Smart Nonlinear Energy Feedback Power Electronic Load to Simulate Power Quality Problems

This paper proposes a design approach for an energy feedback power electronic load (PEL) based on back-to-back voltage source PWM converters. It can simulate not only constant linear loads but also a class of nonlinear loads to simulate power quality problems. The rectifier side and the inverter side are decoupled in control by using a DC capacitor. The rectifier side is responsible for tracking designed current commands to achieve the designed load characteristics, and based on the characteristics of the current commands, a multi-resonant controller plus state feedback controller was designed by using the internal model principle. The inverter side employs an LCL filter with active damping to control the phase of the grid-connected current, enabling energy to feed back to the grid. A simulation model of this PEL is established in Simulink software, and the simulation results verify the effectiveness of the proposed control strategy.


Introduction
PEL was initially designed to assess the load capacity of power supplies [1].Once a traditional resistive load is designed and finalized, its voltage, current, capacity, and other parameters are fixed, making it suitable only for a specific voltage level and exhibiting low automation.Moreover, resistive loads consume a significant amount of electrical energy, and in most cases, this energy is dissipated as heat, resulting in substantial energy wastage.This has led to the emergence of PEL, which can meet various testing requirements for power devices [2][3][4].These loads utilize grid-connected inverters to feed back the energy absorbed from the power source to the grid, enabling the conservation and recycling of the energy.Given the aforementioned advantages of PEL, this paper aims to design a PEL capable of generating specified harmonics.This load could be used to simulate different power quality problems.
This paper first analyses and determines the topology of a back-to-back voltage source PWM converter with a DC link in the middle.In terms of rectifier design, this paper focuses on converting the load simulation into a current tracking problem.While methods for simulating linear loads or constant currents are well-established, various approaches exist for simulating nonlinear loads with harmonics.In [5], traditional dual-loop control is indicated that it has a limited ability to suppress nonlinear loads since the current inner loop requires high speed.In addition, this control method exhibits significant steady-state errors under heavy load conditions.The improved dual-loop strategy using voltage RMS feedback operates between 0 frequency and the base frequency but is less effective for tracking high-frequency harmonic signals when simulating nonlinear loads.In [6], hysteresis current control is utilized to calculate the reference current algorithmically by using numerical methods, but it can only simulate loads with impedance or constant sinusoidal currents.It struggles to track nonlinear loads with significant harmonic currents.In [7], a PI controller is employed to combine 2 with a multi-resonant voltage controller to suppress voltage harmonics.However, in [8], it is pointed out that using a multi-resonant controller with excessive gain leads to system instability.Thus, combining virtual harmonic impedance with the multi-resonant controller partially mitigates the voltage drop across the output harmonic impedance.However, these methods are designed for a limited number of harmonic frequencies, which imposes certain requirements on the quality of the harmonics themselves.In [9], a repetitive plus PI controller is used to track the current, aiming to address the poor performance of traditional PI controllers in tracking currents with rich harmonic content.However, the compensation stage required for system stability in repetitive control is complex, and its dynamic response performance is poor.There is likely no universally applicable solution for tracking the current of nonlinear loads.This paper starts from the characteristics of a specific type of nonlinear load and designs a tracking method based on the internal model principle that conforms to certain characteristics.
In terms of inverter design, grid-connected inverters can generally be classified into two types based on the output filter: L-type and LCL-type.The former belongs to a first-order system with simple control, but it has limited ability to suppress high-frequency harmonics and usually requires larger inductance to meet harmonic standards.The latter belongs to a third-order system and exhibits stronger suppression of high-frequency harmonics, requiring smaller inductance [10].Therefore, LCL filters are commonly used in the research of grid-tied inverters.To address the resonance issues in LCL filters, a resistor can be added in series with the capacitor branch to directly dampen the resonance peak.However, the introduced actual resistance significantly increases energy losses.Thus, in [11], active damping is introduced to reduce the resonance peak of LCL filters, which is equivalent to modifying the feedback loop of the transfer function.The added resistance is virtual and does not incur additional losses.In [12], the value range of the grid current regulator parameters and capacitor current feedback coefficient is determined based on requirements for grid current steady-state error, system loop phase margin, and gain margin.Suitable closed-loop control parameters are then selected from this range.It is worth mentioning that the design of grid-connected inverters differs from that of the rectifier side.Grid-connected inverters have higher requirements for stability and grid-connected current quality.Although the PR controller can achieve zero steady-state tracking error for sinusoidal commands, it has a bad influence on system stability.On the other hand, while the PI controller cannot eliminate steady-state error, its quality is sufficient to meet the quality requirements of grid-connected current.However, both of these controllers are designed for stable DC conditions or minor disturbances, making it challenging to ensure strict tracking of the current phase when there are continuous long-term variations on the DC side.This paper improves the design method for LCL gridtied inverters with active damping and develops an inverter control strategy for continuous changes on the DC side.
The simulation results demonstrate that the PEL designed in this paper can simulate continuous variations in the load, allowing for continuous adjustment of impedance and current.It can also simulate nonlinear loads with certain harmonics.Additionally, the energy absorbed from the voltage source is fed back into the grid, enabling the conservation and recycling of the energy.

PEL topology
The input of PEL is the AC voltage source and the output of PEL is the low-voltage grid.On the input side, the right part of the PEL should exhibit the desired impedance value.On the output side, it is necessary to control the phase of grid-connected current to track the phase of grid voltage.In this study, two voltage source PWM rectifiers are employed to operate in rectification and inversion states respectively, with a DC capacitor in between to form an AC-DC-AC conversion structure.The structure is shown in Figure 1.

Figure 1. PEL topology
In Figure 1, U is the AC voltage source, and Z1 is the impendence of the input side, which consists of an inductor and its parasitic resistance.The inductor is used as the energy storage element of the boost circuit in the voltage source rectifier to realize the voltage boost on the DC side.S11~S14 act as control devices for simulating different characteristics of the load.C is the support capacity of the DC side, and it acts as a bridge between the rectifier side and the inverter side.When the PEL is working, the DC voltage is constant, and the input power and output power are exchanged and balanced here [13].S21~S24 act as control devices for the grid-connected converter.L1, L2, and C1 consist of the LCL filter, Zg is the equivalent impedance of the grid side, and Ug is the voltage of the grid [14].
The rectifier side and the inverter side are coupled.The input power from the voltage source will inevitably increase the voltage of the capacitor.However, the inverter side reduces the voltage of the capacitor by feeding back energy to the grid.The interaction between the PWM converters on both sides ensures that the voltage of the DC side capacitor ultimately remains at a relatively fixed value.A large-capacity capacitor can suppress the DC voltage variation caused by energy changes on both sides, thus approximately achieving the decoupling of the rectifier and inverter side controls.Although the control variables in this article are AC quantities on two sides, their control quality is closely related to the DC voltage.The specific design of the DC capacitor is not the focus of this article.Therefore, it is advisable to use the capacitor with the largest possible capacity to suppress DC voltage fluctuations while satisfying certain system requirements for rapid response.

Control strategy for the rectifier
VSR control strategy generally adopts a dual-loop control with the voltage outer loop and current inner loop [15].The current control is subordinate to voltage control.However, in this paper, the control object is current, so it is not appropriate to use the general dual-loop control strategy.Based on Figure 1, the KVL equation for the rectification side can be derived: Therefore, by knowing the impedance characteristics of the desired load, the corresponding current values can be analytically obtained.This allows the simulation of the load to be converted into tracking a given current command.This means that theoretically if any current value can be fully tracked, it is possible to simulate loads with any characteristics.However, according to the internal model principle, there is no such universal controller.Therefore, it is necessary to design different controllers for different load characteristics to achieve better current tracking performance.
When simulating a constant impedance load, the required current to be tracked is sinusoidal.The control flow diagram could be represented in Figure 2.
It can be observed that the transfer function has an infinite gain at the frequency  , which achieves a unity closed-loop transfer function.However, it has a small phase margin and narrow bandwidth, which can easily cause stability issues.Therefore, in practical control, a quasi-PR controller is commonly used, where the transfer function is modified as follows: where c  is a control parameter which is related to the system's crossover frequency and bandwidth.
The infinite magnitude at the fundamental frequency of the PR controller is unnecessary.The quasi-PR controller compromises through a relatively large magnitude at the fundamental frequency to achieve better stability of the system.It is worth mentioning that the impedance of the load cannot be arbitrarily chosen because the rectifier side and the inverter side are coupled.The arbitrariness of the control on the rectifier side can affect the inability of the inverter side to achieve the unity power factor grid connection.The power transmission of the rectifier is carried out from the AC side grid to the DC side capacitor.It is the precise direction of this power transmission that determines the limitation on the choice of load values.From a power balance perspective, the input power Pin is equal to the power consumed by the capacitor: It can be viewed as a first-order differential equation for Ud 2 .We ignore the load R and integrate both sides: This equation consists of three parts: the steady-state DC part, the steady-state sinusoidal part at twice the frequency, and the changing DC part.The above equation is obtained by ignoring the load resistance R, assuming that the capacitor no longer has a continuous discharge link to the load R. Therefore, the DC part should increase over time, meaning that the coefficient of changing the DC part is greater than 0: 1 arccos( ) This limit of current simulation corresponds to the limit of load simulation: When the load is linear, the PR controller can effectively track the sinusoidal current.However, if the load is nonlinear and the current contains a significant amount of harmonics, using the PR controller will excessively suppress the harmonic components outside the fundamental frequency, making it unable to simulate the nonlinear load accurately.It is assumed that the desired tracking nonlinear current command can be represented as the sum of a periodic command and a non-periodic command, so the state-space expression of the rectifier is given by: As to the non-periodic command, the model of reference input can be equivalent as: If there exists r (n) (t)=0, we will consider the tracking error e=y-r.If e (n) =y (n) -r (n) =Cx (n) , we will get: ( [ , ,..

., ]
n T e e e   P We have: (1) According to the state feedback design principles [16,17], there exists a kind of state feedback that stabilizes the system and achieves zero steady-state error tracking: (1) We integrate the above equation n times: This implies that the internal model of the input command can be introduced to form the feedback control law for achieving zero steady-state tracking error of the command [18].This control law can be denoted as Gu.Similarly, for periodic nonlinear commands, they can always be represented by Fourier transform as follows: To achieve zero steady-state tracking of Equation ( 17), the controller should have tremendous gain in the multiples of the fundamental frequency range.The transfer function of the controller can be expressed as Equation (18).However, it should be noted that this controller design method is intended for specific harmonics with significant magnitudes.If there are multiple harmonics with small magnitudes, the controller's order will increase, and the error signals in various frequency ranges will also be amplified accordingly.
Therefore, the overall control strategy designed based on this internal model principle should be as shown in the block diagram in Figure 3.

Control strategy for the inverter
If the control on the rectifier side focuses on high-quality tracking of the reference current command, the inverter side is responsible for precise control of the grid current phase.The inverter side is responsible for feeding energy back to the grid, which imposes higher requirements on the control quality of the grid current.At this point, it differs significantly from general current control strategies, mainly in the coupling between the DC capacitor voltage and the control relationship on the input side of the rectifier.This is because the amplitude of the grid current essentially reflects the power provided by the rectifier side.The larger the power provided by the rectifier side is, the more energy the inverter side needs to feed back to the grid, resulting in a higher amplitude of the grid current.
The amplitude of the grid current is determined by the simulated load on the rectifier side, and they are numerically coupled through power conservation, making it hard to know their exact values in advance.Therefore, what needs to be strictly controlled here is making the grid current phase equal to the grid voltage phase.Only when the system transitions to a steady state and the capacitance value is chosen such that the DC voltage fluctuation is small under steady-state conditions, can it be approximately assumed that the amplitude of the grid current remains constant.Therefore, the control of PEL on the inverter side needs to transform the control of constant current in a conventional inverter into the control of constant current phase with variable current amplitude.The control of current amplitude essentially reflects the control of the DC voltage.
Therefore, its control strategy is similar to the control of current on the rectifier side, but the amplitude of the current needs to be determined by the actual value of the DC voltage.If the actual value of the capacitor voltage is lower than the setpoint, it indicates that the capacitor is discharging too rapidly, and the amplitude of the grid current needs to be reduced to decrease the power fed back to the grid.Figure 4 below shows the control block diagram for constant current control [19].The feedback loop H1 is added to suppress the resonant peak of the LCL filter and it incorporates active damping, which effectively introduces a virtual resistor to avoid energy losses associated with the addition of a real resistor.In terms of controller selection, although the PR controller can track sinusoidal commands with zero steady-state error, it adversely affects system stability.Furthermore, in power electronic load design, fluctuations in the DC voltage can further exacerbate system instability, making the use of a PR controller undesirable.On the other hand, the PI controller, while inevitably introducing steady-state error, can still meet the requirements for grid current quality through proper parameter design.Regarding the reference current setpoint for power electronic loads, considering that the current amplitude varies with the DC voltage, the reference current amplitude can be determined accordingly.This results in a modified version of Figure 4, as shown in Figure 5.

Simulation and analysis
A simulation model of the aforementioned PEL was developed by using Simulink.The specific parameters of the model are listed in Table 1.While simulating the constant linear load, three typical impedance types, namely inductive and resistive load, capacitive and resistive load, and resistive load, are denoted as ZL1=4.5-j3.0,ZL2=4.5+j3.0, and ZL3=5.2+j0,respectively.The current waveform on the rectifier side is shown in Figure 6, while the grid-connected current on the inverter side is shown in Figure 7.It can be observed that the current can achieve nearly zero error tracking, and the grid-connected current meets the phase requirements.According to equation, ( 9)considering the approximate limitation of the simulated load, the current tracking is shown in Figure 8 and the grid-connected current on the inverter side is shown in Figure 9.It can be observed that the waveform of the grid-connected current is severely distorted.This is because in this extreme case, the DC voltage is difficult to maintain, and the large fluctuations on the DC side will significantly affect the control quality on the inverter side.When simulating a nonlinear load with a tracking current consisting mainly of 3rd and 5th harmonic components, the current tracking of the rectifier side is shown in Figure 10.The gridconnected current on the inverter side is shown in Figure 11.It shows that the current can achieve nearly zero error tracking, and the phase of the grid-connected current meets the requirements.

Conclusion
This paper presents the design of an energy feedback PEL that can simulate nonlinear loads to emulate power quality issues.The following conclusions are drawn: The PEL designed above can accurately simulate load characteristics over a wide range, ensuring high-quality load simulation.
 By employing the internal model principle for controller design, tracking of a specific class of nonlinear commands can be achieved with high quality;  By controlling the current on the simulated load side, a harmonic power source can be easily simulated, thus replicating a certain level of power quality issues;  The LCL grid-connected inverter, utilizing an improved grid current control strategy, achieves high-quality grid connection throughout all operating stages;  This research expands the application of PEL and achieves a high level of automation, enabling their use not only as a common load but also as harmonic current sources that can generate predefined power quality issues.

Figure 2 .
Figure 2. Block diagram of sinusoidal current tracking

Figure 3 .
Figure 3. Block diagram of complex nonlinear current tracking strategy

Figure 4 .
Figure 4. Block diagram of the inverter

Figure 5 .
Figure 5. Modified block diagram of the inverter

Figure 6 .
Figure 6.Rectifier side current tracking Figure 7. Grid-connected current with with constant linear load constant linear load

Figure 8 .
Figure 8. Rectifier side current tracking Figure 9. Grid-connected current under load limitation under load limitation

Figure 10 .Figure 11 .
Figure 10.Rectifier side current tracking under nonlinear load PR controller can realize zero steady-state error tracking of the sinusoidal command, and its transfer function can be expressed as follows: