An Electric Spring Control Strategy Based on Finite Control Set-Model Predictive Control

The development of renewable energy sources (RES) has accelerated the reform of the energy structure. However, the intermittency and unsustainability of RES have brought various power quality problems to the power supply and distribution network [1, 2]. Proposed by Hui et al. [3], electric spring (ES) provides a new solution to such problems. Unlike the traditional circuit compensation device, the ES operates in a distributed manner on the load side, making the power consumption more reliable on the load side [4]. Compared with the static var compensator (SVC), the ES boasts small capacity and high reliability [5]. Different from the traditional measures to alleviate RES supply fluctuations (e.g. controlling load switches), the ES improves the comfort and happiness of power users, without hindering the normal power consumption on the load side [6].

The original ES structure can only realize reactive power compensation: the device is connected in series with the non-critical loads (NCLs), which are insensitive to voltage changes, forming smart loads (SLs); then, the ES transfers the voltage fluctuations on the bus to the NCLs, thereby stabilizing the bus voltage. In this way, the voltage of the critical loads (CLs) on the bus, which are sensitive to voltage changes, will also be stabilized. Through years of research, the ES has been proved to be feasible in frequency stabilization, power factor correction, and filtering [7, 8].

Over the years, the ES control has become increasingly robust, nonlinear, and intelligent. The control devices have evolved from the proportional-integral (PI) controller suitable for ES-1 topology [9, 10], to quasi-proportional resonant (PR) controller for ES-2 topology [11], and to intelligent controllers like fuzzy controller [12]. The increase of nonlinear loads, coupled with various circuit interferences, raise stricter requirements on the normal operation of the ES. In alternative current (AC) ES control system, the PI controller cannot achieve error-free tracking of the AC reference signal, and the parameter debugging of the traditional linear controller is time-consuming and laborious. To solve the problem, more and more nonlinear control theories have been proposed and applied to inverter systems, such as hysteresis control, sliding mode control, repetitive control, and adaptive control [13].

Model predictive control (MPC) is an algorithm that uses the system model to predict the controlled quantity and approximate it to the reference quantity [14]. The algorithm supports fast dynamic response and transient response, and adapts well to nonlinear constraints. However, the MPC theory was not widely accepted at the beginning, because the control system needs a high-performance hardware platform to perform the numerous calculations in each sampling period. Thanks to the development of computer technology, the MPC has become a research hotspot in the control of power electronic converters [15, 16]. For instance, the finite control set-model predictive control (FCS-MPC) [17, 18] can achieve flexible and multivariable control, eliminating the need for pulse width modulation (PWM) of the output. It is suitable for nonlinear systems with constraints. Previous studies [19, 20] have confirmed the feasibility of the MPC in the control of single-phase and three-phase grid-connected inverters. Unlike other control methods, the MPC takes the predicted circuit quantities in future as the reference for the current actions of the system.

This paper introduces the FCS-MPC into ES control system, with the aim to improve the anti-interference ability of the ES and to optimize the dynamic and static ES performance under complex load environment. Firstly, a load-side circuit prediction model was established and analyzed. On this basis, the prediction formula of ES output voltage and the calculation formula of ES reference formula were created, as well as the performance indicator function. Finally, the proposed control strategy was proved feasible and effective through MATLAB simulations and dSPACE physical experiments, and the simulation and experimental results were given in the presence of interference in the circuit.

This paper constructs a control system based on the ES system model (Figure 3). The principle of the control system goes as follows: the CL reference voltage U_ref is generated based on the CL voltage U_cl collected by the voltage sensor, and imported to the MPC to output a control signal for switch tubes S₁~S₄ on the inverter bridge; through the control, the output voltage of the ES U_C approaches the reference voltage U_Cref, that is, the error between the two voltages is minimized.

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Figure 3. The block diagram of our control system

The performance indicator function, which is often adopted in prediction models, measures the optimization trend of system indices. In MPC rolling optimization, the function is the key player that decides the controller output, and the enabler of the MPC to solve multivariable constrained problems. To simplify the control structure and reduce the calculation amount of the control system, the square of the difference g_min between U_Cref and the predicted value was selected as the performance indicator function:

${{\text{g}}_{min}}\text{=}{{\left( {{U}_{Cref}}\left( k+1 \right)-{{\overset{\wedge }{\mathop{U}}\,}_{C}}\left( k+1 \right) \right)}^{2}}$   (1)

The calculation of g_min depends heavily on three parameters: the CL reference voltage U_ref, the predicted ES output voltage $\widehat{U}_{c}(k+1)$ , and the reference value of the output voltage U_Cref. Therefore, the calculation process of the three parameters is detailed below.

3.1 Generation of U_ref

The U_ref was generated by the method shown in Figure 4, for the stability of the amplitude and phase of CL voltage.

When ES is not needed for circuit regulation, the Swi signal is set to low level, and the Tri signal (Figure 2) to high level. Then, switch Q is closed, the voltage sensor collects U_cl, and the phase of U_cl is obtained through the phase-locked loop. Next, channel 1 in the switch is selected for signal transmission, and the phase is passed through the sin module and booster m to generate the CL reference voltage U_ref. But U_ref has not effect at this time.

When ES is needed for circuit regulation, the Swi signal is set to high level, channel 2 in the switch is selected for signal transmission, and the Tri signal is set to low level. Then, switch Q is opened, and ES is connected in parallel to the bus. The phase signal of U_ref will change periodically after passing the time delay module. The delay is an integer number of power frequency cycles.

The U_ref generation method in Figure 4 keeps U_ref unchanged regardless of circuit conditions, eliminates the interference of external factors on CL, and minimizes the dependence on external circuit parameters, thereby improving the robustness of our control system.

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Figure 4. The structure of U_ref generator

3.2 Prediction of $\widehat{U}_{C}(k+1)$

To analyze the load-side model, Figure 2 was further simplified in the ES working state into Figure 5, where i₁ is the current input to a single load unit on the bus, i₂ is the CL current, U_cl is the CL voltage, i₃ is the SL current, i_L and i_c are respectively the inductance and capacitance currents of the filter, U_c is the ES output voltage, U_ncl is the NCL voltage, uV_in is the inverter bridge output voltage, with u being the switching function. If u=1, S₁ and S₄ (Figure 2) are closed; if u=0, S₁ and S₂ are closed, or S₃ and S₄ are closed; if u=-1, S₂ and S₃ are closed.

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Figure 5. The simplified structure of the load side during ES operation

To reduce the influence of the sensor in the circuit on the predicted value, this paper only collects the inductor L current and the capacitor C voltage to predict the ES output voltage:

$L\frac{d{{i}_{L}}}{dt}=u{{V}_{in}}-{{U}_{C}}$   (2)

$C\frac{d{{U}_{C}}}{dt}={{i}_{L}}-{{i}_{3}}$   (3)

Let T_s be the discrete sampling period of the ES control system. From formula (2) and the forward difference formula [24], the current of the inductor L at time k can be predicted by:

${{\overset{\wedge }{\mathop{i}}\,}_{L}}(k)={{i}_{L}}(k-1)+\frac{Ts}{L}(u{{V}_{in}}(k)-{{U}_{C}}(k))$   (4)

From formula (3) and backward difference formula [25], the ES output voltage at time k+1 can be predicted by:

${{\overset{\wedge }{\mathop{U}}\,}_{C}}\left( k+1 \right)={{U}_{C}}\left( k \right)+\frac{Ts}{C}\left( {{\overset{\wedge }{\mathop{i}}\,}_{_{L}}}\left( k \right)-{{\overset{\wedge }{\mathop{i}}\,}_{3}}\left( k \right) \right)$      (5)

For convenience, it is assumed that i₃ does not change in a sampling period T_s:

${{\overset{\wedge }{\mathop{i}}\,}_{3}}(k)\approx \overset{\wedge }{\mathop{{{i}_{3}}}}\,(k-1)$     (6)

From formula (3), formula (6), and forward difference formula [24], we have:

$\overset{\wedge }{\mathop{{{i}_{3}}}}\,(k-1)\text{=}{{i}_{L}}(k-1)-\frac{C}{Ts}\left( {{U}_{C}}\left( k \right)-{{U}_{C}}\left( k-1 \right) \right)$    (7)

Substituting formulas (4), (6), and (7) into formula (5), we have:

${{\overset{\wedge }{\mathop{U}}\,}_{C}}\left( k+1 \right)\text{=}\frac{T{{s}^{2}}}{LC}u{{V}_{in}}\left( k \right)+\left( 2-\frac{T{{s}^{2}}}{LC} \right){{U}_{C}}\left( k \right)-{{U}_{C}}\left( k-1 \right)$     (8)

3.3 Calculation of U_Cref

To eliminate the interference of bus current $i_{1}$  on the system, the output voltage control was adopted, with the controlled quantity being volage U_c. When the supply side fluctuates, the bus current changes accordingly. To stabilize U_cl, the amplitude and phase of ES output voltage will both change. The controlled quantity can be derived from the circuit in Figure 5:

${{U}_{C}}\text{=}{{U}_{cl}}+{{U}_{ncl}}$    (9)

Substituting current i₁, and load parameters Z_ncl and Z_cl into formula (9), the reference value U_Cref of ES output voltage U_C at time k+1 can be expressed as:

${{U}_{Cref}}\left( k+1 \right)=\left( 1+\frac{{{Z}_{ncl}}}{{{Z}_{cl}}} \right){{U}_{ref}}\left( k+1 \right)-{{Z}_{ncl}}{{i}_{1}}\left( k+1 \right)$    (10)

where, U_ref is the reference voltage of CL, whose effective value remains constant; i₁ is the current flowing from the bus into the load side, which changes in real time under the effects of changes on both supply side and load side. U_Cref also changes in real time.

Before calculating U_Cref by formula (10), it is necessary to obtain the values of i₁ and U_ref at time k+1. Since the change of U_ref is periodic, and i₁ is collected from the circuit, it is assumed that i₁ does not change between two adjacent sampling periods:

${{i}_{1}}\left( k+1 \right)\approx {{i}_{1}}\left( k \right)$    (11)

The i₁ value at time k was measured by installing a current sensor on the outside of the load unit. The calculation method of i₁ in formula (10) enhances the anti-interference ability of the control system, and optimizes the dynamic performance of the system under supply-side fluctuations.

5.1 Simulation and result analysis

To verify the proposed FCS-MPC control system for the ES, a simulation model was established on MATLAB/Simulink to simulate the control effect of the system. The simulation circuit was built in accordance with Figure 2. The circuit parameters and controller parameters are listed in Table 2. Considering the effects of the sampling period on the quality of the controlled quantity in discrete sampling, the simulation was carried out under different sampling periods: 2e-5s, 1e-5s, and 1e-6s. The U_cl values under these periods are compared in Figure 9.

Table 2. The hardware parameters of the simulation circuit

Parameter	Value
Transmission line impedance Zt	0.6Ω+2.86mH
CL Zcl	40Ω
NCL Zncl	4Ω
Filter inductance L	3.6mH
Filter capacitance C	100μF
Battery voltage Vin	360V

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Figure 9. The comparison of U_cl under different sampling periods

As shown in Figure 9, the shorter the sampling period, the smaller the fluctuation of the controlled quantity, and the closer the waveform is to the ideal sine. Because the MPC has no modulation link, the trigger pulse frequency it generates depends entirely on the discrete sampling period of the system. The shorter the sampling period, the higher the accuracy of the prediction result, and the faster the CL will enter the normal working environment in terms of electrical energy. Repeated simulations show that the proposed control system achieved fast calculation speeds under all three sampling periods, and experienced virtually no delay. As a result, the minimum sampling period of 1e-6s was adopted for subsequent simulation.

5.1.1 Simulation at 220V

The next step is to verify whether the ES in the FCS-MPC control system could stabilize the CL voltage quickly and stably. Hence, the amplitude and effective value of U_cl were observed under the drop and rise of supply-side voltage, respectively. After debugging in MATLAB simulation, it is learned that the load-side load voltage is 220V, when the supply-side voltage U_g is 262V, under the circuit parameters in Table 2. Then, the supply-side voltage U_g was decreased and increased by 10% to 235.8V and 288.2V, respectively, and maintained for 0.1s. The simulation time was set to 0.4s. The time variations of U_g and Tri signal are recorded in Table 3.

As shown in Table 3, the supply-side voltage remained normal in 0~0.2s, and resumed the normal state in 0.3~0.4s after the voltage dropped or rose in 0.2~0.3s. Figures 10 and 11 present the variations in U_cl amplitude during voltage drop and voltage rise, respectively. It can be seen that, when supply-side voltage fluctuated, the U_cl amplitude in the circuit with ES remained basically unchanged in 0~0.5s, ensuring the stable and good power environment at both ends of Z_cl; the U_cl amplitude in the circuit without ES fluctuated significantly in 0.2~0.3s.

Table 3. The time variations of U_g and Tri signal

Time/s	Ug/V	Tri
0~0.1	262	1
0.1~0.2	262	0
0.2~0.3	235.8 (drop)/288.2 (rise)	0
0.3~0.4	262	0

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Figure 10. The U_cl amplitude at the drop of supply-side voltage

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Figure 11. The U_cl amplitude at the rise of supply-side voltage

Figures 12 and 13 display the variations in the effective value of U_cl during voltage drop and voltage rise, respectively. It can be observed that the effective value of U_cl in the circuit with ES fluctuated slightly after 0.1s, and then stabilized quickly at the reference value, owing to the sudden switching of the ES. When the supply-side voltage fluctuated, the U_cl always remained stably at about 220V, regardless of the voltage drop or rise in the circuit with ES, providing a stable and reliable power environment for Z_cl. By contrast, the effective value of U_cl in the circuit without ES fluctuated with the supply-side voltage. Overall, the ES under FCS-MPC performs excellently in stabilizing U_cl voltage during voltage rise and voltage drop, which demonstrates the feasibility of the FCS-MPC control system and its strong adaptability to voltage fluctuations.

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Figure 12. The effective value of U_cl at the drop of supply-side voltage

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Figure 13. The effective value of U_cl at the rise of supply-side voltage

5.1.2 Simulation at 30V

To facilitate the comparison between the simulation results and the experimental results, a MATLAB simulation was carried out at a low voltage (30V), with basically the same circuit and controller settings as the simulation at 220V. The only difference is that the battery voltage Vin was changed from 360V to 48V, and the  $U_{\text {ref}}$ was reduced from 220V to 30V. The supply-side voltage was also increased and decreased for the simulation.

Through debugging, it is learned that the CL voltage U_cl remained at 30V, when the ES was not involved in circuit regulation and the supply-side voltage U_g was set to 35.69V. Therefore, the supply-side voltage was decreased and increased by 10% to 32.04V and 39.16V. The simulation time was set to 0.5s. The time variations of U_g and Tri signal are recorded in Table 4.

Table 4. The time variations of U_g and Tri signal

Figures 14 and 15 compare the amplitudes and effective values of U_cl in circuits with and without ES, respectively. In 0~0.1s, the supply-side voltage was normal, the Tri signal was 1, and the switch Q was closed, i.e. the ES was out of service. At 0.1s, the Tri signal was 0, and the switch Q was opened, i.e. the ES was put into use.

As shown in Figure 14, the U_cl amplitude in the circuit without ES fluctuated with the dropping or rising voltage, while that in the circuit with ES remained stable. As shown in Figure 15, the circuit with ES achieved a good performance in voltage stabilization. Regardless of voltage drop or voltage rise, the circuit quickly stabilized the effective value of U_cl near 30V. This proves the robustness and adaptability of the ES under the control of FCS-MPC.

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Figure 14. The variations in U_cl amplitude at the drop or rise of supply-side voltage

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Figure 15. The variations in the effective value of U_cl at the drop or rise of supply-side voltage

Figure 16 compares the phases of U_cl and U_ES at 30V. In 0~0.1s, the Tri signal was 1, and the switch Q was closed, i.e. the ES was out of service. In this case, the ES voltage output was 0. At 0.1s, the Tri signal was 0, and the switch Q was opened, i.e. the ES was put into use. At 0.2s, the supply-side voltage dropped. In 0.2-0.3s, the U_C lagged U_cl by about 90°; the ES worked in capacitive mode under the voltage drop, and injected negative reactive power into the bus to stabilize the voltage across Z_cl. In 0.3-0.4s, the supply-side voltage increased, and the grid voltage exceeded the reference value; in this case, the U_C led U_cl by about 90°; the ES worked in capacitive mode under the voltage rise, and injected positive reactive power into the bus to stabilize the voltage across $Z_{c l}$. The above results further confirm the correctness and effectiveness of our control system.

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Figure 16. The phase comparison between U_cl and U_ES

5.2 Physical experiment and result analysis

Figure 17 illustrates the physical experiment platform. For the safety of physical experiment, four 12V/24Ah batteries were connected in series to simulate the 48V DC power supply. The supply-side power source of the grid was simulated with a voltage regulator, with the effective value of U_ref at 30V. The control part was simulated with a dSPACE DS1202 controller. The CL and NCL were simulated by adjustable resistors. The controller parameters in dSPACE were kept consistent with those in the simulation at 30V, except that the step length was adjusted to 5e-5s. To prevent short circuiting, the PWM dead time of the controller output was set to a sampling period.

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Figure 17. The dSPACE-based ES experimental platform

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Figure 18. The waveforms of U_cl amplitude, U_cl effective value, and U_C at voltage drop

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Figure 19. The waveforms of U_cl amplitude, U_cl effective value, and U_C at voltage rise

Figures 18 and 19 compare the U_cl amplitudes, U_cl effective values, and U_C as the supply-side voltage dropped or increased at time t₁, respectively. It can be seen that U_cl amplitude did not fluctuate greatly, while the effective value of U_cl stabilized quickly after slight volatility, during the fluctuations of the supply-side voltage.

The ES was put into use after the switch Q was opened, and started to regulate the circuit at time t₁. During ES regulation, the U_C lagged U_cl at the drop of supply-side voltage, but led U_cl at the increase of supply-side voltage. This is because, the ES worked in capacitive mode under the voltage drop, and injected negative reactive power into the bus to stabilize the voltage across Z_cl; the ES worked in capacitive mode under the voltage rise, and injected positive reactive power into the bus to stabilize the voltage across Z_cl.

To clearly observe the ES working states at different voltage fluctuations, the U_C phases at the drop and rise of supply-side voltage are compared in Figure 20. As the supply-side voltage started to fluctuate at time t₁, there was a 180° difference between the U_C phases at the drop and rise of supply-side voltage. The reason is that the U_C lagged U_cl by 90° at voltage drop, but led the latter by 90° at voltage rise. The above analysis shows that the results of physical experiment agree with the simulation results. This further proves the feasibility and effectiveness of our control system.

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Figure 20. The U_C phases at the drop and rise of supply-side voltage