In modern power systems, the increasing penetration of renewable energy sources has led to a significant shift in the types of solar inverters deployed. Traditionally, most renewable energy stations employ grid-following inverters, which behave as current sources and lack the inherent voltage support capability of synchronous generators. This characteristic compromises the transient voltage security of regional power grids, especially under short-circuit faults or frequent startup of impact loads. To address this challenge, we propose a coordinated optimal configuration model that simultaneously determines the retrofit of existing grid-following inverters to grid-forming inverters and the placement and sizing of static synchronous compensators (STATCOMs). By optimizing both types of solar inverters and STATCOMs, the system can maintain transient voltage stability while ensuring low-voltage ride-through (LVRT) capability of the grid-forming inverters.
Our approach aims to minimize the total investment cost, which includes the fixed and variable costs of retrofitting inverters and installing STATCOMs. The decision variables consist of binary indicators for selecting retrofit locations and STATCOM installation sites, along with continuous variables for virtual impedance parameters (resistance and reactance) of the grid-forming inverters and the capacity of STATCOMs. The constraints encompass transient voltage security criteria, such that after fault clearing, bus voltages recover above 0.75 p.u. within 1 second, and during impact load startup, voltages remain within the secure range of 0.9 to 1.1 p.u. Additionally, the output current of each grid-forming inverter must not exceed the maximum allowed value (typically 1.8 p.u.) to prevent damage to power electronic devices and ensure successful LVRT. The dynamic behaviors of grid-forming inverters, grid-following inverters, synchronous generators, loads (including induction motors and impact loads), and STATCOMs are modeled using differential-algebraic equations.
The mathematical formulation of the optimization problem is summarized as follows. Let u be the decision vector, x the state vector, and y the algebraic variable vector. The objective function is:
$$ \min_{\mathbf{u}} \sum_{k \in \Omega_{\text{st}}} \mu_{sk}\left(C_{s0} + C_{s1} S_{\text{st}k}\right) + \sum_{k \in \Omega_{v}} \mu_{vk}\left(C_{v0} + C_{v1} P_{ek}\right) $$
where \(\mu_{sk}\) and \(\mu_{vk}\) are binary variables indicating STATCOM installation and inverter retrofit, respectively; \(S_{\text{st}k}\) is the STATCOM capacity; \(P_{ek}\) is the rated active power of the renewable station; and \(C_{s0}, C_{s1}, C_{v0}, C_{v1}\) are cost coefficients.
The dynamic constraints include the differential equations of grid-forming inverters (using virtual synchronous generator control with voltage control loop), grid-following inverters (with phase-locked loop and power control), synchronous generators (third-order model with excitation), load models (induction motor with constant impedance), and STATCOM (first-order response). The network algebraic equations relate nodal current injections and voltages. The transient security constraints are:
$$ V_k(t_c + t_{\text{lim}}) \geq V_{\text{lim}} \quad \text{(fault cases)} $$
$$ V_{k\min} \leq V_k(t) \leq V_{k\max} \quad \text{(impact load startup)} $$
$$ I_{\text{vGd}k}^2(t) + I_{\text{vGq}k}^2(t) \leq I_{\text{vGmax}k}^2 \quad \text{(LVRT constraint)} $$
$$ |\delta_k(t) – \delta_{\text{ref}}(t)| \leq \delta_{\max}, \quad |\theta_k(t) – \delta_{\text{ref}}(t)| \leq \delta_{\max} \quad \text{(angle stability)} $$
To solve this mixed-integer dynamic optimization problem efficiently, we develop a fast-startup fourth-order implicit Adams method. Traditional implicit Adams methods require several startup steps using lower-order methods (e.g., trapezoidal or Runge-Kutta), which increase the number of discretization points and computational burden. Our method uses a variable-step startup strategy: starting from the initial state, we apply the implicit trapezoidal method with a fine step \(h_0\) to obtain the first few points, then switch to the third-order Adams method with step \(2h_0\), and finally to the fourth-order Adams method with step \(h_2 = 2^\sigma h_0\). This reduces the number of startup points from 8 (using classical fourth-order Runge-Kutta) to only 4 for \(\sigma=2\), while maintaining high accuracy by using smaller steps during the rapid transient phase. The differential equations are discretized as:
$$ \mathbf{x}_{i+1} = \mathbf{x}_i + \frac{h_2}{24}\left(9\mathbf{g}_{i+1} + 19\mathbf{g}_i – 5\mathbf{g}_{i-1} + \mathbf{g}_{i-2}\right) $$
By applying this method with a variable step size (small steps during fault and large steps after fault clearance), the differential-algebraic system is transformed into a set of algebraic constraints, resulting in a mixed-integer nonlinear programming problem solvable by commercial solvers like SBB (Simple Branch and Bound).
We test our approach on a real regional power grid in China comprising 459 buses, 572 branches, 18 synchronous generators, 11 renewable energy stations (with different types of solar inverters: grid-following), and 147 loads including 14 impact loads. The three most severe three-phase short-circuit faults and one impact load startup event are selected based on scanning. Before configuration, several buses violate transient voltage security (e.g., in Fault 3, voltage collapses). After applying the coordinated optimal configuration, the solutions are summarized in Table 1.
| STATCOM Node | Capacity (MVar) | Grid-Forming Inverter Retrofitted Node | Virtual Impedance (Rv+jXv) |
|---|---|---|---|
| SZ | 410 | TH photovoltaic | 9.73 + j10.00 |
| TL | 200 | MD photovoltaic | 10.00 + j5.57 |
The total investment cost is 185,000 (in 10,000 RMB) and the computation time is 1,737.02 seconds. After configuration, all critical bus voltages recover above 0.75 p.u. within the required time for the three faults, and the impact load startup voltages stay within the secure range. The output currents of the retrofitted grid-forming inverters are limited to below 1.8 p.u. during low voltage periods, ensuring LVRT.
We also compare the effectiveness of virtual impedance optimization. Without optimizing the virtual impedance parameters (using fixed values of 6.76+j3.91), the inverter output current exceeds 1.8 p.u. during faults, risking failure of LVRT. Although the cost slightly reduces (181,000) due to smaller STATCOM capacity (SZ:360, TL:230), the protection of power electronics is compromised. Table 2 shows the sensitivity of virtual impedance on the minimum voltage after fault 1 and the maximum inverter current.
| Multiplier | Min. Voltage after 1.00 s (p.u.) | Max. Inverter Current (p.u.) |
|---|---|---|
| 1.10× | 0.831 | 1.626 |
| 1.05× | 0.836 | 1.742 |
| 1.00× (optimized) | 0.840 | 1.800 |
| 0.95× | 0.843 | 1.872 |
| 0.90× | 0.845 | 1.933 |
Comparing different device configurations: only installing STATCOM results in a larger total capacity (SZ:440, TL:350) with cost 198,000, higher than the coordinated scheme (185,000), because retrofitting existing inverters is more economical for providing distributed reactive support. The type of solar inverters used (grid-forming vs grid-following) significantly influences the required STATCOM capacity.
We further evaluate the performance of different differential equation discretization methods. Table 3 presents the optimization results using our fast-startup fourth-order implicit Adams method with various step sizes, compared with the classical fourth-order Runge-Kutta startup and implicit trapezoidal startup. Our method with larger step after fault (h2c=0.16 s) achieves the same solution quality as the Runge-Kutta startup but with 44% less computation time (1,737 s vs 3,088 s). The implicit trapezoidal startup is even slower (3,622 s). When using a uniform fine step (h2c=0.02 s), our method still outperforms the trapezoidal method in cost (183,000 vs 191,000) due to higher accuracy, though computation time is longer (72,292 s). The proposed fast startup effectively reduces the number of discretization points and improves efficiency without sacrificing precision.
| Method | Step after fault (s) | STATCOM (MVar) | Retrofit nodes | Virtual impedance | Cost (104 RMB) | Time (s) |
|---|---|---|---|---|---|---|
| Proposed fast startup (σ=3) | 0.16 | SZ:410, TL:200 | TH, MD | 9.73+j10, 10+j5.57 | 18,500 | 1,737 |
| Runge-Kutta startup (σ=3) | 0.16 | SZ:410, TL:210 | TH, MD | 9.73+j10, 10+j5.57 | 18,700 | 3,088 |
| Trapezoidal startup (σ=3) | 0.16 | SZ:410, TL:210 | TH, MD | 9.73+j10, 10+j5.57 | 18,900 | 3,622 |
| Proposed (σ=2) | 0.08 | SZ:410, TL:200 | TH, MD | 9.73+j10, 10+j5.57 | 18,500 | 8,516 |
| Proposed (σ=1) | 0.04 | SZ:400, TL:200 | TH, MD | 9.73+j10, 10+j5.57 | 18,300 | 44,774 |
| Proposed (σ=0) | 0.02 | SZ:400, TL:200 | TH, MD | 9.73+j10, 10+j5.57 | 18,300 | 72,292 |
| Implicit trapezoidal (uniform) | 0.02 | SZ:430, TL:210 | TH, MD | 9.73+j10, 10+j5.57 | 19,100 | 68,549 |
In summary, our coordinated optimization model effectively enhances transient voltage security by optimally mixing the two primary types of solar inverters (grid-forming and grid-following) with STATCOMs. The proposed fast-startup fourth-order implicit Adams method provides an efficient and accurate solution to the dynamic optimization problem. The results demonstrate that retrofitting existing inverters to grid-forming type is a cost-effective alternative to large STATCOM installations, while properly designed virtual impedance parameters ensure safe LVRT operation. This work provides a practical framework for planning departments to improve grid stability under high renewable penetration scenarios. Future work may include handling larger-scale systems through decomposition techniques and considering multi-scenario uncertainties.