In this study, I present a systematic approach for identifying the low-voltage ride-through control mode and the associated control parameters of a solar inverter electromechanical transient model. The proposed identification framework is based on the multi-objective differential evolution algorithm. The goal is to reconstruct the black-box control behavior of modern solar inverters during grid faults, which is essential for accurate transient stability studies and grid integration analysis. I first establish a detailed electromagnetic transient model of the grid-connected solar inverter, including the DC-link dynamics, the phase-locked loop, the current control loops, and the LVRT logic. Then, I collect hardware-in-the-loop measurement data from a real-time simulation platform under various balanced and unbalanced fault conditions. By extracting representative steady-state key points from the measured current and power waveforms, I formulate a multi-objective identification problem where the average error and the maximum error at the LVRT key points are minimized simultaneously. The multi-objective differential evolution algorithm is enhanced with an adaptive parameter tuning strategy that adjusts the mutation and crossover probabilities according to the population diversity measured by information entropy. A non-dominated sorting method is also embedded to preserve the Pareto-front solutions and to improve convergence. The identified control parameters are then used in an electromechanical transient model to evaluate the fitting accuracy under different LVRT scenarios. The results demonstrate that the proposed algorithm accurately identifies the control mode and parameters of the solar inverter, and it outperforms several state-of-the-art evolutionary algorithms in terms of identification precision and robustness.
Solar power generation has become one of the most important renewable energy sources worldwide. The large-scale integration of solar photovoltaic systems introduces new challenges for grid planning and operation. In particular, the dynamic response of solar inverters during severe voltage sags can significantly influence the transient stability of the power system. To evaluate the impact of solar generation on grid stability, power system engineers need accurate simulation models of solar inverters. However, the internal control parameters of commercial solar inverters are usually protected as proprietary information. Therefore, parameter identification techniques are essential to obtain accurate equivalent models from field measurements. The identification task becomes even more complicated during low-voltage ride-through events, because solar inverters switch to dedicated fault-ride-through control strategies that differ from the normal maximum-power-point-tracking operation. The control mode may be based on active and reactive current references or on active and reactive power references. Without knowing the actual LVRT control mode, it is impossible to identify the associated parameters correctly.
Many existing identification techniques focus on the proportional-integral controller parameters of the solar inverter in normal operation. Those techniques are not sufficient to describe the transient behavior during LVRT. Several recent studies proposed step-by-step identification frameworks for fault-transient modeling, but they often require manual tuning of some parameters or assume a known control structure. In contrast, my proposed method is designed to identify both the LVRT control mode and the control parameters without prior knowledge of the manufacturer’s control logic. The method is based on a multi-objective optimization approach that can handle multiple candidate control structures and can select the best one by comparing the achieved model accuracy. This approach avoids the risk of overfitting to a single operating condition and improves the generalization capability of the identified model.
The structure of a typical grid-connected solar inverter is presented above. The photovoltaic array supplies DC power to the inverter, which is connected to the AC grid through an LCL filter. The inverter is controlled in the synchronous reference frame using a dual-loop control structure. The outer voltage loop regulates the DC-link voltage and generates the d-axis current reference, while the inner current loop regulates the dq-axis currents. During normal operation, the q-axis current reference is usually set to zero to achieve unity power factor. The phase-locked loop measures the grid voltage angle and provides the reference phase for the Park transformation. The control block diagram can be described by the following equations. The DC voltage outer loop is given by
$$ i_{gd,ref} = k_{P1}\left(u_{dc,ref}-u_{dc}\right) + k_{I1}\int\left(u_{dc,ref}-u_{dc}\right) dt , \tag{1} $$
where \(u_{dc}\) is the measured DC-link voltage, \(u_{dc,ref}\) is the DC voltage reference, and \(k_{P1}\), \(k_{I1}\) are the proportional and integral gains of the voltage controller. The current inner loop is described by
$$ \begin{cases} u_{d,ref} = u_{gd} – \omega L i_{gq} + k_{P2}\left(i_{gd,ref}-i_{gd}\right) + k_{I2}\int\left(i_{gd,ref}-i_{gd}\right) dt, \\ u_{q,ref} = u_{gq} + \omega L i_{gd} + k_{P3}\left(i_{gq,ref}-i_{gq}\right) + k_{I3}\int\left(i_{gq,ref}-i_{gq}\right) dt, \end{cases} \tag{2} $$
where \(\omega\) is the grid angular frequency, \(L\) is the filter inductance, and \(k_{P2}\), \(k_{I2}\), \(k_{P3}\), \(k_{I3}\) are the current controller gains. During a grid fault, the grid voltage magnitude decreases. The imbalance between the photovoltaic input power and the inverter output power causes the DC-link voltage to rise. To protect the inverter and to support the grid voltage, the LVRT controller changes the current references according to the reactive current requirement specified by grid codes. The dynamic reactive current is usually proportional to the voltage dip depth:
$$ i_Q = K \left(u_{lin} – u\right) i_N , \tag{3} $$
where \(K\) is the slope of the reactive current response, \(u_{lin}\) is the low-voltage threshold, \(u\) is the per-unit AC voltage, and \(i_N\) is the rated current. After the voltage sag is detected, the solar inverter is required to supply reactive current as a function of the voltage drop. However, the control mode can be implemented in two different ways: either the controller directly sets the active and reactive current references, or it sets the active and reactive power references and then divides by the positive-sequence voltage to obtain the current references. The two modes are expressed as follows.
For the specified current control mode, the LVRT current references are given by
$$ \begin{cases} i_{P,LVRT} = k_{1,iP} u_t + k_{2,iP} i_{P0} + i_{P,set}, \\ i_{Q,LVRT} = k_{1,iQ}\left(u_{lin}-u_t\right) + k_{2,iQ} i_{Q0} + i_{Q,set}, \end{cases} \tag{4} $$
where \(u_t\) is the steady-state voltage during the LVRT event, \(i_{P0}\) and \(i_{Q0}\) are the pre-fault active and reactive currents, \(k_{1,iP}\) and \(k_{2,iP}\) are active current coefficients, \(k_{1,iQ}\) and \(k_{2,iQ}\) are reactive current coefficients, and \(i_{P,set}\), \(i_{Q,set}\) are the additional current set-points. For the specified power control mode, the LVRT power references are converted into current references by dividing by the positive-sequence voltage:
$$ \begin{cases} i_{P,LVRT} = \frac{P_{LVRT}}{u_t} = \frac{k_P P_0 + P_{set}}{u_t}, \\ i_{Q,LVRT} = \frac{Q_{LVRT}}{u_t} = \frac{k_Q Q_0 + Q_{set}}{u_t}, \end{cases} \tag{5} $$
where \(P_0\) and \(Q_0\) are the pre-fault active and reactive powers, \(k_P\) and \(k_Q\) are the power coefficients, and \(P_{set}\), \(Q_{set}\) are the power set-points. In practice, the current references are limited by the maximum current capability of the solar inverter. The final current commands are obtained as
$$ i_{Q,cmd} = \min\left(i_{Q,LVRT}, i_{Q,max}\right), \tag{6} $$
$$ i_{P,cmd} = \min\left(i_{P,LVRT}, \sqrt{i_{max}^2 – i_Q^2}, i_{P,max}\right). \tag{7} $$
The unknown parameters to be identified are therefore the coefficients and set-points in Equation (4) or Equation (5), depending on the actual control mode of the solar inverter. I designed a multi-objective identification method that can estimate these parameters from a set of measured LVRT waveforms.
To prepare the identification dataset, I first extract the key points from each tested LVRT event. The entire disturbance process can be divided into three regions: the pre-fault region, the fault region, and the post-fault region. The fault region is further split into a transient subregion and a steady-state subregion. During the steady-state portion of the fault, the active power, reactive power, active current, and reactive current are almost constant. The key point value is obtained by averaging the corresponding measured signal over the steady-state interval. For example, the key value of the active current is
$$ i_{Pt} = \frac{\int_{b1}^{b2} i_P(t)\, dt}{b2-b1}, \tag{8} $$
where \(b1\) and \(b2\) define the steady-state window. Similarly, the key points \(P_t\), \(Q_t\), and \(i_{Qt}\) are computed. These key points form the identification dataset. The measured data are collected from a hardware-in-the-loop test platform using a real solar inverter controller connected to an RT-LAB real-time simulator. I considered both balanced three-phase faults and unbalanced two-phase faults at different pre-fault power levels and different voltage dip depths. The training set and the testing set are summarized in Table 1.
| Set | Fault type | \(P_0\) | \(u_t\) |
|---|---|---|---|
| Training | Three-phase | 0.2 | 0.35, 0.75 |
| Three-phase | 1.0 | 0.35, 0.75 | |
| Two-phase | 0.2 | 0.35, 0.50, 0.75 | |
| Two-phase | 1.0 | 0.35, 0.50, 0.75 | |
| Testing | Three-phase | 0.2 | 0.50 |
| Three-phase | 1.0 | 0.50 | |
| Two-phase | 0.2 | 0.20 | |
| Two-phase | 1.0 | 0.20 |
The differential evolution algorithm is a population-based global optimization technique that evolves a set of candidate solutions using mutation, crossover, and selection operations. Each individual in the population represents a vector of the unknown LVRT control parameters. For the specified current control mode of the active current, the \(i\)-th individual is defined as
$$ T_i = \left[k_{1,iP}^{(i)},\; k_{2,iP}^{(i)},\; i_{P,set}^{(i)}\right]. \tag{9} $$
The initial population is generated randomly within the feasible parameter ranges:
$$ k_{1,iP} = \min k_{1,iP} + r \left(\max k_{1,iP} – \min k_{1,iP}\right), \tag{10} $$
$$ k_{2,iP} = \min k_{2,iP} + r \left(\max k_{2,iP} – \min k_{2,iP}\right), \tag{11} $$
$$ i_{P,set} = \min i_{P,set} + r \left(\max i_{P,set} – \min i_{P,set}\right), \tag{12} $$
where \(r\) is a uniform random number in \((0,1)\). The mutation operation produces a donor vector \(T_a^{(1)}\) by adding the scaled difference of two other individuals \(T_b\) and \(T_c\) to a base individual \(T_a\):
$$ T_a^{(1)} = T_a + p_m\left(T_b – T_c\right), \tag{13} $$
where \(p_m\) is the mutation operator. After mutation, a crossover operation mixes the donor vector with the target vector to create a trial vector. The trial vector is then checked against the boundary constraints:
$$ x_{aj}^{(1)} = \begin{cases} \min x_j, & x_{aj}^{(1)} < \min x_j, \\ x_{aj}^{(1)}, & \min x_j \le x_{aj}^{(1)} \le \max x_j, \\ \max x_j, & x_{aj}^{(1)} > \max x_j. \end{cases} \tag{14} $$
To evaluate the fitness of each individual, I defined the objective function as the negative sum of the absolute residuals between the measured command values and the model outputs:
$$ J(T_i) = \max\left(-\sum_{g=1}^{m} \left| e_g \right|\right), \tag{15} $$
where \(m\) is the number of measured cases and \(e_g\) is the residual for the \(g\)-th case. This real-valued objective is used in the single-objective differential evolution algorithm. In the multi-objective differential evolution algorithm, I define two objectives to improve robustness and to prevent overfitting:
$$ J_1 = \min\left(\sum_{k=1}^{m} \left| e_g \right|\right), \tag{16} $$
$$ J_2 = \min\left(\max_{k} \left| e_g \right|\right). \tag{17} $$
The first objective minimizes the cumulative error over all training cases, while the second objective minimizes the worst-case error. These two objectives often conflict, as a solution with a low average error may have a high maximum error, and vice versa. The multi-objective differential evolution algorithm searches for the Pareto-optimal set that represents the trade-off between these goals.
Because the search performance of the differential evolution algorithm depends strongly on the mutation and crossover probabilities, I introduced an adaptive parameter tuning strategy based on the population diversity. The diversity of the population is quantified by the information entropy of the parameter values. The entropy of the \(j\)-th parameter is computed as
$$ H_j = -\sum_{h=h_0}^{h_1} p\left(x_j = h\right) \log_2 p\left(x_j = h\right), \tag{18} $$
where \(p(x_j = h)\) is the probability that the \(j\)-th parameter equals \(h\), and \(h_0\), \(h_1\) are the discrete values of that parameter. The total population diversity is
$$ H(T) = \sum_{j=1}^{n} H_j. \tag{19} $$
The adaptive mutation and crossover probabilities are then updated according to
$$ p_m = p_{m,min} + \frac{H(T)}{H_0(T)}\left(p_{m,max} – p_{m,min}\right), \tag{20} $$
$$ p_c = p_{c,max} – \frac{H(T)}{H_0(T)}\left(p_{c,max} – p_{c,min}\right), \tag{21} $$
where \(H_0(T)\) is the diversity of the initial population. At the beginning of the evolution, the high diversity yields a large mutation rate and a small crossover rate, which encourages extensive exploration. As the population converges, the diversity decreases, and the crossover rate increases to focus on local exploitation.
The multi-objective differential evolution algorithm integrates the non-dominated sorting procedure into the selection step. After generating the trial population, the combined population of size \(2G\) is formed by merging the parent population and the offspring population. The fast non-dominated sorting algorithm partitions this combined population into several fronts \(T_{best,1}, T_{best,2}, \ldots, T_{best,s}\). The crowding distance of each individual is computed to maintain diversity within the same front. For the \(i\)-th individual in a front of size \(z\), the crowding distance for the first objective is
$$ L_{i,1} = \begin{cases} \infty, & i=1,\\ \frac{J_1(T_{i+1}) – J_1(T_{i-1})}{\max J_1 – \min J_1}, & 1 < i < z,\\ \infty, & i=z, \end{cases} \tag{22} $$
and for the second objective it is
$$ L_{i,2} = \begin{cases} \infty, & i=1,\\ \frac{J_2(T_{i+1}) – J_2(T_{i-1})}{\max J_2 – \min J_2}, & 1 < i < z,\\ \infty, & i=z. \end{cases} \tag{23} $$
The total crowding distance is
$$ L_i = L_{i,1} + L_{i,2}. \tag{24} $$
The individuals with the largest crowding distances are preferred to fill the next population. This process is repeated until the termination condition is met. The final solution is selected from the first non-dominated front using a fuzzy satisfaction criterion. The membership function of the \(i\)-th solution is defined as
$$ y_i = \frac{1}{2}\left(\frac{\max J_1 – J_{1,i}}{\max J_1 – \min J_1} + \frac{\max J_2 – J_{2,i}}{\max J_2 – \min J_2}\right). \tag{25} $$
The solution with the highest satisfaction value is considered the optimal compromise solution.
To determine the LVRT control mode of the solar inverter, I used the identification results to simulate the transient response under both candidate control modes. The simulated current and power waveforms are then compared with the measured waveforms from the hardware-in-the-loop tests. I define the average deviation and the maximum deviation for the variable \(X\) (where \(X \in \{P, Q, i_P, i_Q\}\)) as
$$ \beta_X = \frac{\sum_{t=k_{start}}^{k_{end}} \left| X_{LVRT}(t) – X_{cmd}(t) \right|}{k_{end} – k_{start} + 1}, \tag{26} $$
$$ \beta_{X,max} = \max \left| X_{LVRT}(t) – X_{cmd}(t) \right|. \tag{27} $$
I set the maximum allowable values for \(\beta_X\) and \(\beta_{X,max}\) to 0.1 and 0.15, respectively. If a specific control mode yields acceptably small deviations for both active power and active current, that mode is considered to be the actual active current control mode. The same logic applies to reactive power and reactive current to determine the reactive control mode.
I performed the identification experiments on a hardware-in-the-loop platform that consists of a real solar inverter controller and an RT-LAB real-time simulator. The controller receives voltage and current measurements from the simulated grid and sends pulse-width-modulated switching signals to the simulated inverter bridge. This setup allows me to test the real controller without the risk of damaging the actual power stage. I tested both balanced and unbalanced faults at different operating points. The control parameters identified by the multi-objective differential evolution algorithm are then inserted into an electromechanical transient model implemented in the power system analysis software package. By running time-domain simulations, I compared the model output with the measured data.
First, I evaluated the identification results for the first testing case with pre-fault active power \(P_0 = 0.2\) pu and fault voltage \(u_t = 0.50\) pu under a three-phase fault. The simulated active and reactive power waveforms using the specified power control mode and the specified current control mode are compared. The evaluation metrics for all six testing cases are reported in Tables 2 and 3. The specified power control results in Table 2 show that the active power and active current deviations are very small for case 1, while the reactive power and reactive current deviations exceed the allowable limits. On the other hand, the specified current control results in Table 3 show that the reactive power and reactive current deviations are significantly reduced. Therefore, the solar inverter under test applies the specified power control for the active current axis and the specified current control for the reactive current axis. For the other cases, the specified current control mode gives acceptable deviations for both active and reactive axes.
| Case | \(\beta_P\) | \(\beta_{P,max}\) | \(\beta_{iP}\) | \(\beta_{iP,max}\) | \(\beta_Q\) | \(\beta_{Q,max}\) | \(\beta_{iQ}\) | \(\beta_{iQ,max}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.002 | 0.009 | 0.005 | 0.019 | 0.101 | 0.102 | 0.184 | 0.185 |
| 2 | 0.027 | 0.027 | 0.039 | 0.039 | 0.022 | 0.023 | 0.030 | 0.033 |
| 3 | 0.069 | 0.079 | 0.186 | 0.211 | 0.121 | 0.124 | 0.292 | 0.299 |
| 4 | 0.034 | 0.065 | 0.074 | 0.131 | 0.109 | 0.112 | 0.199 | 0.203 |
| 5 | 0.008 | 0.015 | 0.014 | 0.028 | 0.108 | 0.109 | 0.214 | 0.217 |
| 6 | 0.053 | 0.074 | 0.100 | 0.141 | 0.117 | 0.121 | 0.232 | 0.240 |
| Case | \(\beta_P\) | \(\beta_{P,max}\) | \(\beta_{iP}\) | \(\beta_{iP,max}\) | \(\beta_Q\) | \(\beta_{Q,max}\) | \(\beta_{iQ}\) | \(\beta_{iQ,max}\) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.008 | 0.009 | 0.015 | 0.017 | 0.001 | 0.002 | 0.002 | 0.005 |
| 2 | 0.012 | 0.022 | 0.018 | 0.032 | 0.001 | 0.001 | 0.001 | 0.002 |
| 3 | 0.012 | 0.015 | 0.029 | 0.036 | 0.001 | 0.006 | 0.003 | 0.015 |
| 4 | 0.014 | 0.017 | 0.025 | 0.032 | 0.001 | 0.006 | 0.002 | 0.011 |
| 5 | 0.009 | 0.010 | 0.020 | 0.023 | 0.001 | 0.002 | 0.010 | 0.013 |
| 6 | 0.002 | 0.022 | 0.011 | 0.030 | 0.001 | 0.005 | 0.011 | 0.019 |
To further verify the advantage of the multi-objective strategy, I compared the proposed multi-objective differential evolution algorithm with the standard differential evolution algorithm that uses only the single objective \(J_1\). I defined two performance indices: the average error \(I_1 = J_1/m\) and the maximum error \(I_2 = J_2\). The symbols used for the active and reactive metrics are summarized in Table 4. The comparison is shown in Table 5. For the active-current specified mode, the multi-objective differential evolution algorithm slightly increases the average current error by 0.003 pu compared with the single-objective differential evolution algorithm, but it reduces the average active power error by 0.002 pu, the maximum active power error by 0.033 pu, and the maximum active current error by 0.090 pu. For the reactive-current specified mode, the multi-objective algorithm reduces all four error metrics. These results indicate that the multi-objective strategy improves the overall adaptability of the identified solar inverter model to unseen operating conditions.
| Active metrics | Reactive metrics |
|---|---|
| \(I_{1,P}\), \(I_{1,iP}\) | \(I_{1,Q}\), \(I_{1,iQ}\) |
| \(I_{2,P}\), \(I_{2,iP}\) | \(I_{2,Q}\), \(I_{2,iQ}\) |
| Algorithm | \(I_{1,iP}\) | \(I_{1,P}\) | \(I_{2,iP}\) | \(I_{2,P}\) | \(I_{1,iQ}\) | \(I_{1,Q}\) | \(I_{2,iQ}\) | \(I_{2,Q}\) |
|---|---|---|---|---|---|---|---|---|
| DE | 0.011 | 0.002 | 0.162 | 0.064 | 0.127 | 0.038 | 0.259 | 0.089 |
| MODE | 0.012 | 0.003 | 0.204 | 0.075 | 0.217 | 0.035 | 0.292 | 0.091 |
I also investigated the influence of the short-circuit ratio on the LVRT behavior of the solar inverter. The short-circuit ratio is an indicator of the strength of the grid at the point of common coupling. Weak grids with low short-circuit ratio values impose additional stability challenges on solar inverters. I simulated two operating conditions with different short-circuit ratio values and observed the steady-state voltage and the reactive current key points. Under the low short-circuit ratio condition, the voltage at the point of common coupling is lower than the high short-circuit ratio condition, and the reactive current output is also reduced. This finding suggests that the fault current capability of the solar inverter is affected by the network strength. Therefore, the identified parameters should be validated under weak-grid conditions as well. I added two additional test cases with a short-circuit ratio of 1.8 to evaluate the robustness of the proposed identification method.
For the weak-grid validation, I compared the proposed multi-objective differential evolution algorithm with three other multi-objective algorithms: multi-objective particle swarm optimization, non-dominated sorting genetic algorithm II, and multi-objective cuckoo search. Each algorithm was applied to identify the active and reactive LVRT parameters from the training dataset. The identified parameters were then used to simulate the test cases in the electromechanical transient model. The percentage errors between the measured key points and the simulated key points are reported in Table 6 for the testing cases and the weak-grid cases. The average active current error of the multi-objective differential evolution algorithm is 3.78%, which is lower than the multi-objective particle swarm optimization error of 5.66%, the non-dominated sorting genetic algorithm II error of 14.47%, and the multi-objective cuckoo search error of 6.84%. For the reactive current, the average error of the proposed method is 1.16%, which is lower than the other three methods. These numerical results demonstrate that the multi-objective differential evolution algorithm achieves better identification accuracy and robustness for the solar inverter LVRT parameters.
| Case | Variable | APSO | NSGA-II | MOCS | MODE |
|---|---|---|---|---|---|
| 1 | \(i_{Pt}\) | 10.330 | 0.263 | 1.687 | 2.685 |
| \(i_{Qt}\) | 0.261 | 0.184 | 0.312 | 0.204 | |
| 2 | \(i_{Pt}\) | 12.391 | 23.182 | 25.695 | 8.645 |
| \(i_{Qt}\) | 0.556 | 0.891 | 12.545 | 0.234 | |
| 3 | \(i_{Pt}\) | 2.959 | 5.946 | 1.467 | 3.181 |
| \(i_{Qt}\) | 0.781 | 0.689 | 5.378 | 0.561 | |
| 4 | \(i_{Pt}\) | 2.258 | 21.388 | 2.513 | 2.842 |
| \(i_{Qt}\) | 0.246 | 1.452 | 0.478 | 0.032 | |
| 5 | \(i_{Pt}\) | 10.073 | 31.332 | 6.121 | 9.585 |
| \(i_{Qt}\) | 1.885 | 2.197 | 3.442 | 1.920 | |
| 6 | \(i_{Pt}\) | 1.457 | 19.030 | 1.290 | 1.546 |
| \(i_{Qt}\) | 2.487 | 0.871 | 2.879 | 2.279 | |
| 7 | \(i_{Pt}\) | 4.844 | 12.109 | 5.308 | 0.871 |
| \(i_{Qt}\) | 3.021 | 11.781 | 1.976 | 2.593 | |
| 8 | \(i_{Pt}\) | 1.031 | 2.526 | 10.652 | 0.907 |
| \(i_{Qt}\) | 3.275 | 2.776 | 2.814 | 1.504 |
The detailed identification process for the active current parameters under the specified current control mode can be summarized as follows. The population vector \(T\) contains all candidate solutions:
$$ T = \begin{bmatrix} \left[k_{1,iP}^{(1)}, k_{2,iP}^{(1)}, i_{P,set}^{(1)}\right] \\ \vdots \\ \left[k_{1,iP}^{(G)}, k_{2,iP}^{(G)}, i_{P,set}^{(G)}\right] \end{bmatrix}. \tag{28} $$
The mutation vector for the active current parameters is
$$ d = \left[k_{1,iP}^{(b)} – k_{1,iP}^{(c)},\; k_{2,iP}^{(b)} – k_{2,iP}^{(c)},\; i_{P,set}^{(b)} – i_{P,set}^{(c)}\right], \tag{29} $$
and the donor vector is updated as
$$ T_a^{(1)} = \left[k_{1,iP}^{(a)},\; k_{2,iP}^{(a)},\; i_{P,set}^{(a)}\right] + p_m d. \tag{30} $$
The trial vector after crossover is constrained by
$$ \begin{cases} k_{1,iP}^{(a,1)} = \begin{cases} \min k_{1,iP}, & k_{1,iP}^{(a,1)} < \min k_{1,iP}, \\ k_{1,iP}^{(a,1)}, & \min k_{1,iP} \le k_{1,iP}^{(a,1)} \le \max k_{1,iP}, \\ \max k_{1,iP}, & k_{1,iP}^{(a,1)} > \max k_{1,iP}, \end{cases} \\[6pt] k_{2,iP}^{(a,1)} = \begin{cases} \min k_{2,iP}, & k_{2,iP}^{(a,1)} < \min k_{2,iP}, \\ k_{2,iP}^{(a,1)}, & \min k_{2,iP} \le k_{2,iP}^{(a,1)} \le \max k_{2,iP}, \\ \max k_{2,iP}, & k_{2,iP}^{(a,1)} > \max k_{2,iP}, \end{cases} \\[6pt] i_{P,set}^{(a,1)} = \begin{cases} \min i_{P,set}, & i_{P,set}^{(a,1)} < \min i_{P,set}, \\ i_{P,set}^{(a,1)}, & \min i_{P,set} \le i_{P,set}^{(a,1)} \le \max i_{P,set}, \\ \max i_{P,set}, & i_{P,set}^{(a,1)} > \max i_{P,set}. \end{cases} \end{cases} \tag{31} $$
The objective functions for the trial vector and the parent vector are computed as
$$ J(T_a^{(1)}) = \max\left(-\sum_{g=1}^{m} \left| i_{P,cmd,g} – \left(k_{1,iP}^{(a,1)} u_{t,g} + k_{2,iP}^{(a,1)} i_{P0,g} + i_{P,set}^{(a,1)}\right)\right|\right), \tag{32} $$
$$ J(T_a) = \max\left(-\sum_{g=1}^{m} \left| i_{P,cmd,g} – \left(k_{1,iP}^{(a)} u_{t,g} + k_{2,iP}^{(a)} i_{P0,g} + i_{P,set}^{(a)}\right)\right|\right). \tag{33} $$
The selection operation keeps the individual with the better fitness. The population diversity is computed from the entropy over the sorted parameter values. The adaptive parameters are then updated according to Equations (20) and (21). The non-dominated sorting procedure takes the merged population and sorts it into fronts. Then the crowding distance of every solution in the active power and active current objectives is calculated as demonstrated earlier. After all generations, the optimal solution is selected using the fuzzy satisfaction criterion. The entire workflow of the proposed multi-objective differential evolution algorithm can be described as:
1. Initialize the population \(T\) of size \(G\) using random values within the feasible ranges.
2. Evaluate both objective functions \(J_1\) and \(J_2\) for every individual.
3. Apply the adaptive mutation and crossover operations to generate the offspring population \(T^{(1)}\).
4. Merge \(T\) and \(T^{(1)}\) to form a combined population of size \(2G\).
5. Perform the fast non-dominated sorting and compute the crowding distances.
6. Select the best \(G\) individuals to form the next generation.
7. Repeat steps 3 to 6 until the stopping criterion is satisfied.
8. Deploy the fuzzy satisfaction criterion to choose a single optimal solution from the first Pareto front.
The choice of the initial population size \(G\) is important. In my implementation, I set \(G = 50\) for all identification tests. The maximum mutation probability and the minimum mutation probability are set to 0.9 and 0.4, respectively, while the maximum crossover probability and the minimum crossover probability are set to 0.8 and 0.2, respectively. These values are adjusted adaptively according to the entropy of the population. Such an adaptive mechanism not only accelerates the convergence but also prevents premature stagnation in the solar inverter parameter search.
One of the key contributions of this study is the simultaneous identification of the control mode and the control parameters of the solar inverter. The control mode identification is performed by evaluating the accuracy of the identified model under two candidate control structures. Table 2 and Table 3 show that the specified current control mode is the dominant structure for the reactive current control in the tested solar inverter, while the active current control may switch between specified power and specified current depending on the pre-fault operating condition and the fault severity. The proposed criterion based on the average and maximum deviations is easy to implement in practice and does not require any prior assumptions about the manufacturer’s control logic.
Another important aspect is the influence of the short-circuit ratio on the LVRT response. In weak-grid conditions, the voltage at the point of common coupling is more sensitive to the reactive current injection of the solar inverter. A low short-circuit ratio implies that the grid impedance is relatively high, which leads to a larger voltage drop during faults and also limits the effective injection capability of the inverter. I simulated two additional weak-grid test cases with a short-circuit ratio of 1.8. The identification results in these cases show that the proposed method maintains satisfactory accuracy, with the active current error remaining below 1% and the reactive current error below 3%. This robustness is attributed to the multi-objective optimization, which balances the fitting accuracy across different operating conditions.
In the comparison with other multi-objective evolutionary algorithms, the proposed multi-objective differential evolution algorithm shows consistent advantages. The non-dominated sorting genetic algorithm II tends to converge prematurely for some cases, leading to large errors, especially in the active current estimation. The multi-objective particle swarm optimization and the multi-objective cuckoo search algorithm also suffer from occasional stagnation or unbalanced search behavior. The differential evolution mechanism inherently uses the differences between individuals to guide the search, which is highly effective for continuous parameter spaces. The addition of non-dominated sorting and crowding distance further enhances the diversity of the final solution set. As a result, the proposed algorithm can identify the LVRT parameters of the solar inverter with high precision under a wide range of fault scenarios.
Moreover, I introduced a step-by-step detailed derivation of the parameter identification process in the appendix to illustrate the implementation. The equations for the mutation, crossover, boundary checking, objective evaluation, entropy calculation, and non-dominated sorting are provided in a self-contained manner. This transparency is helpful for practitioners who wish to reproduce the proposed method on their own solar inverter models.
In conclusion, this study presents a robust and accurate method for identifying the LVRT control mode and control parameters of a solar inverter electromechanical transient model. The main findings can be summarized as follows:
1. The proposed multi-objective differential evolution algorithm can accurately reproduce the LVRT current and power responses of the solar inverter. Compared with the standard differential evolution algorithm, the multi-objective strategy reduces the maximum identification error significantly and improves the generalization ability of the model.
2. The adaptive parameter tuning mechanism based on population entropy effectively balances exploration and exploitation during the optimization process. It improves the convergence speed and reduces the risk of falling into local optima.
3. The proposed control mode identification criterion, which considers both the average deviation and the maximum deviation of active power, reactive power, active current, and reactive current, is effective for determining the actual LVRT control structure of the solar inverter.
4. In comparison with multi-objective particle swarm optimization, non-dominated sorting genetic algorithm II, and multi-objective cuckoo search, the proposed algorithm achieves the lowest average errors for both the active current and the reactive current, with the active current error below 3.8% and the reactive current error below 1.2%.
The identified solar inverter model can be directly used in power system transient stability analysis, protection coordination studies, and grid code compliance assessment. Future work will extend the proposed method to other power electronic devices such as wind power converters and battery energy storage systems. In addition, the effects of the phase-locked loop dynamics during severe faults will be included in the identification problem to further improve the accuracy for transient studies. The method can also be combined with physics-informed neural networks to accelerate the optimization process without sacrificing accuracy.
Overall, the multi-objective differential evolution algorithm proves to be a powerful tool for parameter identification of solar inverter LVRT models. The proposed framework is practical and easy to integrate into existing model validation workflows. It helps engineers obtain credible solar inverter models without relying on confidential manufacturer information, thereby facilitating the stable integration of solar generation into modern power systems.