A Current Control Method for Grid-Connected Inverters

Abstract

It is simple to implement conventional current control with a proportional integral (PI) controller. However, system stability and dynamic performance are not perfect, particularly when operating under unfavorable conditions. In this paper, an improved control method is proposed by introducing a compensation unit. The compensation unit can effectively compensate the system’s phase around the crossover frequency, greatly enhancing the system’s phase margin and stability. It is also capable of handling weak-grid conditions. In this paper, the concept of the proposed compensation unit is explained first. Then, the corresponding mathematical model for the current control loop is built, and system bode diagrams for the conventional and proposed methods are compared. Furthermore, the effect of the parameters for the compensation unit is investigated, and an optimization method is proposed to determine optimal parameters. In addition, to handle weak-grid conditions, the proposed scheme is expanded by including the compensation unit in the grid’s feed-forward loop. Finally, an experimental platform is constructed, and the experimental results are presented to validate the proposed method.

1. Introduction

As part of the global green mission, more and more renewables are being installed. As renewable energy sources gain traction, power electronic converters are becoming increasingly popular [1]. To reduce harmonics and improve grid-current quality, LCL or LC filters are commonly used between the inverter and the grid. Since the grid current injected into the grid must be of high quality, many researchers proposed various methods to control the current and suppress harmonics [2,3].

Linear controllers of four types are commonly used for grid current control. The first type is a stationary PI controller, in which the current is controlled in an a-b-c frame and PI controllers are used for three phases [4]. The main disadvantage is that the reference is AC (alternating current) rather than DC (direct current), and thus, open-loop gain is not infinite, affecting system performance. The second category, on the other hand, is synchronous PI controller [5]. It is applied in a d-q-0 frame rather than an a-b-c frame, with PI controllers for the d-axis, q-axis, and 0-axis, respectively. The open-loop gain is infinite because the references are DC rather than AC, resulting in improved system performance. The third option is proportional resonant (PR) controller, which is a more advanced version of the PI controller. The ability to achieve a high gain around resonance frequency, which can result in improved harmonic suppression performance, is one of the advantages of a PR controller [4,5]. The main disadvantage is that the controller is relatively complex, especially when multiple resonance frequencies are present. The fourth type of controller is a predictive and deadbeat controller [6], which forecasts error at the start of each sampling period. The reference is then generated to reduce the forecasted error. It is worth noting that the predictions require system parameters, and thus, the accuracy is dependent on the system parameters. For nonlinear controllers, the hysteresis controller is normally applied [7], where the reference and grid current feedback are compared to generate inverter signals. The hysteresis controller band is defined by the minimum and maximum error. The disadvantage of this method is that the switching frequency is not fixed, making filter design difficult.

LCL filters are commonly used in grid-connected converters to improve harmonics suppression. The control for LCL filter systems can be generally divided into three categories [8]. The first is inverter-side inductor current sensing for current control. The second method is to control current using grid-side inductor current sensing. The third is current control, which is achieved by sensing both the grid and the inverter current. Furthermore, several approaches for suppressing the resonance peak in LCL filters are proposed. One of these is passive dampening, and one frequent way is to add resistors to the filter capacitor branch. Furthermore, active damping techniques are often used [9]. Existing active damping techniques include filter-based active damping, feedback-based active damping with feedback of extra voltage and current states and weighted average management of the two inductor currents. Filter-based active damping improves stability by adding a digital filter next to the current controller [10]. In feedback-based active damping, the state used for feedback can be the capacitor current, the capacitor voltage, the inverter-side current, the grid current, or a combination of many states [11]. Proportional feedback of the capacitor current has been actively investigated among feedback-based active damping approaches due to its easy implementation [12,13]. Furthermore, a weighted average control is proposed, where the LCL resonance peak is eliminated by taking the weighted average of the grid current and the inverter-side current [14]. An alternative solution is the split-capacitor method, which divides the filter capacitors into two capacitors [15]. The current between the two capacitors is measured. In the control diagram, a three-order system can be reduced to a single-order system using the split capacitor method. Thus, this method greatly optimizes system performance by simplifying the controller’s design.

While current-control-based inverters perform well in strong grids, their control capability deteriorates dramatically in weak-grid conditions [16]. This is because grid-following inverters rely on phase-locked loops (PLLs), which can cause instability in weak grids. With the ongoing and rapid penetration of renewables and the global effort to phase out coal-fired power plants, this problem becomes more difficult. To address the shortcomings of grid-following inverters, several PLL-less control approaches and grid-forming technology are being developed for grid-connected inverters. For example, a voltage-modulated direct power control (VMDPC) is presented in [17]. Another modern PLL-less control approach is linear parameter varying power synchronized control [18]. In contrast to traditional current control methods, VMDPC does not require a PLL. As a result, the instability issues associated with PLLs are eliminated [19]. However, in a weak grid, the VMDPC’s performance can suffer because it still needs the point-of-common-coupling (PCC) voltage to manage the power exchange with the grid, which has a significant impact on PCC voltage. Grid-forming inverters have recently gained popularity [20]. The most commonly used grid-forming inverter functions are droop control functions, virtual oscillator control functions, and virtual synchronous generator functions [21], which can be used for providing voltage, frequency, and inertia support to power grids.

It is known that it is simple to implement conventional current control with a PI controller. However, system performance is not perfect, particularly under unfavorable operating conditions. In this paper, an improved control method is proposed by introducing a compensation unit. The novelty is that the compensation unit can effectively compensate the system’s phase around the crossover frequency, greatly enhancing the system’s phase margin and stability. It is also capable of handling weak-grid conditions. The main contributions are as follows.

(1)

By introducing the compensation unit, the proposed control method can greatly enhance the system’s phase margin and stability. Also, a mathematical model is constructed, and the system bode diagrams for the conventional and proposed controls are compared.

(2)

The effect of two parameters in the compensation unit is investigated. Furthermore, an optimization method is proposed to determine optimal parameters.

(3)

To handle weak-grid conditions, the proposed scheme is expanded by including the compensation unit in the grid’s feed-forward loop.

2. System Setup and Method

2.1. System Setup and Conventional Control Method

Figure 1 depicts the system setup and conventional current control architecture. The conventional method is implemented in a d-q frame and PI controllers are applied [22,23]. To begin, sample the output current i_g and set it as i_α. Shift the current i_α by 90° to obtain the current i_β. The currents i_α and i_β are transformed using αβ/dq transformation to obtain the currents i_d and i_q. The currents i_d and i_q are compared to their respective references I_d^* and I_q^*. The differences are sent to PI controllers, which are then combined with the grid feed-forward values v_gd and v_gq. After that, the dq/abc transformation is used to transform the results. Finally, the duty cycle for the converter d is calculated. With the modulation, four switches (S_a1, S_a2, S_b1, S_b2) can be controlled.

From Figure 1, the mathematical model can be obtained, which is shown in Figure 2. The PI controller transfer function is represented by G_PI(s). G_d(s) denotes control delay. V_dc is the dc-bus voltage that represents the transfer function from the duty cycle d to the inverter output. G_Z(s) is the transfer function from the inverter output to the current i_d or i_q. H(s) is the sampling transfer function, which is equal to 1 in our case. The expressions for G_PI(s), G_d(s), and G_Z(s) are presented in (1)–(3).

$$G_{PI}\left(s\right)=k_p+\frac{k_i}{s}$$

(1)

$$G_d\left(s\right)=e^{-T_{d}s}$$

(2)

$$G_Z\left(s\right)=\frac{1}{sL+s{L}_g//\frac{1}{sC}}$$

(3)

This conventional method is straightforward in terms of implementation. However, the control effect is not perfect. Total harmonic distortion (THD) and dynamic response may not meet the requirements in unfavorable operating conditions. Therefore, we propose an improved control method to improve system performance indicators such as dynamic response, THD, and others.

2.2. Proposed Control Method

Figure 3 presents the proposed current control architecture. Compared with the conventional method, the proposed control architecture adds a compensation unit G_c. The expression of the compensation unit G_c is shown in (4). It is worth mentioning that compensation unit G_c has a lead-lag compensation effect. Overall, the system performance can be greatly improved with this modification.

$$G_C(s)=\frac{T_{c}s+1}{k_c\times T_{c}s+1}$$

(4)

From Figure 3, the corresponding mathematical model is shown in Figure 4.

From Figure 4, the open-loop function can be obtained, which is shown in (5). Also, it is known that close-loop stability can be observed from open-loop behavior.

$$G\left(s\right)=G_c\left(s\right)G_{PI}\left(s\right)G_d\left(s\right)V_{dc}{G}_z\left(s\right)$$

(5)

Based on (5), the open-loop bode diagrams for the conventional and proposed control are shown in Figure 5 with the following set of parameters: k_p = 0.07, k_i = 1, T_d = 10⁻⁵, V_dc = 380 V, L = 200 μH, L_g = 20 μH, C = 4.7 μF, T_c = 5 × 10⁻⁵ and k_c = 0.1. As shown in the blue line in Figure 5, the system is unstable when the conventional control is used. This is because there is no phase margin from the diagram using the conventional control. On the other hand, as shown in the red line in Figure 5, when the proposed control is used, the system is stable, the crossover frequency is about 5 kHz, and the phase margin is increased to 39°.

From Figure 5, it can be seen that the compensation unit G_c can effectively compensate the system’s phase around the crossover frequency. Therefore, by introducing the compensation unit G_c, the proposed control method can greatly enhance the system’s phase margin and stability. Therefore, the performance of the system can be greatly enhanced with the proposed method.

3. Results and Discussion

3.1. Simulation Result Comparison

Figure 6 depicts the simulation results for the conventional and proposed control methods with the parameters in the last section. From Figure 6a, the system is unstable when the conventional control is used. From Figure 6b, the system is stable and the output current i_g is of good quality when the proposed control is applied.

When k_p varies and the rest of the parameters remain the same, Figure 7 depicts the gain and phase margins using conventional and proposed control methods. From Figure 7a,b, it is clear that the compensation unit G_c can significantly increase the system’s gain margin and phase margin, leading to higher stability. From Figure 7c, with the compensation unit G_c, the crossover frequency can also be greatly increased, resulting in improved system dynamic performance.

When we set the phase margin to >35° and the gain margin to >15 dB, k_p is 0.025 for the proposed control and 0.007 for the conventional control. As a result, G_c can increase k_p by 3.5 times. This can result in improved low frequency band gain and THD performance.

3.2. Discussion on Impact of Compensation Unit Parameters

From Equation (4), there are two parameters for the compensation unit G_c. One is T_c and the other one is k_c. The following section analyzes the impact of these two parameters.

3.2.1. T_c Varies

Figure 8 shows the bode diagram of the compensation unit G_c with k_c = 0.1 and T_c varying. When T_c increases from 5 × 10⁻⁵ to 25 × 10⁻⁵, the diagram shifts toward the left-hand side. From the last section, the compensation unit G_c is used to increase the phase margin. Therefore, the diagram should be centered on the crossover frequency and cannot be too far to the left. Furthermore, if the diagram is too far to the left, the gain will be lifted too high, reducing the gain margin.

Figure 9 presents the system bode diagram when k_c = 0.1 and T_c varies. From it, the crossover frequency increases as T_c increases from 5 × 10⁻⁵ to 25 × 10⁻⁵. In terms of the phase margin, it is greatest when T_c is 10 × 10⁻⁵. The gain margin decreases as T_c increases from 5 × 10⁻⁵ to 25 × 10⁻⁵.

Figure 10 presents the gain margin, phase margin, and crossover frequency for different T_c. When T_c increases, the gain margin rises first and then falls. Also, the phase margin increases at first, then decreases. The maximum of the phase margin occurs roughly at T_c of 10 × 10⁻⁵. The crossover frequency increases monotonically, and it increases from 2.6 kHz to 4.75 kHz when T_c is increased from 5 × 10⁻⁵ to 15 × 10⁻⁵.

Figure 11 depicts the simulation results for T_c = 5 × 10⁻⁵ and T_c = 15 × 10⁻⁵, where the d-axis current reference I_{d_ref} steps at 25 ms. If T_c is increased from 5 × 10⁻⁵ to 15 × 10⁻⁵, the dynamic performance, including response time and current overshoot, is improved. This is because the crossover frequency can be increased from 2.6 k to 4.75 k, and the phase margin increases by 5°.

3.2.2. k_c Varies

Figure 12 shows the bode diagram of the compensation unit G_c with T_c = 5 × 10⁻⁵ and k_c varying. For the phase compensation, the smaller the k_c, the greater the corresponding phase compensation. In terms of the gain compensation, the smaller the k_c, the greater the magnitude can be lifted. Note that, taking the frequency of 5 kHz, when k_c is less than 1/6, the gain changes very little. This implies that the impact on the crossover frequency is small when k_c is less than 1/6.

Figure 13 presents the system bode diagram when T_c is 5 × 10⁻⁵ and k_c varies. From it, the crossover frequency increases when k_c is changed from 1/2 to 1/12, but the overall change is very small. In terms of the phase margin, when k_c changes from 1/2 to 1/12, the phase margin increases accordingly, which can bring higher stability.

Figure 14 presents the gain margin, phase margin, and crossover frequency for different k_c. From Figure 14a, the gain margin does not change significantly. From Figure 14b, it decreases monotonically in terms of the phase margin. In general, the smaller the k_c, the greater the phase margin and the greater the stability. From Figure 14c, the crossover frequency decreases monotonically, but the overall change is small.

3.3. Discussion on Parameter Optimization

According to the above analysis, the change trend of parameters k_c and T_c does not have a simple monotonic relationship with the system performance indexes and, thus, cannot be determined directly. As a result, the following describes a method for determining parameters, specifically for G_c (including k_c and T_c) and the PI controller (k_p and k_i). The detailed procedures are shown in Figure 15.

First, determine the range of the parameters. For example, the variation range of k_c is 0.005~0.05; the variation range of T_c is 0.5 × 10⁻⁵~50 × 10⁻⁵; the variation range of k_p is 0.003~0.1. Next, determine the open-loop function G(s) for different parameters. Obtain the corresponding phase margin PM, gain margin GM, crossover frequency f_cross, etc. If the stability requirement is set to the phase margin PM larger than 40° and the gain margin GM larger than 20 dB, eliminate the parameters which do not meet this requirement. Then, among the remaining parameters, find parameters with a higher crossover frequency, a larger k_p, and a larger phase margin. This results in a system with greater bandwidth, faster response, greater gain in low-frequency bands, and less overshoot.

In order to find the optimized parameters with higher crossover frequency, higher k_p, and higher phase margin, the following is an optimization method.

To begin, normalize the performance indicators, which are shown in (6). In (6), f_{cross_base}, k_{p_base}, and PM_base are the bases for crossover frequency, k_p, and phase margin, respectively. F_cross, k_p, and PM are the crossover frequency, k_p value, and phase margin under different parameters, respectively.

$$\begin{gathered} f_{cross}^{\prime }=\frac{f_{cross}}{f_{cross\_base}} \\ {k}_p^{\prime }=\frac{k_p}{k_{p\_base}} \\ {PM}^{\prime }=\frac{PM}{{PM}_{base}} \end{gathered}$$

(6)

(7) presents an optimization function. Apply the normalized values in (6) to (7). Then, find the values that correspond to the highest value of Y. Those corresponding parameters are the optimal parameters we try to find.

$$Y=\sqrt{{f_{cross}^{\prime }}^2+{k_p^{\prime }}^2+{{PM}^{\prime }}^2}$$

(7)

Table 1. Parameters before and after optimization.

Items	Before Optimization	After Optimization
Parameters	kc = 0.1, Tc = 5 × 10−5, kp = 0.014, ki = 1	kc = 0.01, Tc = 4 × 10−5, kp = 0.03, ki = 1
Phase Margin	45.7°	42°
Gain Margin	20.2 dB	20.4 dB
Crossover frequency	1790 Hz	2770 Hz
Gain at 150 Hz	27.1 dB	33.7 dB

presents the parameters before and after the optimization. From it, the parameters before and after the optimization both meet the stability requirement of phase margin PM ≥ 40° and gain margin GM ≥ 20 dB. More importantly, the crossover frequency is increased after the optimization, which enhances dynamic performance. Additionally, the low frequency band’s gain is increased, which can improve the current THD.

Figure 16 presents the system bode diagram before and after the optimization. The phase and gain margins are nearly identical before and after the optimization. The crossover frequency is increased after the parameters are optimized, which can bring about a shorter response time. In addition, the gain of the low frequency band is increased, which can bring about lower harmonics.

Figure 17 presents current THD results before and after the optimization. After the optimization, the gain in the low frequency band is increased, and thus, the current THD can be improved. From Figure 17, THD is 1.5% before the optimization, and it reduces to 1.19% after the optimization.

Figure 18 presents the simulation results where d-axis current reference I_{d_ref} steps at 25 ms. From Figure 18, the response time is shortened after the optimization because the crossover frequency is increased after the optimization. Furthermore, there are no significant changes in the current overshoot after the optimization. This is due to the fact that the phase margin is nearly identical before and after the optimization.

4. Method Extension for Weak-grid Conditions

The previous scheme can be expanded by including a compensation unit in the grid’s feed-forward loop to handle weak-grid conditions, as shown in Figure 19. The grid feed-forward v_gd and v_gq is sent to G_c, then obtains v_gd’ and v_gq’, which are then added to the original current loop. This can significantly improve the phase margin, which can then significantly improve the system stability in weak grid scenarios. The method is simple and effective to handle weak-grid conditions.

From Figure 19, the corresponding mathematical model is shown in Figure 20.

From Figure 20, the open-loop function G(s) can be obtained, which is shown in (8).

$$G(s)=\frac{G_c(\mathrm{s})G_{PI}\left(\mathrm{s}\right)G_d\left(\mathrm{s}\right)V_{dc}}{\left(s{L}_2+s{L}_g\right)\left(1+s^2{L}_{1}C\right)+{sL}_1-{sL}_g{G}_c\left(s\right)G_d\left(s\right)V_{dc}}$$

(8)

Based on (8), the open-loop bode diagrams with and without the compensation unit G_c in the grid’s feed-forward loop are shown in Figure 21 with the following set of parameters: k_p = 0.6, k_i = 1, T_d = 10⁻⁵, V_dc = 380 V, L₁ = 200 μH, L₂ = 20 μH, L_g = 600 μH, C = 4.7 μF, T_c = 5 × 10⁻⁵, and k_c = 0.1. Because the grid-side inductor L_g is relatively large, it is assumed that the grid is weak.

Figure 21 shows the system bode diagram with and without the compensation unit G_c in the grid’s feed-forward loop. From it, it is shown that the gain margin is small for the system without G_c in the grid’s feed-forward loop due to the resonance peak (see red line). In contrast, adding G_c to the grid’s feed-forward loop improves system stability (see blue line). As a result, incorporating the compensation unit G_c into the grid feed-forward can improve system reliability and performance.

5. Experimental Verifications

A platform is built and Figure 22 and Figure 23 are the experimental results. Figure 22 depicts the experimental results for the conventional control and proposed control with the addition of the compensation unit G_c in the current loop. From Figure 22a, the system is unstable when the conventional control is used. There are large oscillations in the inductor current and the output capacitor voltage. From Figure 22b, the system is stable, and the output current is of good quality when the proposed control with the compensation unit G_c added in the current control is applied. Therefore, by introducing the compensation unit G_c, the proposed control method can greatly enhance the system’s stability and performance.

Figure 23 depicts the experimental results with or without the compensation unit G_c in the grid’s feed-forward loop under weak-grid conditions. From Figure 23a, there are large oscillations in the output capacitor voltage, which can trigger the system’s over-current protection. From Figure 23b, the output current is of good quality when the proposed control with the compensation unit G_c added in the grid’s feed-forward loop is applied. Therefore, by introducing the compensation unit G_c in the grid’s feed-forward loop, the proposed control method can greatly enhance the system’s stability and performance.

6. Conclusions

It is simple to implement conventional current control with a PI controller. However, system performance is not perfect, particularly under unfavorable operating conditions. In this paper, an improved control method is proposed by introducing a compensation unit. The compensation unit can greatly enhance the system’s phase margin and stability. It is also capable of handling weak-grid conditions.

• The system bode diagrams for the conventional and proposed controls are compared. From it, the compensation unit can effectively compensate the system’s phase around the crossover frequency, greatly enhancing the system’s phase margin and stability.
• The effect of two parameters, k_c and T_c, is investigated. From the analysis, the change trend of parameters does not have a simple monotonic relationship with the system performance indexes and cannot be determined directly.
• An optimization method is proposed to determine optimal parameters for the compensation unit and PI controller. From the results, after optimization, the crossover frequency and the low frequency band’s gain can be increased.
• To handle weak-grid conditions, the proposed scheme is expanded by including the compensation unit in the grid’s feed-forward loop. From the analysis, the system reliability and performance can be enhanced by incorporating the compensation unit.
• Finally, an experimental platform is constructed. From the experimental results, the system is unstable when the conventional control is used and there are large oscillations in the current and voltage. When the proposed control is applied, the system is stable, and the output current is of good quality. Therefore, the proposed control method can greatly enhance the system’s stability and performance.
• In the future, as renewable energy becomes more popular, more and more grid-connected inverters will be applied. This proposed method incorporates a simple compensation unit into the control loop, which greatly improves system performance. This can increase the inverters’ robustness, allowing them to adapt to more complex operating scenarios. Furthermore, weak-grid conditions are becoming more common, and this control method allows the grid-connected inverters to be more adaptable to weak grids.