The profound transformation of power systems towards “carbon peak and carbon neutrality” necessitates the large-scale integration of renewable energy sources. As a critical interface for grid-connection of photovoltaic and wind power generation, the utility interactive inverter faces significant stability challenges when deployed in weak grid scenarios characterized by high grid impedance. Suboptimal design of control parameters, particularly for the phase-locked loop (PLL) and the current control loop, can induce oscillatory instability or even loss of synchronism, compromising system reliability.
Existing research on parameter design for utility interactive inverters primarily addresses small-signal stability, employing methods such as impedance modeling, eigenvalue analysis, and design based on gain and phase margins. While these approaches ensure stability under minor disturbances, they often overlook the stringent requirements imposed by large-signal transients, such as severe grid voltage sags during fault conditions. This gap necessitates a comprehensive design methodology that concurrently satisfies both steady-state performance metrics and transient stability constraints.
This article proposes a feasible-parameter-domain optimization design method for utility interactive inverters that explicitly incorporates transient stability constraints. The proposed methodology leverages the D-partition technique to establish initial parameter domains respecting classic control performance specifications like gain margin, phase margin, and bandwidth. These domains are subsequently refined and optimized by enforcing stability criteria derived from a singular perturbation model of the inverter system, which captures the distinct dynamics during large disturbances. Hardware-in-the-loop (HIL) experimental results validate that parameters selected from the final optimized domain guarantee both robust control performance and transient stability.
System Modeling and Transient Stability Analysis
The topology of a typical utility interactive inverter connected to a weak grid is illustrated above. The system comprises a voltage source converter with an L-type output filter (\(L_f\)), and the grid is represented by a voltage source \(U_g^s\) behind a line impedance \(Z_g = R_g + j\omega_g L_g\). The control structure includes a synchronous reference frame phase-locked loop (SRF-PLL) for grid synchronization and a PI-based current regulator. During grid voltage sag events, the utility interactive inverter typically enters a low-voltage ride-through (LVRT) mode, where the current reference is directly set to a prescribed value, bypassing the outer power control loops.
Neglecting the relatively small line resistance \(R_g\), the full-order nonlinear model of the utility interactive inverter in the grid synchronous (\(s\)) reference frame can be expressed as:
$$
\begin{aligned}
\frac{d^2\delta}{dt^2} &= \text{Im}\left\{ \left[ k_{p,\text{PLL}}\left(\frac{dU_t^s}{dt} – j\frac{d\delta}{dt} U_t^s e^{-j\delta}\right) + k_{i,\text{PLL}} U_t^s e^{-j\delta} \right] \right\}, \\
\frac{L_g + L_f}{k_{i,c}} \frac{d^2 I_1^s}{dt^2} &= \frac{k_{p,c}}{k_{i,c}} \frac{d(I_{1\text{ref}}^c e^{j\delta} – I_1^s)}{dt} + (I_{1\text{ref}}^c e^{j\delta} – I_1^s) + \\
&\quad j\frac{L_f}{k_{i,c}}\left( \frac{d^2\delta}{dt^2} I_1^s + \frac{d\delta}{dt} \frac{d I_1^s}{dt} \right) – j\frac{\omega_g L_g}{k_{i,c}} \frac{d I_1^s}{dt}, \\
U_t^s &= U_g^s + L_g \frac{d I_1^s}{dt} + j\omega_g L_g I_1^s,
\end{aligned}
$$
where \(\delta = \theta_{\text{PLL}} – \theta_g\) is the PLL angle error; \(k_{p,\text{PLL}}, k_{i,\text{PLL}}\) and \(k_{p,c}, k_{i,c}\) are the PI parameters for the PLL and current loop, respectively; and superscripts \(s\) and \(c\) denote variables in the grid frame and the controller frame, respectively.
Singular Perturbation Model and Stability Criteria
To analyze the transient stability of the utility interactive inverter under large disturbances, a singular perturbation approach is adopted. Defining the small parameter \(\epsilon = 1/k_{i,c} \ll 1\), the system dynamics can be separated into slow and fast subsystems.
The slow subsystem, which primarily governs the PLL synchronization dynamics, is derived as:
$$
\begin{aligned}
I_{1,\text{slow}}^s &= (I_{1\text{ref}d}^c + jI_{1\text{ref}q}^c)e^{j\delta_{\text{slow}}}, \\
\left( \frac{1}{k_{p,\text{PLL}}} – L_g I_{1\text{ref}d}^c \right) \frac{d^2\delta_{\text{slow}}}{dt^2} &= \left(k_{\text{PLL}} U_{gq}^s – U_{gd}^s \frac{d\delta_{\text{slow}}}{dt}\right) \cos \delta_{\text{slow}} + \\
& \frac{d\delta_{\text{slow}}}{dt}\left( k_{\text{PLL}} L_g I_{1\text{ref}d}^c – (k_{\text{PLL}} U_{gd}^s + U_{gq}^s) \right) \sin \delta_{\text{slow}} + \omega_g k_{\text{PLL}} L_g I_{1\text{ref}d}^c,
\end{aligned}
$$
where \(k_{\text{PLL}} = k_{i,\text{PLL}} / k_{p,\text{PLL}}\). The transient stability of this slow subsystem can be assessed via an energy function. The stability criteria are:
$$
\begin{aligned}
\frac{1}{k_{p,\text{PLL}}} – L_g I_{1\text{ref}d}^c &> 0, \\
k_{\text{PLL}} U_{gd}^s &> 0, \\
U_{gd}^s + U_{gq}^s \delta_{\text{slow}} – k_{\text{PLL}} L_g I_{1\text{ref}d}^c &> 0.
\end{aligned}
$$
These inequalities provide direct constraints on the PLL parameters (\(k_{p,\text{PLL}}, k_{i,\text{PLL}}\)) to maintain synchronism during faults, given the grid voltage (\(U_{gd}^s, U_{gq}^s\)) and the current reference \(I_{1\text{ref}d}^c\).
The fast subsystem, describing the current tracking dynamics, is linear and given by:
$$
\frac{d}{dt}
\begin{bmatrix}
I_{1,\text{fast}d}^s \\
I_{1,\text{fast}q}^s \\
\delta_{\text{fast}}
\end{bmatrix} =
\begin{bmatrix}
A_{11} & A_{12} & A_{13} \\
B_{11} & B_{12} & B_{13} \\
C_1 & C_2 & C_3
\end{bmatrix}
\begin{bmatrix}
I_{1,\text{fast}d}^s \\
I_{1,\text{fast}q}^s \\
\delta_{\text{fast}}
\end{bmatrix},
$$
where the matrix elements \(A_{ij}, B_{ij}, C_i\) are functions of system parameters, operating point (\(I_{1,\text{slow}}^s, \delta_{\text{slow}}\)), and controller gains. For the utility interactive inverter to be transiently stable, the eigenvalues of this state matrix must all have negative real parts. This condition imposes constraints on the current loop PI parameters, \(k_{p,c}\) and \(k_{i,c}\).
Parameter Feasible Domain Optimization Design Method
The proposed design method synthesizes classical frequency-domain performance criteria with the novel transient stability constraints derived above. The systematic flowchart is summarized in the following table.
| Step | Action | Key Constraints / Tools | Output |
|---|---|---|---|
| 1-2 | PLL Initial Domain Design | Gain Margin (GM>5dB), Phase Margin (PM=30°-70°), Bandwidth (e.g., 10-50 Hz) via D-Partition. | Initial PLL parameter feasible domain \(\mathcal{D}_{\text{PLL, init}}\). |
| 3 | PLL Domain Refinement via Slow Subsystem | Apply stability criteria from Eq. (X) for a specified severe voltage sag (e.g., 0.2 p.u.). | Refined PLL domain \(\mathcal{D}_{\text{PLL, refined}}\). |
| 4-5 | Current Loop Initial Domain Design | GM>5dB, PM=30°-60°, Bandwidth (e.g., 400-900 Hz) via D-Partition. | Initial current loop parameter domain \(\mathcal{D}_{c, \text{init}}\). |
| 6 | Current Loop Domain Refinement via Fast Subsystem | 1. Enforce \(\epsilon \ll 1 \Rightarrow k_{i,c} > 1000\). 2. Ensure eigenvalues of fast subsystem matrix have negative real parts. |
Refined current loop domain \(\mathcal{D}_{c, \text{refined}}\). |
| Final | Parameter Selection | Select final \((k_{p,\text{PLL}}, k_{i,\text{PLL}}) \in \mathcal{D}_{\text{PLL, refined}}\) and \((k_{p,c}, k_{i,c}) \in \mathcal{D}_{c, \text{refined}}\). | Optimized parameters guaranteeing performance & transient stability. |
D-Partition Method for Initial Domain
The D-partition method is used to map the feasible parameter regions satisfying frequency-domain constraints. Consider a generic PI controller \(G_{PI}(s) = k_p + k_i/s\) in a negative feedback loop with a plant \(G(s)\). The closed-loop characteristic equation is \(D(s, k_p, k_i)=0\). The boundaries of the stable parameter region in the \((k_p, k_i)\) plane are given by:
$$
\begin{aligned}
D(0, k_p, k_i) &= k_i a_0 = 0 \quad \Rightarrow \quad k_i = 0 \quad \text{(singular boundary)}, \\
D(\pm j\omega, k_p, k_i) &= 0 \quad \text{for } \omega \in (0, \infty) \quad \text{(non-singular boundary)}.
\end{aligned}
$$
Letting \(G(j\omega) = R(\omega) + jI(\omega)\), the non-singular boundary is calculated as:
$$
k_p(\omega) = -\frac{R(\omega)}{R^2(\omega)+I^2(\omega)}, \quad k_i(\omega) = -\frac{\omega I(\omega)}{R^2(\omega)+I^2(\omega)}.
$$
The region enclosed by the \(k_i=0\) line and the parametric curve \((k_p(\omega), k_i(\omega))\) constitutes the feasible parameter domain for stability. Gain and phase margin requirements are incorporated by adding a gain-phase tester \(A e^{-j\phi}\) in the loop and solving for \((k_p, k_i)\) that satisfy \(GM = 20\log_{10}A\) and \(PM = \phi\). Bandwidth constraints are applied by requiring the closed-loop magnitude to be \(1/\sqrt{2}\) at the desired cutoff frequency \(\omega_b\), further trimming the feasible domain.
PLL Parameter Design for the Utility Interactive Inverter
For the PLL in a utility interactive inverter, the open-loop transfer function including a low-pass filter (cutoff \(\omega_p\)) is:
$$
G_{o,\text{PLL}}(s) = V_g \frac{\omega_p}{s+\omega_p} \left(k_{p,\text{PLL}} + \frac{k_{i,\text{PLL}}}{s}\right) \frac{1}{s} = \frac{V_g \omega_p (k_{p,\text{PLL}} s + k_{i,\text{PLL}})}{s^3 + \omega_p s^2}.
$$
Applying the D-partition method with constraints \(GM > 5\) dB, \(PM \in [30^\circ, 70^\circ]\), and bandwidth \(f_{b,\text{PLL}} \in [10, 50]\) Hz yields the initial feasible domain \(\mathcal{D}_{\text{PLL, init}}\). For a 50 kW utility interactive inverter with parameters \(V_g=391V\), \(L_g=7.34\) mH, \(L_f=1.5\) mH, this domain is shown below (conceptually).
This domain is then optimized using the slow subsystem stability criteria. For a severe LVRT condition where voltage sags to \(0.2\) p.u. and the d-axis current reference is set to \(I_{1\text{ref}d}^c = 0.35\) p.u., the criteria from Eq. (X) translate to:
$$
\begin{aligned}
k_{p,\text{PLL}} &< \frac{1}{L_g I_{1\text{ref}d}^c}, \\
k_{i,\text{PLL}} &> 0, \\
k_{i,\text{PLL}} &< \frac{U_{gd}^s + U_{gq}^s \delta_{\text{slow}}}{L_g I_{1\text{ref}d}^c} k_{p,\text{PLL}}.
\end{aligned}
$$
These inequalities define a region \(\mathcal{D}_{\text{PLL, transient}}\) in the parameter plane. The final refined PLL domain is the intersection: \(\mathcal{D}_{\text{PLL, refined}} = \mathcal{D}_{\text{PLL, init}} \cap \mathcal{D}_{\text{PLL, transient}}\). Parameters chosen from this intersection ensure the utility interactive inverter’s PLL maintains synchronism during the fault.
Current Loop Parameter Design for the Utility Interactive Inverter
The current control loop of the utility interactive inverter includes computation delay \(G_d(s)=e^{-sT_s}\). Its open-loop transfer function is:
$$
G_{o,c}(s) = \left(k_{p,c} + \frac{k_{i,c}}{s}\right) e^{-sT_s} \frac{1}{s L_f}.
$$
Applying the D-partition method with \(GM > 5\) dB, \(PM \in [30^\circ, 60^\circ]\), and bandwidth \(f_{b,c} \in [400, 900]\) Hz gives the initial domain \(\mathcal{D}_{c, \text{init}}\).
Transient stability refinement for the fast subsystem involves two steps. First, the singular perturbation theory requires \(\epsilon = 1/k_{i,c} \ll 1\), imposing a lower bound: \(k_{i,c} > 1000\). Second, the eigenvalues of the fast subsystem’s state matrix \(\mathbf{A}_{\text{fast}}\) must be in the left-half plane. For a given operating point and chosen PLL parameters, this leads to analytical constraints on \(k_{p,c}\). For the example system, stability requires:
$$
0 < k_{p,c} < 12.863.
$$
The refined current loop domain is thus: \(\mathcal{D}_{c, \text{refined}} = \mathcal{D}_{c, \text{init}} \cap \{ (k_{p,c}, k_{i,c}) | k_{i,c}>1000, 0<k_{p,c}<12.863 \}\).="" a="" by="" control="" current="" disturbance.
Experimental Validation
The proposed parameter design method for the utility interactive inverter was validated using a SpaceR hardware-in-the-loop (HIL) experimental platform. A 50 kW utility interactive inverter model and a weak grid were simulated in the real-time simulator, while the control algorithms were executed on a TMS320F28335 DSP controller. Three distinct parameter sets were tested under a grid voltage sag to 0.2 p.u., with the d-axis current reference stepped from 0.9 p.u. to 0.35 p.u.
| Parameter Set | PLL Parameters \((k_{p,\text{PLL}}, k_{i,\text{PLL}})\) | Current Loop Parameters \((k_{p,c}, k_{i,c})\) | Location Relative to Optimized Domains |
|---|---|---|---|
| Set 1 (Proposed) | (0.3, 4) | (8, 5000) | Within both \(\mathcal{D}_{\text{PLL, refined}}\) and \(\mathcal{D}_{c, \text{refined}}\) |
| Set 2 | (0.3, 18) | (8, 5000) | Outside \(\mathcal{D}_{\text{PLL, refined}}\) (PLL unstable) |
| Set 3 | (0.3, 4) | (2, 5000) | Outside \(\mathcal{D}_{c, \text{refined}}\) (Current loop dynamics poor) |
Results for Set 1: With both PLL and current loop parameters lying within the optimized feasible domains, the utility interactive inverter demonstrated excellent transient performance. The PLL frequency deviation remained bounded (max ~5 Hz) and settled within 0.2 seconds. The grid currents tracked their references smoothly without significant overshoot or distortion.
Results for Set 2: The PLL parameters violated the slow subsystem stability constraint. During the fault, the PLL lost synchronism, leading to diverging frequency oscillations. Consequently, the grid currents and voltages became severely distorted, leading to instability of the utility interactive inverter.
Results for Set 3: While the PLL parameters were acceptable, the current loop proportional gain \(k_{p,c}=2\) was below the optimized range. Although the system remained stable, the transient response was markedly worse. The PLL experienced a larger frequency deviation (max ~8 Hz) with pronounced oscillatory behavior, and the currents exhibited significant distortion immediately after the fault. This validates the importance of the fast subsystem stability constraints in the design of the utility interactive inverter’s current controller.
Conclusion
This article has presented a comprehensive feasible-parameter-domain optimization design method for utility interactive inverters that rigorously incorporates transient stability constraints. The methodology bridges the gap between conventional small-signal design and large-disturbance stability requirements. By first establishing parameter domains using the D-partition method subject to gain margin, phase margin, and bandwidth specifications, and then systematically refining these domains using stability criteria derived from a singular perturbation model, the approach guarantees that the final selected parameters ensure both robust steady-state performance and reliable operation during severe grid faults. The effectiveness of the method was conclusively demonstrated through hardware-in-the-loop experiments on a representative utility interactive inverter system. This design framework provides a valuable and practical tool for engineers to configure utility interactive inverters that are resilient in weak grid environments.
</k_{p,c}