triangle \omega _{n} =\omega _{n}-\omega _{ref}. \tag{42b}\end{align*}
\begin{align*} \dot {X}_{P_{d,ctrl}}&=\mathbf {A_{P}}\;X_{P_{d,ctrl}}+ \,\mathbf {B_{P\omega }}\; \triangle \omega _{n} + \mathbf {B_{P}}\; U_{inv}, \\ Y_{P_{d,ctrl}}&=\mathbf {C_{P}}\;X_{P_{d,ctrl}}+ \,\mathbf {E_{P}}, \tag{43}\end{align*}
\begin{align*} X^{T}_{P_{d,ctrl}}&= \begin{bmatrix} \delta _{n}&P& Q \end{bmatrix}~, \tag{44a}\\ Y^{T}_{P_{d,ctrl}}&= \begin{bmatrix} \omega & {v^{\ast }_{od}}&{v^{\ast }_{oq}} \end{bmatrix}~, \tag{44b}\\ U^{T}_{inv}&= \begin{bmatrix} i_{ld}&i_{lq} & v_{od}&v_{oq}& i_{od}& i_{oq} \end{bmatrix}~, \tag{44c}\\ \mathbf {A_{P}}&= \begin{bmatrix} 0 & -m_{p} & 0\\ 0 & -\omega _{f} & 0\\ 0 & 0 & -\omega _{f} \end{bmatrix},\; \mathbf {B_{P\omega }}= \begin{bmatrix} 1 \\ 0\\ 0 \end{bmatrix}~, \tag{44d}\\ \mathbf {B_{P}} &= \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \omega _{f}\,v_{od} & \omega _{f}\,v_{oq}\\ 0 & 0 & 0 & 0 & -\omega _{f}\,v_{oq} & \omega _{f}\,v_{od} \end{bmatrix}~, \tag{44e}\\ \mathbf {C_{P}}&= \begin{bmatrix} 0 & -m_{p} & 0\\ 0 & 0 & -n_{q}\\ 0 & 0 & 0 \end{bmatrix}, \mathbf {E_{P}}= \begin{bmatrix} \omega _{n} \\ V_{n}\\ 0 \end{bmatrix}, \tag{44f}\end{align*}
b: Voltage Controller
The differential equation that represents the voltage controller dynamics is described as follows:\begin{equation*} \frac {\mathrm {d}\varrho _{d}}{\mathrm {d}t}=v^{\ast }_{od}-v_{od}\;, \quad \frac {\mathrm {d}\varrho _{q}}{\mathrm {d}t}=v^{\ast }_{oq}-v_{oq}~, \tag{45}\end{equation*}
The nonlinear dynamic model of the voltage controller is defined as follows:\begin{align*} &\dot {X}_{\varrho _{d,ctrl}}=\mathbf {A_{v}}\,X_{\varrho _{d,ctrl}} + \mathbf {B_{v1}}\,U_{v} +\mathbf {B_{v2}}\, U_{inv}~, \\ &{Y_{\varrho _{d,ctrl}}}=\mathbf {C_{v}}\;\,X_{\varrho _{d,ctrl}}+ \,\mathbf {D_{v1}}\,U_{v} + \mathbf {D_{v2}}\,U_{inv}~, \tag{46}\end{align*}
\begin{align*} X_{\varrho _{d,ctrl}}&= \begin{bmatrix} \varrho _{d}\\ \varrho _{q} \end{bmatrix},\; Y_{\varrho _{d,ctrl}}= \begin{bmatrix} i^{\ast }_{ld} \\ i^{\ast }_{lq} \end{bmatrix}, \tag{47a}\\ U_{v}&= \begin{bmatrix} v^{\ast }_{od}\\ v ^{\ast }_{oq} \end{bmatrix},\; \mathbf {A_{v}}= \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}, \tag{47b}\\ \mathbf {B_{v1}}&= \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix},~\mathbf {B_{v2}}= \begin{bmatrix} 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ \end{bmatrix}, \tag{47c}\\ \mathbf {C_{v}}&= \begin{bmatrix} K_{iv} & 0 \\ 0 & K_{iv} \end{bmatrix},~\mathbf {D_{v1}}= \begin{bmatrix} K_{pv} & 0 \\ 0 & K_{pv}\\ \end{bmatrix}, \tag{47d}\\ \mathbf {D_{v2}}&= \begin{bmatrix} 0 & 0 & -K_{pv} & -\omega _{n}\,C_{f} & F_{c} & 0 \\ 0 & 0 & \omega _{n}\,C_{f} & -K_{pv} & 0 & F_{c} \end{bmatrix}~. \tag{47e}\end{align*}
In the given equations,
c: Current Controller
To formulate the state-space dynamic model of the current controller, the same approach as the voltage controller is applied. Equations in (48) describe the present controller’s differential equation based on the circuit laws:\begin{equation*} \frac {\mathrm {d}\gamma _{d}}{\mathrm {d}t}=i^{\ast }_{ld}-i_{ld}, \quad \frac {\mathrm {d}\gamma _{q}}{\mathrm {d}t}=i^{\ast }_{lq}-i_{lq}~, \tag{48}\end{equation*}
\begin{align*} &\dot {X}_{\gamma _{d,ctrl}}=\mathbf {A_{c}}\,X_{\gamma _{d,ctrl}} + \mathbf {B_{c1}}\,U_{c} +\mathbf {B_{c2}}\, U_{inv}, \\ & {Y_{\gamma _{d,ctrl}}}=\mathbf {C_{c}}\;\,X_{\gamma _{d,ctrl}}+ \,\mathbf {D_{c1}}\,U_{c} + \mathbf {D_{c2}}\,U_{inv}~, \tag{49}\end{align*}
\begin{align*} X_{\gamma _{d,ctrl}}&= \begin{bmatrix} \gamma _{d}\\ \gamma _{q} \end{bmatrix},~U_{c}= \begin{bmatrix} i^{\ast }_{ld}\\ i ^{\ast }_{lq} \end{bmatrix},~Y_{\gamma _{d,ctrl}}= \begin{bmatrix} v^{\ast }_{inv,d} \\ v^{\ast }_{inv,q} \end{bmatrix}, \tag{50a}\\ \mathbf {A_{c}}&= \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}, \mathbf {B_{c1}}= \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}, \tag{50b}\\ \mathbf {B_{c2}}&= \begin{bmatrix} -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0\\ \end{bmatrix}~, \tag{50c}\\ \mathbf {C_{c}}&= \begin{bmatrix} K_{ic} & 0 \\ 0 & K_{ic} \end{bmatrix},\; \mathbf {D_{c1}}= \begin{bmatrix} K_{pc} & 0 \\ 0 & K_{pc}\\ \end{bmatrix}~, \tag{50d}\\ \mathbf {D_{c2}}&= \begin{bmatrix} -K_{pc} & -\omega _{n}\,L_{f} & 0 & 0 & 0 & 0 \\ \omega _{n}\,L_{f} & -K_{pc} & 0 & 0 & 0 & 0 \end{bmatrix}~. \tag{50e}\end{align*}
In (50e), the PI coefficients for the current controller are
d: LCL Filter
Using the circuit laws, the nonlinear differential equations of the LC filter are given by:\begin{align*} \frac {\mathrm {d}i_{ld}}{\mathrm {d}t}&=\frac {-r_{f}}{L_{f}} i_{ld}+\omega \,i_{lq}+\frac {1}{L_{f}}\,(v_{inv,d}-\,v_{od}), \tag{51a}\\ \frac {\mathrm {d}i_{lq}}{\mathrm {d}t}&=\frac {-r_{f}}{L_{f}} i_{lq}-\omega \,i_{ld}+\frac {1}{L_{f}}\,(v_{inv,q}-\,v_{oq}), \tag{51b}\\ \frac {\mathrm {d}v_{od}}{\mathrm {d}t}&=\omega \,v_{oq}+\frac {1}{C_{f}}\,(i_{ld}-\,i_{od}), \tag{51c}\\ \frac {\mathrm {d}v_{oq}}{\mathrm {d}t}&=-\omega \,v_{od}+\frac {1}{C_{f}}\,(i_{lq}-\,i_{oq}), \tag{51d}\\ \frac {\mathrm {d}i_{od}}{\mathrm {d}t}&=\frac {-r_{c}}{L_{c}} i_{od}+\omega \,i_{oq}+\frac {1}{L_{c}}\,(v_{od}-\,v_{bd}), \tag{51e}\\ \frac {\mathrm {d}i_{oq}}{\mathrm {d}t}&=\frac {-r_{c}}{L_{c}} i_{oq}-\omega \,i_{od}+\frac {1}{L_{c}}\,(v_{oq}-\,v_{bq}). \tag{51f}\end{align*}
Then, the state-space model of the LC filter can be described as follows:\begin{equation*} \dot {X}_{LC_{fltr}}=\mathbf {A_{LC}}\,X_{LC_{fltr}} + \mathbf {B_{LC1}}\,U_{LC1} +\mathbf {B_{LC2}}\, U_{LC2}, \tag{52}\end{equation*}
\begin{align*} X_{LC_{fltr}}&=U_{inv}, \tag{53a}\\ U_{LC1}&= \begin{bmatrix} v_{inv,d} \\ v_{inv,q} \end{bmatrix},\, \quad U_{LC2}= \begin{bmatrix} v_{bd} \\ v_{bq} \end{bmatrix}. \tag{53b}\end{align*}
The other parameter of (54b) (mentioned in bottom of next page) are defined as follows:\begin{align*} \mathbf {A_{LC}}&= \begin{bmatrix} \frac {-r_{f}}{L_{f}} & \omega _{com} & \frac {-1}{L_{f}} & 0 & 0 & 0 \\ -\omega _{com} & \frac {-r_{f}}{L_{f}} & 0 & \frac {-1}{L_{f}} & 0 & 0 \\ \frac {1}{C_{f}} & 0 & 0 & \omega _{com}& \frac {-1}{C_{f}} & 0 \\ 0 & \frac {1}{C_{f}} & -\omega _{com} & 0 & 0 & \frac {-1}{C_{f}} \\ 0 & 0 & \frac {1}{L_{c}} & 0 & \frac {-r_{c}}{L_{c}} & \omega _{com} \\ 0 & 0 & 0 & \frac {1}{L_{c}} & -\omega _{com} & \frac {-r_{c}}{L_{c}} \end{bmatrix}, \tag{54a}\\ \mathbf {B_{LC1}}&= \begin{bmatrix} \frac {1}{L_{f}} & 0 \\ 0 & \frac {1}{L_{f}} \\ 0 & 0\\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix},\, \quad \mathbf {B_{LC2}}= \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ \frac {-1}{L_{c}} & 0 \\ 0 & \frac {-1}{L_{c}} \end{bmatrix}, \tag{54b}\end{align*}
e: State-Space model of an IBR
The three-phase system is balanced and free of harmonics, with a modulation ratio of 1 being assumed. To develop the state-space model for an IBR, the dynamic models of all components are consolidated, as described in [50] and [137]. The combined state-space dynamic model is given by:\begin{align*} \dot {X}_{IBR}&= \mathbf {A_{IBR}}\,X_{IBR} + \mathbf {B_{IBR1}}\,U_{IBR}+\mathbf {B_{IBR2}}+\mathbf {B_{IBR3}}\,\omega _{ref}, \\ Y_{IBR}&=\mathbf {C_{IBR}}\,X_{IBR} + \mathbf {D_{IBR1}}\,\omega _{n}, \tag{55}\end{align*}
\begin{align*} X^{T}_{IBR}&=[\delta \quad P \quad Q \quad \varrho _{d} \quad \varrho _{q} \quad \gamma _{d} \quad \gamma _{q} \quad i_{ld} \quad i_{lq}\; \quad v_{od} \\ &v_{oq} \quad i_{od} \quad i_{oq}]~, \tag{56a}\\ U^{T}_{IBR}&=\bigl [ v_{bD} \quad v_{bQ}\bigr ],~~Y_{IBR}^{T}=[\;\omega _{n} \quad i_{oD} \quad i_{oQ}]~, \tag{56b}\\ \mathbf {B^{T}_{IBR1}}&= \begin{bmatrix} 0_{1\times 11} &-\cos (\delta )/{L_{c}} & \sin (\delta )/{L_{c}} \; \\ 0_{1\times 11} & -\sin (\delta )/{L_{c}} & -\cos (\delta )/{L_{c}} \; \end{bmatrix}~, \tag{56c}\\ \mathbf {B^{T}_{IBR2}}&=\bigl [\omega _{n} \quad 0_{1\times 2}\quad V_{n} \quad 0\quad K_{pv}V_{n} \quad 0_{1\times 7}\bigr ]~, \tag{56d}\\ \mathbf {B^{T}_{IBR3}}&=\bigl [-1 \quad 0_{1\times 12}\bigr ], \;\mathbf {D^{T}_{IBR}}=\bigl [1 \quad 0_{1\times 2}\bigr ]~, \tag{56e}\\ \mathbf {C^{T}_{IBR}}&= \begin{bmatrix} 0_{1\times 11} & 0 & 0\; \\ 0_{1\times 11} & \cos (\delta ) & -\sin (\delta )\\ 0_{1\times 11} & \sin (\delta ) & cos(\delta ) \end{bmatrix}~, \tag{56f}\end{align*}
\begin{align*} \mathbf {A_{IBR}}&= \begin{bmatrix} \mathbf { A_{P}} & 0_{3\times 4} & \qquad \qquad \qquad \qquad \mathbf {B_{P}}\\ \mathbf {B_{v1}}\; \mathbf {C'_{P}} & 0_{2\times 4} & \qquad \qquad \qquad \qquad \mathbf {B_{v2}}\\ \mathbf {B_{c1}}\;\mathbf {D_{v1}}\;\mathbf {C'_{P}} & \mathbf {B_{c1}}\;\mathbf {C_{v}} & 0_{2\times 2} & & \mathbf {B_{c1}}\;\mathbf {D_{v2}}+\mathbf {B_{c2}}\\ \psi \;\mathbf {B_{LCL1}}\;\mathbf {D_{c1}}\;\mathbf {D_{v1}}\;\mathbf {C'_{P}}& \psi \;\mathbf {B_{LCL1}}\;\mathbf {D_{c1}}\;\mathbf {C_{v}} & \psi \;\mathbf {B_{LCL1}}\;\mathbf {C_{c}} & \sigma \end{bmatrix}_{13\times 13} \\ \psi &=\mathcal {M}_{IBR}\;V_{dc}\;,\quad \sigma =\mathbf {A_{LCL}}+\psi \;\mathbf {B_{LCL1}}\;\mathbf {(D_{c1}}\;\mathbf {D_{v2}}+\mathbf {D_{c2}})\;. \tag{57}\end{align*}
f: Network
Based on the circuit laws, the network’s nonlinear dynamic differential equations at PCC are given by:\begin{align*} & \frac {\mathrm {d}i_{D,line}}{\mathrm {d}t}=\frac {-r_{line}\; i_{D,line}}{L_{line}} -\frac {(v_{kD}-v_{bD})}{L_{line}}+\omega \,i_{Q,line}~, \tag{58a}\\ & \frac {\mathrm {d}i_{Q,line}}{\mathrm {d}t}=\frac {-r_{line}\; i_{Q,line}}{L_{line}} -\frac {(v_{kQ}-v_{bQ})}{L_{line}}\,-\omega \,i_{D,line}. \tag{58b}\end{align*}
The state-space dynamic model can be obtained as below:\begin{equation*} \dot {X}_{i_{Ntwrk}}=\mathbf {A_{Ntwrk}}\,X_{i_{Ntwrk}} + \mathbf {B_{Ntwrk}}\,U_{Ntwrk}, \tag{59}\end{equation*}
\begin{align*} X_{i_{Ntwrk}}&= \begin{bmatrix} X_{i_{line1}} \end{bmatrix}\,, U_{Ntwrk}= \begin{bmatrix} v_{bDQ1} \\ v_{bDQ2} \end{bmatrix}. \tag{60a}\\ X_{i_{line}}&= \begin{bmatrix} i_{D,line} \; \\ i_{Q,line} \end{bmatrix},\quad v_{bDQ}= \begin{bmatrix} v_{bD} \\ v_{bQ} \end{bmatrix}~, \tag{60b}\\ \mathbf {A_{Ntwrk}}&= \begin{bmatrix} A_{LINE1} \end{bmatrix}~, \tag{60c}\\ \mathbf {A_{LINE1}}&= \begin{bmatrix} {-r_{line}}/{L_{line}} & \omega _{ref} \; \\ -\omega _{ref} & {-r_{line}}/{L_{line}} \; \end{bmatrix}~, \tag{60d}\\ \mathbf {B^{T}_{Ntwrk}}&= \begin{bmatrix} B^{T}_{LINE1} \end{bmatrix}~, \tag{60e}\\ \mathbf {B_{LINE1}}&= \begin{bmatrix} \frac {1}{L_{line}} & 0 & \frac {-1}{L_{line}} & 0 \; \\ 0 & \frac {1}{L_{line}} & 0 &\frac {-1}{L_{line}} \; \end{bmatrix}, \\ &\;\quad \leftharpoonup node\; b \rightharpoonup \;\quad \leftharpoonup node\; k\rightharpoonup \tag{60f}\end{align*}
g: Load
This analysis takes into account a common sort of constant resistive-inductive (RL) load. Below is a model of the load dynamics behavior at each node. \begin{align*} & \frac {\mathrm {d}i_{D,load}}{\mathrm {d}t}=\frac {-R_{load} \;i_{D,load}}{L_{load}} +\frac {v_{bD}}{L_{load}}+\omega \,i_{Q,load}~, \tag{61a}\\ & \frac {\mathrm {d}i_{Q,load}}{\mathrm {d}t}=\frac {-R_{load}\;i_{Q,load}}{L_{load}} +\frac {v_{bQ}}{L_{load}}-\omega \,i_{D,load}. \tag{61b}\end{align*}
\begin{equation*} \dot {X}_{i_{load}}=\mathbf {A_{load}}\,X_{i_{load}} + \mathbf {B_{load}}\,U_{load}~, \tag{62}\end{equation*}
\begin{align*} &X_{i_{load}}= \begin{bmatrix} i_{D,load} \; \\ i_{Q,load} \end{bmatrix}, U_{load}=U_{LC2}\;[T_{DQ}]= \begin{bmatrix} v_{bD} \\ v_{bQ} \end{bmatrix}~, \tag{63a}\\ &\mathbf {A_{load}}= \begin{bmatrix} -R_{load}/L_{load} & \omega _{ref} \; \\ -\omega _{ref} & -R_{load}/L_{load} \end{bmatrix}~, \tag{63b}\\ & \mathbf {B_{load}}= \begin{bmatrix} {1}/{L_{load}} & 0 \; \\ 0 & {1}/{L_{load}} \; \end{bmatrix}. \tag{63c}\end{align*}
The transfer function for the global
h: Microgrid Model and Mapping Matrices
To measure the voltage of individual nodes, a virtual resistor (\begin{align*} v_{bDQ}=\mathbf {R_{v}}(\mathbf {M_{IBR}}i_{oDQ}+\mathbf {M_{L}}i_{loadDQ}+\mathbf {M_{Ntwrk}}i_{lineDQ}). \\{}\tag{64}\end{align*}
The load (L), line (Ntwrk), and inverter (IBR) models are developed and subsequently integrated to generate the voltage values, as expressed in (64). The
The matrix element corresponding to a node has a value of 1 when the current enters the node and a value of −1 when it exits [50], [137]. The state-space model of the inverter-based microgrid, which incorporates all components, is elaborated in the following equation:\begin{equation*} \dot {X}_{\mathbf {Microgrid}}=\mathbf {A_{Microgrid}}\,X_{\mathbf {Microgrid}} + \mathbf {B}_{\mathbf {Microgrid}}. \tag{65}\end{equation*}
\begin{align*} \mathbf {B_{IBR}}&= \mathbf {B_{IBR2}}-\omega _{n}\,\mathbf {B_{IBR3}}. \tag{66a}\\ {X}_{\mathbf {Microgrid}}&= \begin{bmatrix} {X}_{\mathbf {IBR}} \\ i_{DQ,line} \\ i_{DQ,load} \end{bmatrix}, \mathbf {B_{load}}= \begin{bmatrix} \mathbf {B_{IBR}} \\ 0_{2\times 1} \\ 0_{4\times 1} \end{bmatrix}. \tag{66b}\end{align*}
\begin{align*} \mathbf {A_{Microgrid}}= \begin{bmatrix} \mathbf {A_{IBR}}+\mathbf {B_{IBR}}\mathbf {R_{v}} \mathbf {M_{IBR}}\mathbf {C_{IBRc}} & \mathbf {B_{IBR}}\mathbf {R_{v}} \mathbf {M_{Ntwrk}} & \mathbf {B_{IBR}}\mathbf {R_{v}} \mathbf {M_{L}}\\ \mathbf {B_{Ntwrk}}\mathbf {R_{v}} \mathbf {M_{IBR}}\mathbf {C_{IBRc}} & \mathbf {A_{Ntwrk}}+\mathbf {B_{Ntwrk}}\mathbf {R_{v}} \mathbf {M_{Ntwrk}} & \mathbf {B_{Ntwrk}}\mathbf {R_{v}} \mathbf {M_{L}}\\ \mathbf {B_{load}}\mathbf {R_{v}} \mathbf {M_{IBR}}\mathbf {C_{IBR}c} & \mathbf {B_{load}}\mathbf {R_{v}} \mathbf {M_{Ntwrk}} & \mathbf {A_{load}}+\mathbf {B_{load}}\mathbf {R_{v}} \mathbf {M_{L}} \end{bmatrix}. \tag{67}\end{align*}
2) Numerical Analysis
The case study involves an IBR connected to the primary grid and two loads. Table 4 presents the network parameter. The stability analysis of Krasovoskii’s approach and the SOS-based Lyapunov function are compared and elucidated. The sensitivity analysis of the SOS-based method is investigated for two situations: changes in load and control scenarios.
3) Results and Discussion
The SOS-based technique has the potential to enhance the domain of attraction to a verifiable state space, supporting the accuracy claims mentioned above for the SOS-based Lyapunov function. Figure 10 displays a comparison of the DOA constructed using two techniques, namely Krasovskii’s and the SOS-based methods. The DOA obtained via Krasovskii’s method is exceedingly restricted compared to the SOS-based approach. The power system’s response to contingencies in Krasovskii’s method is rather conservative due to the limited domain of attraction constructed by Krasovskii’s method. The transient stability analysis, such as the domain of attraction, is typically employed to adjust the relay and protection systems. The domain of attraction can be used to determine the critical period required to clear faults and other contingencies, ensuring stable and reliable network operation. A limited domain of attraction results in a conservative adjustment of the protection mechanism, leading to an unnecessary outage or load shedding in the system.
Domain of attractions constructed by SOS-based (contour) and Krasovskii’s (surf) methods.
The Lyapunov function developed must meet the following criteria for stability region based on the Lyapunov-based stability:\begin{align*} \begin{cases} \displaystyle \; V(0)=0\; \Rightarrow \text {the}~V~{{\mathrm {must~be~equal~to~zero~at}}}~{x_{n}=0.}\\ \displaystyle \; V(x_{n}) > 0\; \Rightarrow \text {the}~V~{{\mathrm {must be~positive~in~stability~region}}.}\\ \displaystyle \; \dot {V}(x_{n})\,\leq \,0 \,\Rightarrow \text {the derivative of}~ V~{{\mathrm {must~ indicate~ a~ down}}-}\\ \displaystyle \text {ward slope in stability region.} \end{cases} \\{}\tag{68}\end{align*}
The network’s state-space model is shifted to its equilibrium point or steady-state operation for the first condition, meaning that the network’s equilibrium point is represented by the equation
Figure 11 portrays the SOS-based Lyapunov function and its derivative. Negative contours on the Lyapunov function plot do not satisfy the second Lyapunov-based stability criterion. Hence, the areas in Figure 11a with negative contours are excluded from the stability region. The negative and zero contours in Figure 11b satisfy the third requirement; i.e., the negative derivative of the Lyapunov function, which excludes the light-colored regions (positive contours) from the stability region.
In the subsequent section, the impact of load variation events and control modifications on the stability region in inverter-based microgrids is examined. These scenarios are frequently used as an inevitable action of a smart system, they get modified during both temporary and permanent changes in the system to guarantee stable and reliable operation. The goal of analyzing these scenarios is to ensure that the microgrid can adapt to altering conditions while remaining stable, thereby avoiding any interruptions or system failures. By adapting to changing conditions and implementing appropriate control modifications, the stability region can be sustained, ensuring that the microgrid operates reliably and effectively.
a: Load Changing
Load reconfiguration is a solution that can enhance the reliability of a system during contingencies, thereby inducing a change in the stability region. Proper load reconfiguration or reduction can amplify the stability region, leading to an enhancement of the network’s ability to withstand transient faults that may occur. By adjusting the load appropriately, the system’s resilience to unexpected changes can be strengthened, thereby mitigating any disruptions or failures, and ultimately boosting its overall performance.
In addition, the impact of load increasing on the stability region is demonstrated in Figure 12. By increasing Load 1, the contour lines for the same
b: Control Modification
Inverter switching systems implement various control techniques to ensure reliable and stable operation. A smart control system is designed to modify the control parameters during both normal and contingency operations. The control system employs several control objects to make appropriate decisions for the switching system, and the region of attraction can be an additional criterion for modifying the control parameters. Therefore, the control system can adjust the control parameters in response to changes in the system’s behavior, ensuring that the system remains stable and reliable, ultimately improving its operational efficiency.
By increasing the
Discussion and Conclusion
This paper has provided a comprehensive critical review of large-signal stability analysis methods and models for microgrids. Numerical examples with critical analyses are provided to compare the performances of existing methods and models. Sections V-B and V-A have presented a comparative analysis of two different state-space models for an inverter-based microgrid. The study developed a nonlinear dynamic model to investigate a microgrid connected to the main grid, and the results demonstrated that (a summary is provided in Table 5):
Utilizing the Lyapunov-based stability method based on the sum of squares can produce a stability region that is less conservative than those generated by Krasovskii’s and Popov-Lure’s methods. This precise stability region helps in accurately determining the critical clearing time for given contingencies.
The type of chosen nonlinear dynamic model for the accuracy and authenticity of the stability assessment is crucial as an appropriate dynamic model allows for a more precise presentation of the microgrid’s dynamic performance concerning the grid it connects to.
The results emphasize the importance of choosing the appropriate stability analysis method, as it can significantly affect the accuracy of the predictions. The SOS-based Lyapunov method is superior in this regard and should be considered for future stability analyses of inverter-based microgrids. This can be particularly helpful in designing and optimizing the performance of such systems to improve the stability and reliability of power grids.
It is worth mentioning that even though the Lyapunov function based on the SOS-based method has shown to be effective in analyzing the stability of microgrids, its computational process can be time-consuming, especially for higher-order state-space models. Additionally, in some cases, it may not converge to an acceptable result. Furthermore, for larger systems, using higher-degree Lyapunov functions with SOSTOOL can lead to a substantial rise in the number of optimization decision variables, presenting a challenge in terms of computational resources.
To overcome the challenges associated with using the SOS-based method for stability analysis of large-scale systems, it is recommended to employ a simplified and smaller-order dynamic model for IBRs. This approach can reduce the computational demands of the SOS-based method, resulting in improved efficiency and effectiveness in analyzing the system’s stability. However, it is crucial to ensure that the simplified models can still capture the critical dynamics and characteristics of the inverter-based microgrid being analyzed while having reasonable approximations. One model that has been shown to be both accurate and acceptable for inverters is the 5-order state-space model presented in [46]. Another low-order state-space model, known as the second-order or virtual synchronous generator model [39], can only provide frequency and angle as the states of the dynamic model for grid-connected microgrid and IBRs. Therefore, selecting an appropriate simplified model for the inverter-based microgrid being analyzed is critical to balance the need for accuracy and computational efficiency.
The selection of a model for stability analysis requires a trade-off between the number of states that can be monitored and the computational complexity of the model. Selecting the appropriate model should balance computational efficiency and the accuracy of the stability analysis results. The decision will depend on the specific needs and requirements of the stability analysis application. If accuracy is the top priority, then the 5-order state-space model may be the preferred option since it provides more states that can be monitored. However, if computational efficiency is the priority, then the second-order or VSG model may be the better choice for the stability analysis of inverter-based microgrids. Nevertheless, it is essential to recognize that simplified models may not fully capture all of the complex dynamics of the microgrid, and as such, the stability analysis results should be interpreted with caution. In conclusion, the selection of the model should consider the trade-offs between accuracy and computational efficiency, as well as the specific needs of the stability analysis.
In contrast with the SOS-based Lyapunov method, Krasovskii’s and Popov-Lure’s methods are less precise but more scalable and have reasonable computation times for complex systems that have numerous IBRs and distributed generators. Therefore, for systems that have limited computational resources or when using a detailed and meticulous dynamic model of inverters, Krasovskii and Popov-Lure’s methods may be more practical and feasible options for stability analysis.
It is important to strike a balance between accuracy and scalability, and the limitations of the chosen method should be considered when interpreting the results of the stability analysis. Ultimately, the selection of stability analysis methods and the desired dynamic model should be made with careful consideration of the specific aims of the project and case study, the type of application (whether it’s online or post-fault analysis), system constraints, and the states and parameters to be analyzed.