A Simplified Design Strategy for Multi-Resonant Current Control of a Grid-Connected Voltage Source Inverter with an LCL Filter

Abstract

A distorted grid voltage or nonlinear behavior in the current control loop can cause low frequency current harmonics in a grid-connected voltage source inverter (VSI). Many efforts have been made to mitigate such phenomena, including hardware and/or control structure improvements. A well-known suitable strategy to reduce current harmonics in a selective manner is to apply a Proportional Multi-Resonant (PMR) current controller. Inverter-grid stability is another common issue when dealing with grid-connected VSI. Stability is influenced by the inverter impedance, which depends on the controller parameters. This paper presents a simplified tuning strategy for the PMR controller, taking into consideration the inverter-grid stability issue. The obtained controller was implemented and tested in a 10 kW three-phase inverter with a passively damped LCL filter. A significant reduction of current harmonics emission from the inverter up to 650 Hz was achieved without any hardware modification. The limits of PMR controllers to mitigate current harmonics were studied, and the influence of the grid impedance was verified.

1. Introduction

Distributed energy generation is increasing with the exploitation of renewable energy sources. Current-controlled voltage source inverters (VSIs) are commonly used to connect those sources to the AC utility grid. The inverters cause current harmonics at both low (few hundred hertz) and medium frequencies (from a few kHz up to 100 kHz). PWM (pulse-width-modulated) inverters produce current harmonics at their switching frequency and its multiples. Typically, passive or active output filters are used to reduce these perturbations in the kHz range [1].

However, low frequency harmonics also exist as a consequence of

• nonlinear behavior in the current control loop such as the switching dead time [2],
• grid voltage disturbances [1,3], and
• DC-link voltage harmonics [3].

A constant DC voltage is assumed in this manuscript. They generally occur at multiples of the grids fundamental frequency and therefore in the lower frequency range, up to 2 kHz.

This paper focuses on low frequency current harmonics mitigation for a three-phase, three-wire inverter with a passively damped LCL filter, without any hardware intervention. Several solutions have already been studied in the literature to mitigate those harmonics. A non-exhaustive review is presented here.

Current controllers with a single proportional integral (PI) per axis are commonly used to control three-phase grid-connected inverters. Typically, these controllers are implemented in the synchronous dq-reference frame and are sufficient when dealing with a balanced and mainly sinusoidal electrical grid. However, when grid voltages are unbalanced and distorted, advanced current controller and/or additional hardware are required. A good strategy to achieve a zero steady state error at the grid’s fundamental frequency is to use proportional resonant (PR) controllers in the stationary αβ-reference frame [4]. The mathematical relation between the rotating frame PI controller and the stationary frame PR controller was derived, and a damped PR controller was proposed to increase system stability in [5,6]. It has been demonstrated that an ideal PR controller corresponds to a pair of PI controllers, synchronized with the grids’ positive and negative sequence components [1,7]. A controller with additional resonances was proposed in [8] to cancel current harmonics. This type of controller modifies the output impedance of the inverter by integrating a model of the periodic disturbances.

The influence of proportional multi-resonant (PMR) controllers on the output impedance of the inverter was analyzed in [9]. Different structures were studied, e.g., to improve the current distortion under unbalanced grid conditions [10]. In [2], it was demonstrated that multi-resonant control effectively mitigates current harmonics caused by the converter dead time. Its advantage over average theory-based approaches to compensate the influence of the dead-time is that there is no need for current polarity detection.

Unfortunately, due to the phase lag introduced by the PMR controller, closed-loop stability is not an effortless task. In order to tackle this drawback, a repetitive controller (RC) has been proposed [11]. An RC rejects all grid harmonics disturbances; however, from an implementation point of view, an RC requires more memory resources compared to a PMR controller. Moreover, a PMR controller allows specific and selective control of non-harmonic components (sub- and inter-harmonics). A cascaded PI controller can also realize current harmonics mitigation [1]. From an implementation point of view, a cascaded PI controller requires more computational resources and memory. A Z-active power filter (Z-APF) aims to sink the harmonics current from the grid by generating a current reference directly from the grid voltage and the desired impedance type and value. In a grid-connected application, the R-APF type is mostly employed to damp out currents harmonics and stabilize the grid [12]. Predictive controllers (PCs), such as deadbeat or finite control set models, have been investigated as well [13]. PCs require a strong knowledge of the inverter model, plant, and grid to predict dynamic behavior, meaning that PCs are inherently sensitive to model and parameter mismatches. Robust control using H-infinity methods based on a PI plus resonant controller and based on an RC have been proposed [14,15]. Many other non-linear controllers such as swarm-optimization-based harmonic elimination [16] and fuzzy logic [17] have been studied. In [18,19], a direct Lyapunov-based control technique is presented for active power filtering of the utility grid. Others system topologies have been studied to attenuate current harmonics, such as Z-source inverters [20]. In [21], an active power filter approach is presented. In [22], an auxiliary power converter is devoted to the control of the current harmonics of a high power VSI.

None of the above strategies/methods benefit from a perfect harmonic cancellation, low computation, no losses, no specific position in the grid, and no additional cost.

The aim of this paper is to present a design strategy of the inverter current controller to attenuate current harmonics perturbations in the frequency range below 2 kHz. This can be done with a PMR controller. In order to reduce the number of parameters to be tuned, harmonic resonant coefficients have been imposed to be inversely proportional to the value of the fundamental harmonic coefficient. More precisely, they will follow an inverse proportional law as a function of the harmonic order. The design of the current controller then takes into account the inverter-grid stability issue by using sensitivity values. This approach of a simplified choice of coefficients of a PMR controller, together with the inverter-grid stability approach, is novel. The obtained current controller was implemented in a 10 kW three-phase grid-connected VSI with an LCL filter. Tests were executed under varying grid voltage conditions.

The paper is organized as follows. The experimental setup is described in Section 2. Section 3 presents the inverter model. In Section 4, a description of the PMR controller tuning procedure is given, with particular emphasis on the influence of the different optimization criteria. Finally, in Section 5, the experimental results are presented.

2. Experimental Setup

The experimental setup has been presented in a previous paper [23]. For ease of reading, the key elements, along with some of the design criteria, are summarized in this section. A principle schematic of the inverter is shown in Figure 1.

The main system specifications are listed in

Table 1. Inverter nominal values.

Parameter	Symbol	Value
DC-link voltage	UDC	700 V
Nominal power (overload 200%, 60 s)	P	10 kW
Utility grid nominal voltage (line-to-line)	Vg	400 V
Utility grid nominal frequency	f	50 Hz
Inverter switching frequency	fsw	15 kHz
Switching dead time	Ta	3.2 µs
Inverter side inductor	L1	3 mH
Grid side inductor	L2	600 µH
Filter Capacitor	C1	1.41 µH
Damping resistor	Rf	1 Ω
Inductance of the damping circuit	Lf	51 µH

.

The inverter hardware is mainly made of three boards, i.e., power, control, and communication boards, a coupling inductance section, and an EMC filter section. Both power and control boards were custom made, while for the communication an evaluation board was chosen: STKA28-AA, supplied by the TQ group. A picture of the electrical cabinet of the inverter is presented in Figure 2.

2.1. Power Board Hardware Features

The power board bloc-schematic is shown in Figure 3. It is supplied by an external +24 V power supply. All auxiliary power supplies, e.g., +3.3 V and +5 V, are embedded. An intelligent power module has been selected for the switching section. Due to the required overload capability, the PM50RLB120, 50 A/1200 V (Mitsubishi Electric Corporation, Tokyo, Japan) was chosen. The inverter losses were calculated according to [24] and simulated using PLECS^® (Version 4.0.5, Plexim GmbH, Zurich, Switzerland), simulation software for power electronics. In nominal conditions, overall losses of about 400 W were evaluated at the switching frequency f_sw = 15 kHz. The DC link capacitor bank was designed according to [25]. Considering an ambient temperature of 45 °C and a junction virtual temperature Tj = 125 °C, the required thermal resistance of the heat-sink was evaluated as 0.08 K/W. The chosen heatsink was a LA8-300-24 V (Astrel AG, Pfäffikon, Switzerland). The IPM module was directly soldered on the PCB (Printed Circuit Board).

The DC voltage was connected to the intelligent power module (IPM) through two positive and negative overlapping copper layers on the board (equivalent busbar). The DC link voltage and the three phase output currents were measured with HXS20-NP Hall sensors (LEM, Geneva, Switzerland). The leakage current on the DC link was measured with a LEM CTSR 0.3-TP/SP4. The PWM control signals from the DSP were provided via fast optocouplers HCPL-4504 (Broadcom, San Jose, CA, USA—formerly Avago), while the IPM error output was sent back to the control through fast optocouplers HMHA2801 (Fairchild—ON Semiconductor, Phoenix, AZ, USA). The three-phase utility voltages were measured after a three-phase contactor and the filter section to perform grid synchronization independently from the power injection procedure.

2.2. Coupling Inductance and Harmonics Filter

The coupling inductance L₁ value of 3 mH was designed by neglecting the additional differential mode harmonic filter made by C₁ and L₂. A nominal voltage at L₁ of about 5% of utility voltage was selected. Simulated current harmonics were used to choose the inductance core. The details of the harmonics filter design are presented in [23].

2.3. Control Board Hardware Features

The inverter can be controlled using the International Electrotechnical Commission standard communication protocol IEC-61850, through a custom designed software called Mat2DSP (Proprietary custom developed software), or through a Labview (2017, National Instruments Corporation, Austin, TX, USA) application. Figure 4 presents the bloc-schematic of the control board including the communication port.

The current controller presented in this paper was implemented in the experimental setup described above.

3. Inverter Model

Figure 5 shows the block diagram of the current control loop.

The transfer functions of the different blocks are defined by Equations (1)–(5). The inverter itself is represented by a gain with a time delay, due to the computation time of the microcontroller and the delay introduced by space vector modulation (SVM). It is represented by the block G_inv(s). In this study, a 4th-order Padé-approximation of the delay was used, in order to model the phase lag accurately up to the resonance frequency of the filter (5.8 kHz).

The transfer function of the PMR controller [1,3,7,10] is shown by Equation (5). It contains the proportional gain k_p and the sum of damped resonance terms. As is well known, a damped resonance was implemented to increase system stability and to avoid discretization issues [1,5,6]. The peak gain of each resonance is defined by the parameter k_r. The angular frequency of each resonance is determined by the parameter ω_n and the damping factor ζ determines their bandwidth. The passively damped LCL filter is represented by the blocks Y_L1, Z_C1, and Y_L2. The lossless admittances of the filter inductors are modeled by the blocks Y_L1 and Y_L2, while the Z_C1 is the impedance of the capacitor branch, including its damping circuit.

$$G_{inv}\left(s\right)=\frac{U_e}{2}{e}^{-s{T}_s}$$

(1)

$$Y_{L1}\left(s\right)=\frac{1}{L_{1}s}$$

(2)

$$Y_{L2}\left(s\right)=\frac{1}{L_{2}s}$$

(3)

$$Z_{C1}\left(s\right)=\frac{L_f{R}_f{C}_1{s}^2+L_{f}s+R_f}{L_f{C}_1{s}^2+R_f{C}_{1}s}$$

(4)

$$G_{reg}\left(s\right)=k_p+{\displaystyle \sum }_n\frac{k_{r,n}2{\zeta }_n{\omega }_{\mathrm{n}}s}{s^2+2{\zeta }_n{\omega }_{\mathrm{n}}s+{\omega }_{\mathrm{n}}{}^2}.$$

(5)

U_e, T_s, R_f, and L_f terms represent respectively the DC-bus voltage, the delay time caused by computation delay and zero-order hold effect, the damping resistor of the LCL filter, and the parallel inductance to the damping resistor in order to reduce ohmic losses.

The grid voltage harmonics of V_g(s) acts as a disturbance on the current control loop. The transfer function of the current controller is modeled by the block G_reg(s). Equation (6) describes the open loop transfer function of the current control loop.

$$G_o\left(s\right)=\frac{i_{L1}\left(s\right)}{i_{ref}\left(s\right)}=\frac{G_{reg}\left(s\right)G_{inv}\left(s\right)}{\frac{1}{Y_{L1}\left(s\right)}+\frac{Z_{C1}\left(s\right)}{Y_{L2}\left(s\right)Z_{C1}\left(s\right)+1}}.$$

(6)

From Equations (1)–(6), the continuous transfer function of inverter admittance can be easily deduced:

$$G_{yv}\left(s\right)=\frac{H_1\left(s\right)Y_{L2}\left(s\right)}{H_1\left(s\right)+H_2\left(s\right)Y_{L2}\left(s\right)}$$

(7)

where

$$H_1\left(s\right)=\frac{G_{reg}\left(s\right)G_{inv}\left(s\right)Y_{L1}\left(s\right)}{1+G_{reg\left(s\right)}{G}_{inv}\left(s\right)Y_{L1}\left(s\right)}$$

(8)

$$H_2\left(s\right)=\frac{H_1\left(s\right)Z_{C1}\left(s\right)}{1+\frac{H_1\left(s\right)Z_{C1}\left(s\right)}{G_{reg}\left(s\right)G_{inv}\left(s\right)}}.$$

(9)

It is worth mentioning that Equation (5) contains a model of the n sinusoidal signal. The internal model principle (IMP) states that, if a permanent and a marginally stable signal need to be rejected, a model of this signal has to be included inside the control loop [26]. By developing Equation (7), it can be demonstrated that its numerator contains a model of the n sinusoidal signal thanks to G_reg(s).

Grid-connected inverters are known to become unstable when dealing with a highly inductive grid. Inverter-grid system stability using only the inverter output impedance and the grid impedance has been investigated in [27]. A grid-connected inverter operation is stable if the ratio of the grid impedance to the inverter output impedance satisfies the Nyquist criterion. As already said, the proposed design of the PMR controller will take into account the inverter-grid system stability.

4. Current Controller Design

4.1. Harmonics Gain Definition for the PMR Controller

The current controller has been designed in the fixed α,β frame, based on the model presented in the section above. According to [27], two sources of low frequency current harmonics must be taken into account: nonlinear effects in the current control loop, mainly the switching dead time [28] and the grid voltage harmonics [29,30] typically produced by nonlinear loads. In a three-phase system, the most annoying harmonics caused by both of the aforementioned effects are the lower order ones. The 5th, 7th, 11th, and 13th harmonics were selected to be compensated by the proposed multi-resonant controller.

The damping factor ζ is related to the quality factor Q of a resonance, and it is an indicator of the ratio between the resonance frequency and the bandwidth B_W as described in Equation (10).

$$Q=\frac{{\omega }_0}{B_W}=\frac{1}{2\zeta }$$

(10)

An identical damping factor ζ of 0.01 was chosen for each selected resonance of the PMR controller. This assures that the bandwidth of each resonance covers a frequency deviation of ±1%. This is the admissible frequency deviation defined by the European standard EN 50160 [31].

The resonance peak of each harmonic compensator was tied to the peak gain of the fundamental resonance, then a reduction of the gain k_r,n inversely proportional to the harmonic order was applied. This is illustrated by Equations (11) and (12).

$${\omega }_n=n{\omega }_0$$

(11)

$$k_{r,n}=\frac{k_{r,1}}{n}.$$

(12)

There are three main reasons for this choice. Firstly, the number of parameters to be separately tuned is reduced, which significantly simplifies the optimization of the controller. Secondly, a decreasing impact on the phase response is achieved, as a resonance is located closer to the cutoff frequency of the system. This leads to an increased phase margin. Thirdly, typically the amplitude of the aforementioned harmonics decreases with increasing order. Therefore, a lower controller gain for compensation of higher order harmonics can be applied.

Figure 6 shows the bode plot of two PMR controllers. The first one has a constant gain k_r,n for each resonance, while the second controller exhibits a reduction of the resonant gain as described in Equations (11) and (12). The decreasing impact of higher order resonances on the phase response in the second case can be noted.

4.2. Sensitivity Functions and Minimum Distances

The presence of multiple resonances must be considered carefully in the design procedure of the PMR controller, since the frequency response can approach 0 dB at several frequencies. The same is true for the phase response and the −180° limit. In order to achieve a robust control design, a procedure relying on the open loop Nyquist plot was applied with the aim to increase the minimum distance of the trajectory to the critical point [32]. The minimum distance of the Nyquist trajectory to the critical point can easily be traced making use of the systems’ sensitivity function described by Equation (13). The minimum distance corresponds to the inverse of the sensitivity functions maximum gain, which is equal to its infinity norm as described by Equation (14).

$$S_0\left(s\right)=\frac{1}{1+G_o\left(s\right)}$$

(13)

$$\frac{1}{{\eta }_0}=\underset{0\le \omega <\infty }{\max }\left|S_0\left(j\omega \right)\right|={\parallel S}_0{\left(s\right)\parallel }_{\infty }.$$

(14)

Similarly, the minimal distance η_z is expressed as follows:

$$S_z\left(s\right)=\frac{1}{1+\frac{Z_g\left(s\right)}{Z_{inv}\left(s\right)}}$$

(15)

$$\frac{1}{{\eta }_z}=\underset{0\le \omega <\infty }{\max }\left|S_z\left(j\omega \right)\right|={\parallel S}_z{\left(s\right)\parallel }_{\infty }.$$

(16)

4.3. Iterative Design Procedure

An iterative design procedure has been developed, as described by the flowchart in Figure 7. It must be highlighted that this procedure can be applied to a PR controller as well.

After the initialization of the optimization parameters (step 1), the grid inductance L_g,ref, the k_p and k_r,1 range and steps, the damping factor ζ value, and the minimal critical distances η_0,min and η_z,min are stated. In a second step, an iterative algorithm for each combination of k_p and k_r,1 checks if the critical distance η₀ is higher than minimum value η_0,min. Then, the algorithm calculates the maximal grid inductance value satisfying the critical distance η_z and the Nyquist criterion (step 3). If a set of parameters agrees with one of the three aforementioned criteria, the algorithm stores the k_p and k_r,1 combination as a valid set for the PMR controller. In the fourth and final step, the results of optimization procedure are displayed in a 2-D surface plot.

4.4. Example of Iterative Design Procedure Results

Figure 8 presents the optimization results for a 130 µH grid inductive impedance and a given set of ζ, η_0,min, and η_z,min. Green lines limit the surface where the values of k_p and k_r,1 respect all the given criteria. As can be seen, multiple choice of k_p and k_r,1 parameters fulfill the design requirements. The final choice is made by choosing the parameter’s pair with the maximum distance from the origin. In this case, this is satisfied by k_p = 0.064 and k_r,1 = 7.42, which guarantees the wider bandwidth. More details on the influence of the three optimization criteria are presented in the next section.

In order to choose the highest values for k_p and k_r,1, the parameters in

Table 2. Table 2. Parameters of the tested controllers.

Parameter	PMR Controller	PR Controller
kp	0.064	0.054
kr,1	7.43	38.6
ζ	0.01	0.01

where selected. The table shows also the parameters obtained with the PMR controller. The PR controller has been designed with the same iterative design procedure.

4.5. Influence of the Optimization Criteria

4.5.1. Influence of η₀

Figure 9 illustrates the influence of the parameters k_p and k_r,1 on the minimum distance η₀ of the open loops transfer function. In this case, we utilize a 3-D representation of the results, namely the minimal distance η₀, k_p, and k_r,1. The black plane represents the minimum of η₀ = 0.3 that was imposed for the design. The distance η₀ is maximum for a proportional gain k_p = 0.04 and a resonant gain k_r,1 = 3.25. However, k_p and k_r,1 should be as high as possible to achieve minimal steady state error at the fundamental frequency, a dynamic transient response, and maximum harmonic rejection. As can be seen, there is a change of the slope of the η₀ surface towards the right corner of the graph. This is due to the fact that the Nyquist plot encircles the critical point for a low proportional gain k_p and increasing resonant gain k_r,1. Of course, these combinations are not valid since they clearly lead to an unstable system.

Figure 10 shows the Bode plot of the open loop transfer function of the VSI with the PMR controller. Below 1 kHz, the transfer function behavior is mainly influenced by the controller. The phase starts to drop significantly above 1 kHz due to the computation delay and the delay caused by the SVM. The resonance at 5.8 kHz is caused by the LCL filter.

4.5.2. Influence of η_z

The ratio Z_g(s)/Z_inv(s) between the grid impedance and the inverter impedance is an indicator of the stability of the grid-connected VSI [27,33]. Figure 11 shows the Bode plot of Z_inv(s) and Z_g(s) for a strong grid with an inductance of 130 µH.

The stability criterion discussed in [27,33] dictates that the phase difference of Z_inv(s) and Z_g(s) must not exceed 180° at the frequency where their magnitudes intersect. The intersection of |Z_inv(s)| and |Z_g(s)| in Figure 11 occurs around the resonance of the LCL filter; i.e., as far as stability is concerned, the filter is the critical element under varying grid conditions. To evaluate the robustness of the system with respect to grid impedance variation, the theory of the minimum distance was applied to the transfer function Z_g(s)/Z_inv(s). Figure 12 shows the minimal distance η_z as a function of k_p and k_r,1. A high gain k_p decreases stability significantly, whereas the influence of k_r,1 on the impedance-based stability criterion is by far less important. Again, the black plane indicates the minimum of 0.3 that was imposed.

4.5.3. Influence of L_g

As mentioned before, grid impedance L_g plays a major role for the inverter-grid system stability. In Figure 13, we assume a grid impedance reference as a pure inductor of 130 µH. According to [27], the theoretical maximal grid impedance value for a stable operation of grid-inverter system is about 320 µH, which is quite low for a grid-connected inverter. The main reason is the limited damping of the passive LCL filter. Figure 13 may be used in conjunction with Figure 8 to select the set of controller parameters, as an alternative to the proposed maximum distance criteria (see section example of the iterative design procedure results).

5. Experimental Results

The aim of the tests was mainly to compare PR and PMR controllers. The controller transfer function was discretized using the Tustin method. In order to reduce the influence of the computational delay, the inverter sampling time has been shifted by half of a PWM period. First of all, PR and PMR controller behavior (see Table 2) was tested with a measured harmonic grid voltage distortion of 2.47%. After that, additional voltage harmonics of the 5th, 7th, 11th, and 13th order were added to the grid voltage using a three-phase low voltage serial regulator [34]. For the second scenario, a harmonic voltage distortion of 6.83% was reached. The tests were carried out with a grid inductance of 130 µH. Figure 14 and Figure 15 show the results of the first and second scenarios. The corresponding current spectra are shown in Figure 16. Even with the linear reduction of the resonance gain for higher order harmonics (see the section on harmonics gain definition), the resonant controller achieves a significant reduction in 11th- and 13th-order harmonics.

Table 3. Harmonic current distortion obtained with the different control strategies.

Voltage Distortion	Current Distortion
	PR	PMR
2.47%	2.41%	1.39%
6.83%	5.34%	1.47%

shows the measured total harmonic distortion with the different control structures considering harmonics up to 2 kHz.

6. Conclusions

The PMR controller is a well-known, easy, and an effective strategy to reduce current harmonics of a current-controlled VSI in a selective manner. It targets both harmonics caused by grid voltage distortion and harmonics caused by non-linear effects in the current control loop of the inverter itself. In order to reduce the number of parameters to be tuned, the choice of harmonic resonant coefficients has been made to be inversely proportional to the value of the fundamental harmonic coefficient. Then the design of the current controller has been performed by taking into account the inverter-grid stability issue. The obtained controller was implemented and successfully tested in a 10 kW three-phase grid-connected VSI with an LCL filter. Depending on grid voltage conditions, current harmonic distortion was reduced by 1.0 or 3.9 percentage points, respectively, compared to a PR controller.