The increasing global demand for electricity, coupled with the challenges of traditional energy shortages and environmental pollution, necessitates a sustainable transformation of power systems. A pivotal strategy involves controlling the total consumption of fossil fuels and actively integrating distributed generation systems based on new energy sources. As the proportion of distributed generation grows, the inherent intermittency and dispersion of new energy resources, such as solar power, pose significant threats to the stability of the main grid. The microgrid concept has emerged as a robust solution to this challenge. By integrating new energy generation units, energy storage, local loads, and the public grid, a microgrid enables the seamless integration of intermittent renewables with minimal adverse impact, enhancing overall system security, reliability, efficiency, and flexibility. Among renewables, solar energy, due to its accessibility, sustainability, and cleanliness, has seen rapid development. To support national carbon neutrality goals, the deployment of distributed photovoltaic (PV) systems integrated with buildings is strongly encouraged. Applying PV generation to large commercial or residential buildings can effectively reduce grid consumption, facilitate feed-in, and provide crucial support to the public grid during peak summer demand.
Many modern sustainable buildings incorporate large-scale PV systems, typically involving multiple solar inverters connected to a common point, forming a PV inverter-based microgrid. The control strategy for such a microgrid must ensure rational power sharing among the inverter units according to load demand. The droop control strategy, which emulates the “power-frequency static characteristic” of synchronous generators in traditional power systems, is widely adopted for autonomous microgrid operation due to its advantages of being communication-free and enabling plug-and-play functionality. However, droop control simplifies the system by ignoring the large inertia and high output impedance of synchronous generators. This omission can lead to reduced system inertia and lower output impedance, making the system more susceptible to disturbances and oscillatory instability. Furthermore, microgrids based on solar inverters are generally less stable than traditional grids, making small-signal stability analysis essential for ensuring reliable operation.
This article focuses on multi-solar inverter microgrids within the context of building-integrated photovoltaics. It employs a control strategy combining outer-loop power droop control (without inter-unit communication) and inner-loop voltage-current feedforward decoupling dual-loop control. The system’s autonomous operation under various conditions is simulated, demonstrating its capability to respond quickly to load changes and allocate power appropriately. To ensure stable operation, the impact of droop control parameters on system stability is analyzed using the eigenvalue method, with results consistent with simulation findings.
1. Architecture of a Multi-Solar Inverter Microgrid
A photovoltaic microgrid system comprises multiple PV generation units connected to an AC bus via voltage-source inverters (VSIs), working alongside energy storage devices to supply power to various loads, including residential and building loads. As shown in the structure below, this system is flexible and independent, capable of operating in two modes: grid-connected during periods of low solar generation (e.g., night) and islanded (autonomous) during sunny periods. This operational flexibility ensures power supply quality under complex conditions while maximizing the use of solar energy and alleviating grid stress during peak demand.
2. Control Structure of a Single Distributed Inverter
Three-phase voltage-source inverters are commonly used as interfaces in PV-based microgrids. In autonomous (islanded) mode, they employ droop-based outer-loop control and dual-loop voltage-current inner-loop control to support the system’s bus voltage and frequency, effectively acting as voltage sources. The droop control determines the required reference setpoints, which are then accurately tracked by the voltage and current controllers, thereby achieving power sharing and stable operation. The detailed topology and control structure of a single solar inverter unit within the microgrid are illustrated in the functional block diagram.
2.1 Power Droop Control
The droop control strategy is prevalent in microgrid inverters, allowing individual units to operate in a “plug-and-play” manner without direct communication links. The form of droop control depends on the nature of the line impedance:
- Inductive impedance: Uses $P-\omega$ (or $P-f$) and $Q-V$ droop.
- Resistive impedance: Uses $P-V$ and $Q-(-\omega)$ droop.
- Capacitive impedance: Uses $P-(-\omega)$ and $Q-(-V)$ droop.
This analysis focuses on the most common case of inductive lines, employing $P-\omega$ and $Q-V$ droop. The control block diagram includes a first-order low-pass filter $G_{LP}(s)$ to remove noise from the sampled power signals. The droop equations for the $i^{th}$ inverter are:
$$
\omega_i = \omega_{0i} – m_{pi}(P_i – P_{0i})
$$
$$
V_i^* = V_{0i} – n_{qi}(Q_i – Q_{0i})
$$
where $\omega_i$ and $\omega_{0i}$ are the output and rated angular frequencies, $V_i^*$ and $V_{0i}$ are the output voltage amplitude reference and rated voltage, $P_i$ and $P_{0i}$ are the output and rated active power, $Q_i$ and $Q_{0i}$ are the output and rated reactive power, and $m_{pi}$, $n_{qi}$ are the active and reactive power droop coefficients.
The instantaneous active and reactive powers ($p_i$, $q_i$) are calculated from the output voltages and currents. A low-pass filter extracts the average power components ($P_i$, $Q_i$) for the droop controller:
$$
P_i = G_{LP}(s) p_i = \frac{\omega_c}{s+\omega_c} p_i, \quad p_i = 1.5(u_{odi}i_{odi}+u_{oqi}i_{oqi})
$$
$$
Q_i = G_{LP}(s) q_i = \frac{\omega_c}{s+\omega_c} q_i, \quad q_i = 1.5(u_{oqi}i_{odi}-u_{odi}i_{oqi})
$$
where $\omega_c$ is the filter cutoff frequency, and $u_{odqi}$, $i_{odqi}$ are the $dq$-axis components of the output voltage and current.
2.2 Voltage-Current Dual-Loop Controller
The inner control loops employ PI regulators in both $d$ and $q$ axes. The voltage loop incorporates the output current $i_o$ as a feedforward term to mitigate the impact of load disturbances on the output voltage, enhancing dynamic response. The control laws are:
Voltage Controller:
$$
i_{1drefi} = F i_{odi} + \left(k_{vp} + \frac{k_{vi}}{s}\right)(u_{odrefi} – u_{odi}) – \omega_i C_f u_{oqi}
$$
$$
i_{1qrefi} = F i_{oqi} + \left(k_{vp} + \frac{k_{vi}}{s}\right)(u_{oqrefi} – u_{oqi}) + \omega_i C_f u_{odi}
$$
Current Controller:
$$
u_{invdi}^* = \left(k_{cp} + \frac{k_{ci}}{s}\right)(i_{1drefi} – i_{1di}) – \omega_i L_f i_{1qi}
$$
$$
u_{invqi}^* = \left(k_{cp} + \frac{k_{ci}}{s}\right)(i_{1qrefi} – i_{1qi}) + \omega_i L_f i_{1di}
$$
Here, $k_{vp}$, $k_{vi}$ and $k_{cp}$, $k_{ci}$ are the proportional and integral gains of the voltage and current controllers, respectively; $F$ is the feedforward coefficient; $\omega_i$ is the inverter’s electrical angular speed; and $L_f$, $C_f$ are the filter inductance and capacitance.
3. Modeling of the Distributed Solar Inverter System
3.1 Coordinate Transformation
Each inverter unit operates in its own $dq$ rotating reference frame, which may differ dynamically. A unified model requires transforming all variables to a common reference frame ($DQ$), chosen here as the frame of the first inverter. The transformation between the common frame $DQ$ and the $i^{th}$ inverter’s frame $dq_i$ is:
$$
\mathbf{f}_{DQ} = \mathbf{T}_i \mathbf{f}_{dqi}, \quad \mathbf{f}_{dqi} = \mathbf{T}_i^{-1} \mathbf{f}_{DQ}
$$
where the transformation matrix $\mathbf{T}_i$ is defined by the phase difference $\delta_i$:
$$
\mathbf{T}_i = \begin{bmatrix}
\cos \delta_i & -\sin \delta_i \\
\sin \delta_i & \cos \delta_i
\end{bmatrix}, \quad \mathbf{T}_i^{-1} = \begin{bmatrix}
\cos \delta_i & \sin \delta_i \\
-\sin \delta_i & \cos \delta_i
\end{bmatrix}
$$
The dynamics of the phase angle difference are given by:
$$
\frac{d\delta_i}{dt} = \omega_i – \omega_{com}
$$
where $\omega_{com}$ is the angular speed of the common $DQ$ frame.
3.2 Power Control Dynamics
Taking the filtered active and reactive powers ($P_i$, $Q_i$) as state variables for the power control loop, their dynamics are derived from the low-pass filter equation:
$$
\frac{dP_i}{dt} = -\omega_c P_i + 1.5\omega_c(u_{odi}i_{odi}+u_{oqi}i_{oqi})
$$
$$
\frac{dQ_i}{dt} = -\omega_c Q_i + 1.5\omega_c(u_{oqi}i_{odi}-u_{odi}i_{oqi})
$$
3.3 Dual-Loop Control Dynamics
Defining auxiliary state variables $\boldsymbol{\phi}_{dqi}$ and $\boldsymbol{\lambda}_{dqi}$ for the integrators of the voltage and current PI controllers simplifies the model:
$$
\frac{d\phi_{di}}{dt} = u_{odrefi} – u_{odi}, \quad \frac{d\phi_{qi}}{dt} = u_{oqrefi} – u_{oqi}
$$
$$
\frac{d\lambda_{di}}{dt} = i_{1drefi} – i_{1di}, \quad \frac{d\lambda_{qi}}{dt} = i_{1qrefi} – i_{1qi}
$$
3.4 LCL Filter Dynamics
Assuming the PWM inverter perfectly tracks the voltage reference ($\mathbf{u}_{inv}^* \approx \mathbf{u}_{inv}$), the dynamics of the $LCL$ filter and coupling inductance are described by the following state equations in the inverter’s $dq$ frame:
Inverter-side inductor current ($\mathbf{i}_{1dqi}$):
$$
\frac{di_{1di}}{dt} = -\frac{R_f}{L_f}i_{1di} + \omega_i i_{1qi} – \frac{1}{L_f}u_{odi} + \frac{1}{L_f}u_{invdi}
$$
$$
\frac{di_{1qi}}{dt} = -\frac{R_f}{L_f}i_{1qi} – \omega_i i_{1di} – \frac{1}{L_f}u_{oqi} + \frac{1}{L_f}u_{invqi}
$$
Filter capacitor voltage ($\mathbf{u}_{odqi}$):
$$
\frac{du_{odi}}{dt} = \omega_i u_{oqi} + \frac{1}{C_f}i_{1di} – \frac{1}{C_f}i_{odi}
$$
$$
\frac{du_{oqi}}{dt} = -\omega_i u_{odi} + \frac{1}{C_f}i_{1qi} – \frac{1}{C_f}i_{oqi}
$$
Output current ($\mathbf{i}_{odqi}$):
$$
\frac{di_{odi}}{dt} = \omega_i i_{oqi} – \frac{R_c}{L_c}i_{odi} – \frac{1}{L_c}u_{odi} + \frac{1}{L_c}u_{bdi}
$$
$$
\frac{di_{oqi}}{dt} = -\omega_i i_{odi} – \frac{R_c}{L_c}i_{oqi} – \frac{1}{L_c}u_{oqi} + \frac{1}{L_c}u_{bqi}
$$
where $R_c$, $L_c$ are the resistance and inductance of the coupling line, and $\mathbf{u}_{bdqi}$ is the voltage at the point of common coupling (PCC).
3.5 Complete Single-Inverter State-Space Model
The state vector for a single solar inverter $i$ is defined as:
$$
\mathbf{x}_{inv_i} = [\delta_i, P_i, Q_i, \boldsymbol{\phi}_{dqi}^T, \boldsymbol{\lambda}_{dqi}^T, \mathbf{i}_{1dqi}^T, \mathbf{u}_{odqi}^T, \mathbf{i}_{odqi}^T]^T
$$
The nonlinear state-space model can be expressed as:
$$
\dot{\mathbf{x}}_{inv_i} = \mathbf{A}_{inv_i} \mathbf{x}_{inv_i} + \mathbf{B}_{inv_i} \mathbf{u}_{bdqi} + \mathbf{o}(\mathbf{x}_{inv_i})
$$
where $\mathbf{A}_{inv_i}$ and $\mathbf{B}_{inv_i}$ are the state and input matrices derived from the equations above, and $\mathbf{o}(\cdot)$ represents higher-order nonlinear terms. Aggregating $N$ inverters gives the multi-inverter model:
$$
\dot{\mathbf{x}}_{INV} = \mathbf{A}_{INV} \mathbf{x}_{INV} + \mathbf{B}_{INV} \mathbf{u}_{bDQ} + \mathbf{o}(\mathbf{x}_{INV})
$$
4. Complete Microgrid Model with Network and Loads
4.1 Network Line Model
The microgrid network, comprising $n$ lines with impedance $Z_{line} = R_{line} + L_{line}s$, is modeled in the common $DQ$ frame. The line current dynamics are:
$$
\dot{\mathbf{i}}_{lineDQ} = \mathbf{A}_{NET} \mathbf{i}_{lineDQ} + \mathbf{B}_{NET} \mathbf{u}_{bDQ}
$$
where $\mathbf{i}_{lineDQ} \in \mathbb{R}^{2n}$ and $\mathbf{u}_{bDQ} \in \mathbb{R}^{2m}$ ($m$ nodes) are the vectors of line currents and node voltages.
4.2 Static Load Model
Assuming $p$ static RL loads, the load current dynamics are:
$$
\dot{\mathbf{i}}_{loadDQ} = \mathbf{A}_{LOAD} \mathbf{i}_{loadDQ} + \mathbf{B}_{LOAD} \mathbf{u}_{bDQ}
$$
4.3 Network Node Model and Complete System
Introducing a large virtual resistor $r_N$ at each node to ground (to establish a well-defined node voltage relationship with minimal impact), the node voltage equation is:
$$
\dot{\mathbf{u}}_{bDQ} = \mathbf{R}_N (\mathbf{M}_{INV}\mathbf{i}_{oDQ} + \mathbf{M}_{LOAD}\mathbf{i}_{loadDQ} + \mathbf{M}_{NET}\mathbf{i}_{lineDQ})
$$
where $\mathbf{M}_{INV}$, $\mathbf{M}_{LOAD}$, $\mathbf{M}_{NET}$ are incidence matrices mapping inverter output currents, load currents, and line currents to the nodes, and $\mathbf{R}_N = r_N\mathbf{I}$.
The complete linearized state-space model of the multi-solar inverter microgrid is formed by combining the models of all components (inverters, lines, loads, nodes):
$$
\frac{d}{dt}
\begin{bmatrix}
\mathbf{x}_{INV} \\
\mathbf{i}_{lineDQ} \\
\mathbf{i}_{loadDQ}
\end{bmatrix}
=
\mathbf{A}_{mg}
\begin{bmatrix}
\mathbf{x}_{INV} \\
\mathbf{i}_{lineDQ} \\
\mathbf{i}_{loadDQ}
\end{bmatrix}
$$
The system matrix $\mathbf{A}_{mg}$ has an order of $(13N + 2n + 2p)$.
5. Stability Analysis and Simulation Verification
A case study of a typical microgrid with three dispatchable PV-based solar inverters and local static loads, operating autonomously, is analyzed. The parameters of the inverters and network are standardized for analysis.
5.1 Eigenvalue Spectrum Analysis
With initial droop parameters $m_p = 1\times10^{-5}$ and $n_q = 1\times10^{-4}$, the eigenvalues of the linearized system matrix $\mathbf{A}_{mg}$ are computed. The spectrum reveals three distinct clusters corresponding to low-frequency (LF), medium-frequency (MF), and high-frequency (HF) oscillatory modes, as summarized in the table below.
| Mode Cluster | Eigenvalue Pair | Real Part | Imaginary Part | Freq. (Hz) | Dominant States |
|---|---|---|---|---|---|
| Low-Frequency (LF) | $\lambda_{1,2}$ | -113.13 | ±146.98 | 23.4 | Power ($P_i$, $Q_i$), Phase ($\delta_i$) |
| $\lambda_{3,4}$ | -139.98 | ±210.34 | 33.5 | ||
| Medium-Frequency (MF) | $\lambda_{5,6}$ | -1301.10 | ±1739.40 | 276.8 | Voltage/Current Controller Integrators ($\boldsymbol{\phi}$, $\boldsymbol{\lambda}$), Capacitor Voltage ($\mathbf{u}_o$) |
| $\lambda_{7,8}$ | -1399.80 | ±2103.40 | 334.8 | ||
| High-Frequency (HF) | $\lambda_{9,10}$ | -2803.70 | ±4247.10 | 675.9 | LCL Filter Currents ($\mathbf{i}_1$, $\mathbf{i}_o$) |
| $\lambda_{11,12}$ | -2815.60 | ±6844.60 | 1089.4 |
5.2 Root Locus Analysis with Variable Droop Parameters
The impact of droop coefficients on stability is investigated via root locus analysis.
Varying Active Droop Coefficient $m_p$: As $m_p$ increases from $1\times10^{-5}$ to $4\times10^{-4}$, the LF eigenvalues move towards the right-half plane. The locus shows that increasing $m_p$ reduces damping and increases the oscillatory frequency of the LF mode. Beyond a critical value, the eigenvalues cross the imaginary axis, indicating the onset of small-signal instability.
Varying Reactive Droop Coefficient $n_q$: A similar trend is observed for the LF modes when $n_q$ increases from $1\times10^{-4}$ to $3\times10^{-3}$. Notably, the MF modes also show significant sensitivity to changes in $n_q$, moving closer to the instability boundary.
The mathematical trend for a dominant LF eigenvalue $\lambda = \sigma \pm j\omega_d$ can be approximated as:
$$
\sigma(m_p) \approx \sigma_0 + \alpha m_p, \quad \omega_d(m_p) \approx \omega_{d0} + \beta m_p
$$
where $\alpha > 0$ and $\beta > 0$ for typical system parameters, confirming the reduction in damping and increase in frequency with larger $m_p$.
5.3 Time-Domain Simulation Validation
Simulations in MATLAB/Simulink validate the theoretical analysis. With initial droop settings ($m_p=1\times10^{-5}$), the system is stable but exhibits slow power sharing dynamics, taking over 1.3 seconds to settle.
Case 1: Improved Dynamics. Increasing $m_p$ to $1.1\times10^{-4}$ significantly improves the transient response, reducing the settling time for active power by approximately 70%.
Case 2: Onset of Oscillations. With $m_p = 2.1\times10^{-4}$, the system exhibits sustained low-frequency oscillations in active power output, corresponding to eigenvalues with very low damping (near the imaginary axis).
Case 3: Instability. With $m_p = 3.1\times10^{-4}$, the system becomes unstable, showing diverging oscillations in active power, frequency, and output voltage. This directly correlates with the root locus prediction where the dominant eigenvalues have moved into the right-half plane ($\sigma > 0$). The system’s response can be characterized by the unstable mode:
$$
P_i(t) \sim e^{\sigma t} \cos(\omega_d t + \varphi), \quad \sigma > 0
$$
The simulation results across all cases show high consistency with the stability boundaries predicted by the eigenvalue and root locus analysis of the developed model.
6. Conclusion
This article presented a detailed modeling methodology for analyzing the stability of building-integrated multi-solar inverter microgrids operating autonomously under droop control. The model accurately captures the dynamics of the power controllers, dual-loop voltage-current regulators, LCL filters, and network interactions within a unified reference frame. Analysis of a three-inverter case study revealed distinct low, medium, and high-frequency oscillatory modes inherent to such systems. The root locus investigation demonstrated a critical trade-off: while increasing droop coefficients ($m_p$, $n_q$) improves power sharing speed and transient response, it simultaneously reduces the damping of low-frequency modes, potentially leading to small-signal instability. Time-domain simulations confirmed the accuracy of the linearized model and the stability limits identified through eigenvalue analysis. This modeling and analysis framework is essential for the proper design and tuning of solar inverter-based microgrids to ensure robust, stable, and efficient operation in sustainable building applications.