Abstract
An nonlinear model predictive controller (NMPC) is proposed in this paper for compensations of single line-to-ground (SLG) faults in resonant grounded power distribution networks (RGPDNs), which reduces the likelihood of power line bushfire due to electric faults. Residual current compensation (RCC) inverters with arc suppression coils (ASCs) in RGPDNs are controlled using the proposed NMPC to provide appropriate compensations during SLG faults. The proposed NMPC is incorporated with the estimation of ASC inductance, where the estimation is carried out based on voltage and current measurements from the neutral point of the distribution network. The compensation scheme is developed in the discrete time using the equivalent circuit of RGPDNs. The proposed NMPC for RCC inverters ensures that the desired current is injected into the neutral point during SLG faults, which is verified through both simulations and control hardware-in-the-loop (CHIL) validations. Comparative results are also presented against an integral sliding mode controller (ISMC) by demonstrating the capability of power line bushfire mitigation.
FIRE risks in distribution networks are becoming a crucial challenge to system operators due to increases in power line bushfires around the world, especially in Australia and the USA [
The switching signals of the RCC inverter are regulated by the control schemes similar to traditional inverter controllers. Two preliminary methods based on switched parallel arc suppression coils (ASCs) or switched parallel resistors are presented in [
Controllers with dual-loop structures overcome the limitations of single-loop controllers [
The linearized models of RGPDNs are used in [
A recently designed nonlinear BSC (NBSC) in [
Based on the facts arising from existing literature on the control of RCC inverters in the RGPDN, the key limitations of these methods can be summarised as follows.
1) The model-free and model-based linear controllers cannot compensate SLG faults in RGPDNs to a level that can mitigate power line bushfires.
2) The model-free and model-based linear controllers do not consider high-impedance faults.
3) The nonlinear controllers significantly reduce the fault current to a safe level though they require complicated SS and exact parameters for BSCs.
This paper proposes an NMPC for the RCC inverter in an RGPDN to compensate the faulty phase voltage and fault current under the SLG faults. Here, the value of the adjustable inductor is estimated using the voltage and current measurements from the neutral point of the distribution network. The optimization problem for the proposed NMPC is formulated by incorporating this estimated parameter, which is then solved to achieve the optimal control performance, i.e., the desired level of fault compensation. Therefore, the proposed NMPC will overcome the limitations (especially the bounded uncertainty and non-optimality) of existing nonlinear SMCs. Another important feature of the proposed NMPC is that it determines the optimal control parameters without increasing undesired components in the control signal. The key novel aspects of the proposed NMPC against the previously developed controller in [
1) The value of the adjustable inductor is estimated using the actual voltage and current measurements from the neutral point of the distribution network. In contrast, the existing control techniques utilize the nominal value of the adjustable inductor, which can adversely impact the control performance in practice bacause the value of an adjustable inductor can change due to environmental factors, which would require constant re-tuning of the inductor.
2) The values of leakage parameters (such as the resistance and capacitance) also change during the operation, and it is necessary to constantly monitor these changes, which is an arduous task due to the large size of the distribution network. Similarly, this would require constant re-tuning of the adjustable inductor to achieve the resonance condition.
3) The optimization problem for the proposed NMPC is solved based on the estimated inductor, which ensures higher accuracy and faster convergence of the controlled variable, i.e., the injected current into the neutral point in this case, to its reference value.
The performance of the proposed NMPC is evaluated through both simulation and control hardware-in-the-loop (CHIL) platforms under both low- and high-impedence faults in an RGPDN. The results are benchmarked against the performance criteria in [
(1) |

Fig. 1 A test RGPDN for dynamic modeling of REFCL.
Only the reactive component of the fault current can be compensated through . And when , there is a perfect resonance, which means the reactive component of will be fully compensated. However, in practice, there also exists a significant active component of , which can be enough to keep the resultant fault characteristics and cause bushfires. The RCC inverter driven by a set of switching signals (-) regulates the injected current into the neutral point to compensate and assists to keep the faulty phase voltage to a safe level. The expression of under an SLG fault can be expressed as:
(2) |
where () is the total current flowing through shunt branch on phase X, which can be expressed as:
(3) |
where is the phase voltage.
Hence, by substituting (3) into (2), we can obtain:
(4) |
As shown in
The phase-to-neutral voltage represents the difference between the phase voltage and the neutral voltage , i.e.,
(5) |
(6) |
It is well-known that for a balanced system, based on which the sum of all phase voltages can be written as:
(7) |
Hence, (4) can be simplified as:
(8) |
At this stage, it is required to find the reference value of , which is denoted as . Taking the SLG fault on phase as an example, can be expressed in terms of the faulty phase voltage according to (5), i.e., . Hence, (8) can be written as:
(9) |
The last term on the right side of (9) corresponds to the effect of the SLG fault on phase [
(10) |
Please note that , which is an instantaneous signal, has the sinusoidal behavior, and thus . It can be observed from (10) that the value of does not depend on the fault impedance. If is set to be through the control action, the faulty phase voltage can be quickly controlled to be zero, and thus setting the fault current to be zero as per in (1). The injected current into the neutral point by the RCC inverter needs to be controlled by controlling the changes in , i.e., , which can be obtained by applying Kirchhoff’s laws to the neutral point of the distribution network shown in
(11) |
In this paper, the modulation index is the modulating signal or control input for the RCC inverter, which is designed using an NMPC.
For an RCC inverter, its modulation index m is its output voltage divided by its input voltage. For a single switching instant in an RCC inverter, the input voltage is . Considering the output voltage as (or simply ), can be represented as [
(12) |
Taking as the controlled variable for , the value of can be rewritten as (13) by substituting (12) into (11).
(13) |
By replacing with and with , (13) can be generalized as:
(14) |
(15) |
where the subscript 0 means the initial value.
The discrete-time system can be obtained by considering the sampling instant as k (), where N is the horizon, and the sampling period can be obtained with a regular sampling time . In order to make a distinction between the continuous- and discrete-time systems, in the continuous-time system is represented by in the discrete-time system. For this reason, is the original control signal to drive the RCC inverter. Therefore, the equivalent discrete-time model of the RCC inverter in an RGPDN can be written as:
(16) |
If and are constants over a period of , (16) can be simplified as [
(17) |
From (17), it is clear that the dynamic injected current depends on the inductor . Therefore, by obtaining the value of from the measurements of and , the proper control of the injected current can be ensured.
Initially, the parameter is completely unknown. Letting , the discrete-time model in (17) can be written as:
(18) |
From
(19) |
where is the complex operator, and is the angular frequency. Denote as the estimated value of , which can be derived from (19) as:
(20) |
This estimated value of is used for the proposed NMPC rather than its nominal value.
As indicated earlier, the design of the proposed NMPC involves the formulation of the optimization problem and the solution of this problem. A cost function is defined initially for a particular prediction horizon , and this cost function should include the information related to the state (), control input (), and estimated value of the unknown parameter (). The main control objective is to make sure that the injected current accurately tracks its reference and therefore, the cost function is designed to penalize the error between the state and its reference . With all these considerations, the associated objective or cost function can be defined as:
(21) |
where corresponds to the reference current ; denotes the instant of the current state; and is a weighting parameter considering the influence of the control input on the cost function , which makes sure that the control input is effective in zeroing the error between and .
The performance index for a prediction horizon of can be computed as [
(22) |
where gives the predicted trajectory of the state .
(23) |
(24) |
where .
Using the value of obtained from (21), (24) can be rewritten as:
(25) |
Furthermore, from (22), the following equations are valid:
(26) |
Using (26), (25) can be written as:
(27) |
The performance index in (27) is minimized only with respect to because does not have any effects on the trajectories of and , and the concept of receding horizon (i.e., using the first horizon as the control signal) is used in this paper. Hence, the optimal condition for can be expressed by equating the partial derivative of with respect to to zero, i.e., , which can be calculated from (27) as:
(28) |
(29) |
From (29), the optimal value of , i.e., , can be determined as:
(30) |
And the optimal control law of the proposed NMPC can be represented as:
(31) |
where represents the optimal control input when . And then, calculated using (20) is fed to (31) to optimally compensate the fault characteristics.
The overall flowchart of the proposed NMPC is shown in

Fig. 2 Overall flowchart of proposed NMPC.

Fig. 3 Block diagram for implementation of proposed NMPC.
The performance analysis of the proposed NMPC is presented in the next section. An ISMC is utilized to compare with the proposed NMPC, and its brief overview is presented in the Supplementary Material.
In this work, the key control objective of the proposed NMPC is to inject the desired current to the neutral point using the RCC inverter in an RGPDN. Subsequently, this current assures that the faulty phase voltage and fault current are compensated to their desired levels for power line bushfire mitigation as given in [
The performance of the proposed NMPC is analyzed based on the RGPDN shown in
The RGPDN is basically a 22 kV (line-to-line) network with an RCC inverter, which is mounted on the neutral point through transformer to step-up the voltage. The values of and are taken as 28 k and F, respectively. It is worth mentioning that these values are selected based on the current industry standard for RGPDN in bushfire prone areas. The RGPDN in
The control parameter for the proposed NMPC is selected as to ensure that both faulty phase voltage and fault current are adequately compensated. This value was found through trial-and-error of the simulations although artificial intelligence (AI) based techniques can be incorporated to fine-tune these values and improve the current and voltage responses obtained from the simulations, which is beyond the scope of this paper. The control parameters and for the ISMC are selected in a similar way, which are 700 and 500, respectively. The performance of the proposed NMPC is analyzed by considering these control parameters and it is not highly sensitive to the variations in these control parameters easing the tedious parameter selection process.
Most of the existing literature does not compensate both current and voltage, and even if they are both considered, there are no indications about the timeframe and the magnitudes of current and voltage required to extinguish power line bushfires due to SLG faults on distribution networks in bushfire prone zones. The regulatory impact statement in [
To analyze the performance of the proposed NMPC, the RGPDN is simulated under two SLG faults with and k to cover both low- and high-impedance faults, i.e., a wide range of fault conditions. The performance of the proposed NMPC via both simulation and CHIL validation is analyzed, and the compensation is initiated at the same instant. All the simulation results are assessed against the performance criteria in [
The simulation is first conducted under an SLG fault on phase at s with .

Fig. 4 Instantaneous and RMS values of fault current with proposed NMPC and ISMC with . (a) Instantaneous values. (b) RMS values.

Fig. 5 Instantaneous and RMS values of faulty phase voltage with proposed NMPC and ISMC with . (a) Instantaneous values. (b) RMS values.
Time instant (s) | RMS value of faulty phase voltage (V) | |
---|---|---|
Proposed NMPC | ISMC | |
0.485 | 18.89 | 66.25 |
0.900 | 19.01 | 67.39 |
2.400 | 19.06 | 66.55 |
The estimated value of with is given in

Fig. 6 Estimated value of with .
The performance of the proposed NMPC is further analyzed by observing the instantaneous and RMS values of current injected into neutral point with , as shown in

Fig. 7 Instantaneous and RMS values of current injected into neutral point with proposed NMPC and ISMC with . (a) Instantaneous values. (b) RMS values.
Both of the proposed NMPC and ISMC ensure the proper reference current tracking though there is a small difference between them. The proper tracking is important for compensating the faulty phase voltage because the current injected into neutral point makes the neutral voltage increase up to a magnitude equal to the faulty phase voltage, i.e., , but in the opposite direction. In this way, the faulty phase voltage needs to reduce to a lower value, which in turn eliminates power line bushfire risks. It is calculated that the value of is 47.9 A while the RMS values of with the proposed NMPC and ISMC are 47.76 A and 47.67 A, respectively. Hence, the accuracy of the proposed NMPC is slightly higher than that of the ISMC, which further supports the findings as discussed earlier.
Then, the simulation is conducted under an SLG fault on phase at s with k. As can be observed from

Fig. 8 Instantaneous and RMS values of fault current with proposed NMPC and ISMC with k. (a) Instantaneous values. (b) RMS values.
The instantaneous and RMS values of faulty phase voltage in

Fig. 9 Instantaneous and RMS values of faulty phase voltage with proposed NMPC and ISMC with k. (a) Instantaneous values. (b) RMS values.
The estimated value of with k is given in

Fig. 10 Estimated value of with k.
From the instantaneous and RMS values of the current injected into neutral point in

Fig. 11 Instantaneous and RMS values of current injected into neutral point with proposed NMPC and ISMC with k. (a) Instantaneous values. (b) RMS values.
Figures

Fig. 12 Instantaneous and RMS values for output current of RCC inverter with . (a) Instantaneous values. (b) RMS values.

Fig. 13 Instantaneous and RMS values for output current of RCC inverter with k. (a) Instantaneous values. (b) RMS values.

Fig. 14 Active and reactive components of output current of RCC inverter with . (a) Active component. (b) Reactive component.

Fig. 15 Active and reactive components of output current of RCC inverter with k. (a) Active component. (b) Reactive component.
Control method | Fault impedance (Ω) | Compensated fault current (A) | Compensated faulty phase voltage (V) |
---|---|---|---|
Closed-loop | 50-2000 | 0.067-0.026 | 3.4-50.2 |
PI | 50 | 7 | |
PI+advanced topology | 2000 | 7 (single-phase), 1 (three-phase) | |
PI+advanced modulation | 10-100 | 4.14-4.67 | |
PR+PI | Load variation | 21 | |
PI+PI | 25 | 0.006 | 0.16 |
Lag and PI | 10-100 | 0.16-0.32 | |
PI+PR and P | 100-10000 | 5-15 | |
Multiple PI and hysteresis | 65 | 0.6 | |
H-infinity | 0.2 | ||
FC-MPC | 10-1000 | ||
BSC | 1-100 | ||
BSC-SOGI-PLL | 1-1000 | 0.0026-0.0600 | |
NBSC | 350-16000 | 0.0023-0.1286 | 43-45 |
IBSC | 50-26000 | 0.003-0.161 | 23-50 |
ISMC | 100-26000 | 0.00658-0.56000 | 56-171 |
A-SMC | 120-25400 | 0.0025-0.1510 | 19-44 |
NT-SMC | 250-26000 | 0.0021-0.4200 | 50-95 |
NFT-SMC | 400-25400 | 0.0023-0.0920 | 37-46 |
PI-NFT-SMC | 300-25400 | 0.0019-0.1130 | 33.3-36.4 |
GT-SMC | 80-26000 | 0.00044-0.01000 | 10.4-11.54 |
NMPC | 400-26000 | 0.0016-0.0780 | 31.36-42.54 |
Note: BSC-SOGI-PLL is short for BSC with SOGI-PLL; FC-MPC is short for MPC with finite control; and NT-SMC is short for nonsingular terminal SMC.
CHIL validations are carried out with the real-time simulator OPAL-RT OP57075XG to validate the feasibility of the proposed NMPC. CHIL validations has an advantage over the scaled-down experimental setups because the CHIL validations can be performed in real time with the actual distribution network voltages. This provides a clear insight into the performance of the proposed NMPC and how effective it is in meeting the bushfire mitigation guidelines in [

Fig. 16 Working process of CHIL validation.

Fig. 17 Experimental setup with OPAL-RT simulator.
CHIL validation results are provided for the proposed NMPC under both the low- and high-impedance faults.

Fig. 18 Instantaneous and RMS values of fault current with for CHIL validation. (a) Instantaneous values. (b) RMS values.

Fig. 19 Instantaneous and RMS values of faulty phase voltage with for CHIL validation. (a) Instantaneous values. (b) RMS values.

Fig. 20 Estimated value of with for CHIL validation.
Figures

Fig. 21 Instantaneous and RMS values of fault current with for CHIL validation. (a) Instantaneous values. (b) RMS values.

Fig. 22 Instantaneous and RMS values of faulty phase voltage with for CHIL validation. (a) Instantaneous values. (b) RMS values.

Fig. 23 Estimated value of with for CHIL validation.
An NMPC is designed to compensate the SLG faults in RGPDNs. The proposed NMPC ensures the proper current injected into the neutral point by using the RCC inverter in an optimal way. The proposed NMPC has the following characteristics compared with other existing controllers.
1) The dynamic change in the adjustable inductor parameter is continuously monitored through the voltage and current measurements from the neutral points.
2) The fault compensation capability follows the timeframe as stated in the regulatory impact statement for the power line bushfire mitigation.
3) There is no requirement for a confined SS to ensure the robustness against parametric uncertainties.
The proposed NMPC utilizes the voltage and current measurements from the neutral point of the distribution network to estimate the value of the adjustable inductor, which is found to be slightly different from its nominal value and assists to improve the accuracy of injected current. This improved accuracy further helps compensate the faulty phase voltage and fault current. The rigorous analysis of results obtained from simulation and CHIL validations clearly highlights that the proposed NMPC is suitable to ensure the desired levels of fault compensation and it is practically applicable. Further research will also focus on automating the gain selection process using the AI based techniques.
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