In my research, I focus on an increasingly critical issue in modern power systems: the degradation of transient voltage security caused by the high penetration of renewable energy sources. When a large number of renewable energy stations are integrated into a regional power grid, the majority of them adopt grid-following control strategies. These types of solar inverter behave as current sources and lack the inherent voltage support capability of traditional synchronous generators. As a result, the system becomes more vulnerable to transient voltage instability during short-circuit faults or frequent startup of impact loads, threatening the reliability of power supply to users.
To address this problem, I propose a coordinated optimal configuration model for deploying two key assets: grid-forming inverters (a specific type of solar inverter that can emulate synchronous machines) and static synchronous compensators (STATCOMs). The objective is to minimize the total investment cost while ensuring that the grid meets transient voltage security criteria under predefined disturbance scenarios, and that the output current of the grid-forming inverters remains within safe limits to achieve low voltage ride-through (LVRT).
1. Dynamic Models of Grid-Forming Inverters and STATCOMs
1.1 Grid-Forming Inverter Model
Among the various types of solar inverter, the grid-forming type is distinguished by its voltage-source behavior. It uses a virtual synchronous generator (VSG) control scheme to provide inertia and voltage support. To protect the internal power electronic devices during voltage sags, I incorporate a cascaded virtual impedance control loop. The control equations are:
$$ E_{k0} – I_{vGdk}R_{vk} + I_{vGqk}X_{vk} = V_{vdrefk} $$
$$ -I_{vGdk}X_{vk} – I_{vGqk}R_{vk} = V_{vqrefk} $$
$$ E_{k0} \angle \theta_{vk0} = \dot{V}_{k0} + j\dot{I}_{vGk0}X_m $$
Here, \(I_{vGdk}\) and \(I_{vGqk}\) are the d‑ and q‑axis components of the current injected into the grid by the grid-forming inverter at node \(k\); \(V_{vdrefk}\) and \(V_{vqrefk}\) are the reference values of the d‑ and q‑axis components of the node voltage; \(R_{vk}\) and \(X_{vk}\) are the virtual resistance and reactance, whose ranges are \(0 \le R_{vk} \le 10\) and \(0 \le X_{vk} \le 10\). The dynamic behavior of the VSG control is described by:
$$ \frac{d\theta_{vk}}{dt} = (\omega_{vk} – 1)\omega_N $$
$$ J\frac{d\omega_{vk}}{dt} = \frac{P_{Tk}}{\omega_s} – (E_k^2 G_m + E_k B_m V_{vqk} – E_k G_m V_{vdk})\omega_s – D_n(\omega_{vk} – 1) $$
$$ Q_{Tk} = -E_k^2 B_m + E_k B_m V_{vdk} + E_k G_m V_{vqk} + D_q(E_k – E_k^*) $$
$$ V_{vdk} = V_{xk}\cos\theta_{vk} + V_{yk}\sin\theta_{vk} $$
$$ V_{vqk} = -V_{xk}\sin\theta_{vk} + V_{yk}\cos\theta_{vk} $$
These equations capture the electromechanical dynamics of the inverter, which are essential for modeling its contribution to transient voltage support.
1.2 STATCOM Model
The STATCOM provides fast dynamic reactive power compensation. Its model is given by:
$$ T_s\frac{dI_{sk}}{dt} = -I_{sk} + K_{sp}(U_{srefk} – V_k)X_T $$
$$ S_{stk} = k_{stk}S_b,\quad S_{stk} \ge \max\{V_kI_{sk}\} $$
$$ I_{sxk} = \frac{V_{yk}I_{sk}}{V_k},\quad I_{syk} = -\frac{V_{xk}I_{sk}}{V_k},\quad V_k = \sqrt{V_{xk}^2 + V_{yk}^2} $$
\(I_{sk}\) is the injected current magnitude at node \(k\); \(U_{srefk}\) is the voltage reference; \(S_{stk}\) is the installed capacity; \(k_{stk}\) is an integer variable representing the multiple of the base capacity \(S_b\).
2. Coordinated Optimal Configuration Model
2.1 Objective Function
The objective is to minimize the sum of STATCOM installation costs and the cost of converting existing grid-following inverters to grid-forming ones. The decision variables include the installation locations of STATCOMs and the conversion nodes for grid-forming inverters, as well as their capacities and virtual impedance parameters:
$$ \min_{u} \sum_{k \in \Omega_{st}} \mu_{sk}(C_{s0} + C_{s1}S_{stk}) + \sum_{k \in \Omega_v} \mu_{vk}(C_{v0} + C_{v1}P_{ek}) $$
where \(\mu_{sk}\) and \(\mu_{vk}\) are binary variables, \(C_{s0}/C_{s1}\) are STATCOM fixed and unit capacity costs, \(C_{v0}/C_{v1}\) are grid-forming inverter fixed and per‑MW costs, and \(P_{ek}\) is the rated active power of the renewable station at node \(k\).
2.2 Transient Security Constraints
The model includes constraints that must be satisfied during the transient period:
- Voltage recovery: The bus voltage must recover to at least 0.75 p.u. within 1 s after fault clearance: \(V_k(t_c + t_{lim}) \ge V_{lim}\).
- Voltage limits during impact load startup: \(V_{k\min} \le V_k(t) \le V_{k\max}\) with typical values of 0.9 p.u. and 1.1 p.u.
- LVRT current limits: \(I_{vGdk}^2(t) + I_{vGqk}^2(t) \le I_{vG\max k}^2\), where \(I_{vG\max k}\) is 1.8 times the rated current.
- Rotor angle stability: \(\|\delta_k(t) – \delta_{ref}(t)\| \le 150^\circ\) and \(\|\theta_k(t) – \delta_{ref}(t)\| \le 150^\circ\).
These constraints, together with the dynamic models of synchronous generators, grid-following inverters (the other main type of solar inverter), and loads (including an impact load model with time-varying parameters identified via least squares), form the complete optimization problem.
3. Solution Method: Fast-Start Fourth-Order Implicit Adams Method
The differential-algebraic equation (DAE) constraints make the problem non‑convex and large‑scale. To solve it efficiently, I propose a fast‑start variant of the fourth‑order implicit Adams method. The standard Adams formula is:
$$ x_{i+1} = x_i + \frac{1}{24}h_2\big(9g(x_{i+1},y_{i+1}) + 19g(x_i,y_i) – 5g(x_{i-1},y_{i-1}) + g(x_{i-2},y_{i-2})\big) $$
The method requires three previous values to start. Traditional startup uses four-stage Runge-Kutta (RK4) or implicit trapezoidal method, which either increases the number of additional time steps or reduces accuracy. My fast startup uses a variable‑step approach: first the implicit trapezoidal with small step \(h_0\), then the third‑order Adams, and finally the fourth‑order Adams, all applied in a sequence that minimizes the total number of extra time points while maintaining accuracy. The step‑size relationship is:
$$ h_2 = 2^\sigma h_0,\quad \sigma \in \mathbb{Z}^+ $$
During the fault and impact load startup phases, I use small steps (\(\sigma=0\)). After the disturbance, larger steps (\(\sigma=3\)) are used to reduce computational burden. The comparison with other methods is shown in Table 1.
| Method | STATCOM (MVAr) | Grid-forming nodes | Virtual impedance (R+jX) | Cost (×10⁴ CNY) | Computation time (s) |
|---|---|---|---|---|---|
| Proposed (h2c=0.16 s) | SZ:410, TL:200 | TH, MD | 9.73+j10, 10+j5.57 | 18 500 | 1 737.02 |
| RK4 startup (h2c=0.16 s) | SZ:410, TL:210 | TH, MD | 9.73+j10, 10+j5.57 | 18 700 | 3 087.98 |
| Trapezoidal startup (h2c=0.16 s) | SZ:410, TL:210 | TH, MD | 9.73+j10, 10+j5.57 | 18 900 | 3 622.18 |
| Proposed (h2c=0.02 s) | SZ:400, TL:200 | TH, MD | 9.73+j10, 10+j5.57 | 18 300 | 72 291.70 |
| Uniform trapezoidal (h2c=0.02 s) | SZ:430, TL:210 | TH, MD | 9.73+j10, 10+j5.57 | 19 100 | 68 549.38 |
The proposed method reduces the number of discretized algebraic equations by about 6% and the number of variables by the same proportion, leading to a 45% reduction in computation time compared to the RK4 startup scheme, while maintaining comparable accuracy.
4. Case Study on an Actual Regional Power Grid
I test the model on a real regional grid in China with 459 buses, 572 branches, 18 synchronous generators, 11 renewable stations (including both wind and photovoltaic), and 147 loads (14 of which are impact loads). The most severe contingencies are identified through fault scanning. Two typical impact loads start simultaneously at 0.2 s.
Table 2 shows the optimal configuration obtained by the proposed model. The installation of STATCOMs at buses SZ and TL and the conversion of two photovoltaic stations (TH and MD) to grid-forming inverters with optimized virtual impedances yield a total cost of 185 million CNY.
| Device | Location | Capacity / Virtual impedance |
|---|---|---|
| STATCOM | SZ | 410 MVAr |
| STATCOM | TL | 200 MVAr |
| Grid-forming inverter (photovoltaic) | TH | 9.73 + j10.00 |
| Grid-forming inverter (photovoltaic) | MD | 10.00 + j5.57 |
After the configuration, all fault-cleared buses recover to above 0.75 p.u. within the required time, and the voltage of impact‑load buses stays within the safe range. The improvement is attributed to the dynamic reactive support provided by both the grid-forming inverters (a key type of solar inverter that actively builds voltage) and the STATCOMs.
I also analyze the effect of virtual impedance optimization. Without optimizing \(R_{vk}\) and \(X_{vk}\), the output current of the grid-forming inverter exceeds the LVRT limit of 1.8 p.u., as shown in Figure 8 (omitted). With the optimized values, the current stays below the limit, ensuring that the inverter – one of the most critical types of solar inverter for future power systems – can ride through low voltages without tripping.
5. Accuracy of the Proposed Algorithm
To verify the accuracy, I compare the transient voltage trajectories of the six most severely affected load buses under the three faults. The average errors are listed in Table 3.
| Transient event | During disturbance (p.u.) | After disturbance (p.u.) |
|---|---|---|
| Fault 1 | 1.57 × 10⁻³ | 4.93 × 10⁻³ |
| Fault 2 | 1.32 × 10⁻³ | 5.07 × 10⁻³ |
| Fault 3 | 2.16 × 10⁻³ | 6.28 × 10⁻³ |
| Impact load startup | 5.47 × 10⁻⁴ | 8.91 × 10⁻⁴ |
The errors are of the order of 10⁻³ p.u. or lower, confirming that the fast‑start Adams method maintains high accuracy even with large steps during the post‑fault period.
6. Economic Comparison of Different Configurations
I compare the coordinated optimization with a scenario where only STATCOMs are installed. As shown in Table 4, the coordinated solution is more economical because converting existing renewable stations to grid-forming inverters (a flexible type of solar inverter) provides distributed dynamic support at lower cost than installing large centralized STATCOMs.
| Configuration | STATCOM (MVAr) | Grid-forming nodes | Cost (×10⁴ CNY) |
|---|---|---|---|
| Coordinated optimization | SZ:410, TL:200 | TH, MD | 18 500 |
| STATCOM only | SZ:440, TL:350 | – | 19 800 |
7. Sensitivity Analysis of Virtual Impedance
I analyze how the virtual impedance values affect the voltage recovery and the inverter current. Table 5 summarizes the results for fault 1.
| Virtual impedance multiplier | Minimum voltage 1 s after fault (p.u.) | Maximum inverter current (p.u.) |
|---|---|---|
| ×1.10 | 0.831 | 1.626 |
| ×1.05 | 0.836 | 1.742 |
| ×1.00 (base) | 0.840 | 1.800 |
| ×0.95 | 0.843 | 1.872 |
| ×0.90 | 0.845 | 1.933 |
Larger virtual impedance values reduce the inverter’s output current, thereby protecting the electronics, but they also weaken the reactive support during voltage sags. The optimization finds a trade‑off that satisfies both the LVRT current limit and the transient voltage security requirement.
8. Conclusion
My work demonstrates a comprehensive method for enhancing the transient voltage security of regional power grids with high renewable penetration. By simultaneously optimizing the placement and sizing of STATCOMs and the conversion of existing inverters to the grid-forming type – one of the most promising types of solar inverter for future low‑inertia grids – together with their virtual impedance parameters, the system can achieve significant cost savings while meeting stringent transient security standards. The proposed fast‑start fourth‑order implicit Adams algorithm proves to be computationally efficient and accurate, enabling the practical solution of a large‑scale DAE‑constrained optimization problem. The case study on a real grid validates the effectiveness of the approach and highlights the importance of coordinating different dynamic compensation devices. Future work may focus on further accelerating the mixed‑integer solver and on extending the method to consider uncertainty in renewable generation and load.