Dynamic Performance Improvement of a Single-Phase VSI with Digital Implementation of an On-Line Optimal Trajectory Control Algorithm

Zaman, Shah; Zhang, Yan; Liu, Jinjun; Yang, Danni; Li, Zhenchao

doi:https://doi.org/10.1155/2023/6902796

International Transactions on Electrical Energy Systems

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

High-frequency Converters: Design, Control, and Applications

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Research Article | Open Access

Volume 2023 | Article ID 6902796 | https://doi.org/10.1155/2023/6902796

Dynamic Performance Improvement of a Single-Phase VSI with Digital Implementation of an On-Line Optimal Trajectory Control Algorithm

Shah Zaman,¹Yan Zhang,¹Jinjun Liu,¹Danni Yang,¹and Zhenchao Li¹

Academic Editor: Anjanee Kumar Mishra

Received01 Dec 2022

Revised23 Jan 2023

Accepted05 Apr 2023

Published20 Apr 2023

Abstract

A single-phase voltage source inverter (VSI) still suffers from a long settling time and significant voltage overshoot/undershoot under the abrupt step-change of load current. This article analyzes the comprehensive large-signal dynamic process and limitation of the linear controller during step change in load current. The combination of linear and nonlinear control is presented which is a realistic, simple, and low-cost technique for enhancing the dynamic response of the single-phase VSI. A nonlinear controller based on the simplified capacitor charge balance algorithm is employed to drive the inductor current and capacitor voltage to attain the target value by precisely following the projected optimal trajectory during the transient process in a brief period. A linear PI controller is utilized during the steady-state operations such as input voltage variations, temperature drifts, and component ageing. The complete mathematical derivation as well as a design method is presented to guide the practical hardware implementation for the given main circuit parameters and identified load step change. Even though accurate time instants can be obtained using off-line numerical calculations, reasonable simplifications are provided for real-time practical engineering applications in DSP. Finally, simulation and experimental verifications prove that the single-phase VSI achieves a substantial settling time decrease and reduced voltage oscillations.

1. Introduction

Single-phase alternating current power supplies are commonly used in applications such as active power filters, solar power generating systems, and laboratory testing, including power equipment research and development and electrical manufacturing. When delivering a load that fluctuates too quickly between many steady-state locations, most single-phase VSIs still have a lengthy settling period and severe voltage overshoot/undershoot. Under a 10% to 80% load step change, the usual transient recovery time is larger than 2 ms, which does not match the high-precision test requirement. As a result, significant research efforts have been directed into improving the dynamic performance of high-precision single-phase AC power sources delivering the impact load in recent years [1, 2].

Traditional linear control techniques, such as PI and PR [3–7], are best suited for situations with gradual load changes. However, in applications that need quick and precise current monitoring, the dynamic response of these kinds of current controllers is often poor. A nonlinear control technique is intensively investigated to increase the dynamic performance because it responds too rapidly to transients. Nevertheless, the majority of them may lead to undesirable performance, such as nonzero steady-state error and varying switching frequency. For example, the critical parameters of deadbeat [8] are developed considering the particular information of load conditions. It is challenging to calculate these characteristics when there are large fluctuations in the load. The hysteresis control [9–11] provides virtually instantaneous reference tracking and is impervious to instability problems but at the expense of variable switching frequency and higher control complexity. Dead time affects current-tracking precision, switching frequency, and duty cycle range for parabolic current-controlled VSIs [12, 13]. The advantages of sliding mode control (SMC) [14–16] include rapid dynamic reaction, rejection of external disturbances, and insensitivity to parameter fluctuations. However, it is difficult to obtain a suitable sliding surface. In addition, boundary control (BC) [17–19] and H-infinity [20] are options for enhancing the dynamic performance. BC and H-infinity are effective but hard to implement.

Combining linear and nonlinear control techniques (PI + non-linear) is used to fully use their benefits, using a nonlinear controller during the transient and a linear controller in the steady state to speed up the transient behaviour and preserve strong steady-state performance. This nonlinear control approach forecasts the best trajectory for single-phase inverters under varying loads. It determines the appropriate switching sequences to force the converter to move from one steady state to another to regain lost charge. This can be performed by fixing the duty cycle high (for a step-up change) and low (for a step-down change) for a specified interval and followed by low (for a step-up change) and high (for a step-down change) for the additional interval. Thus, the controller enhances the dynamic response (reduced settling time and minimal output voltage undershoot/overshoot) undergoing fast load changes.

In the literature [21, 22], a straightforward design strategy for traditional DC-DC converters was given. The capacitor charge balance control (CBC) method is used to calculate the on- and off-state time intervals of power devices [21, 23–31]. It has been shown that the quickest transient response may be attained in a single switching period. And its fundamental concept is to estimate the ideal dynamic trajectory and establish the precise switching sequence of the power device to drive the converter from one steady-state to another in response to rapid load shift. Under conditions of rapid load changes, the single-phase VSI’s output voltage and inductor current fluctuate frequently. As a result, enhanced CBC is offered since the traditional CBC strategy cannot be utilized to the single-phase VSI to boost dynamic response. This article first analyzes the comprehensive large signal dynamic model and transient response limitation of the single-phase VSI in Section 2. Section 3 proposes an on-line trajectory approach for properly monitoring the projected ideal trajectory during the transient phase, therefore causing the inductor current to achieve the desired value. The method for designing system control and the precise mathematical derivation are both provided in Section 4. Section 5 provides examples of both simulation and experimental validation. The last portion is the conclusion.

2. Single-phase VSI Load Step Challenges

The basic circuit of a single-phase full-bridge VSI is seen in Figure 1, and it supplies the pure resistive load R_L1 with the help of an LC filter. The additional switch S_d simulates abrupt load change by connecting or disconnecting R_L2.

As an example of fundamental analysis, a single-phase VSI with the bipolar pulse width modulation (PWM) approach is chosen and modelled for the research. Two pairs of switches (S_ap and S_bn) and (S_bp and S_an) work in complementary states. According to the switching state, there are two equivalent circuits, as shown in Figure 2. When S_ap = S_bn = ON and S_bp = S_an = OFF, the energy flows through S_ap and S_bn to supply R_L1 via the LC filter as demonstrated in Figure 2(a). When S_bp = S_an = ON and S_ap = S_bn = OFF, the power source energy flows through S_bp and S_an to provide R_L1 via the LC filter, as shown in Figure 2(b).

(a)

(b)

Figure 3 depicts the transient response waveform of the single-phase VSI with a linear controller when the load current is abruptly increased or decreased. As seen by the red waveform, i_load quickly changes from its starting value I_o1sin(θ) to I_o2sin(θ) when S_d is turned on at θ₀. However, the inductor current i_L cannot fluctuate too rapidly to match the required load current. Cf thus compensates the transitory current. As for the positive load current change shown in Figure 3(a), the error, sampling voltage feedback signal compared with , is amplified to increase i_ref. i_L is slowly growing. Until θ₁ when i_L equals i_load, i_load is still higher than i_L, and continues to decline.

(a)

(b)

After θ₁, i_L keeps growing and is higher than i_loadC_f when it begins recharging. rises and exceeds the reference value. At θ₂, i_L declines and equals to i_load again. C_f begins discharging, and tends to be close to the reference value. Until θ₃, both i_L and are close to I_o2sin(θ) and and the new steady-state single-phase VSI is reached.

Most of the time, this fluctuation process continues for a long time until it recovers from one steady-state to another steady-state after a sudden change in load. A good tradeoff between the voltage/current overshoot/undershoot and dynamic regulation time should be considered for the optimal parameter design of the linear controller. Furthermore, the dynamic regulation time is increased too much due to the sudden load step-change magnitude under the relatively low switching frequency.

3. On-Line Trajectory Control for the Single-Phase VSI

As seen in Section 2, the single-phase VSI with a standard controller still exhibits a long settling period, and substantial voltage overshoots and undershoots in response to rapid changes in load current. To overcome the challenge, an on-line optimal trajectory control as an improved nonlinear predictive digital control method is proposed to improve the dynamic response as the problem terminator [32]. The fundamental concept is identifying optimal trajectory by examining the large signal dynamic process when the load changes abruptly. In response to load variation, the projected ideal waveforms of capacitor voltage and inductor current from the initial steady state to the final steady-state are shown in Figure 4.

(a)

(b)

The suggested controller incorporates two distinct controls: a linear controller for steady-state operation and a nonlinear controller for transient situations. The capacitor current will provide the necessary current since the inductor current does not vary immediately in response to load current variations. The capacitor charge balance theory states that the charge supplied to the capacitor S₁ (discharge) at the end of the transient cycle must equal the charge withdrawn from the capacitor S₂ (charge), as shown in Figure 4(a). As S₁ has been reduced, the recharging part S₂ has also been reduced. The time required to charge S2 must be reduced in order to decrease the settling time. According to Figure 4(a), the time necessary to recharge the capacitor t_ch may be shortened by increasing the height h of the S₂ triangle (defined by θ₀–θ₂); this is achieved by fixing the duty ratio to 100% for θ₀–θ₂ and 0% for θ₂-θ₃.

The recommended controller is intended to do the following:(i)Determine whether there is an abrupt load shift.(ii)Fix the duty ratio to 100% (θ₀–θ₂) and 0% (θ₂-θ₃) for a step-up change in a positive half cycle; for a step-down change, fix the duty ratio to 0% (θ₀–θ₂) and 100% (θ₂-θ₃) in a negative half cycle.(iii)Fix the duty ratio to 0% (θ₀–θ₂) and 100% (θ₂-θ₃) for a step-down change in a positive half cycle; for a step-up change, set the duty ratio to 100% (θ₀–θ₂) and 0% (θ₂-θ₃) in a negative half cycle.(iv)Switch back to the linear controller once again.

Figure 4 shows the projected ideal waveforms of i_L and throughout the transient process with positive and negative load current fluctuations. The trainset regulation process is divided into three-time intervals for detailed analysis.

The red waveform depicts the instantaneous transition in i_load from I_o1sin(θ) to I_o2sin(θ) when Sd is activated at θ₀. i_L cannot vary rapidly to maintain the required load current at θ₀. To speed up the transient, once the positive step change of i_load is detected, as shown in Figure 4(a), S_ap = S_bn = ON and S_an = S_bp = OFF are set instantly to rise i_L. Until θ_1, when i_L = I_o2sin(θ₁), declines to deliver the necessary load current. After θ₁, i_L rises higher than I_o2sin(θ), and begins to rise. The inductor current rising slope during θ₁ and θ₂ may be described using the equivalent circuit depicted in Figure 2(a) as follows:

At θ₂, S_an = S_bp = ON, S_ap = S_bn = OFF, and i_L begin to decline as grows. At θ₃, i_L is close to the load current, and is back to the reference voltage. Both i_L and enter the quasi-steady state. The inductor current falling slope through (θ₂ and θ₃) is derived in (2) using the equivalent circuit depicted in Figure 2(b).

In the same way, the analysis described above can also be performed for single-phase VSIs with the optimal trajectory controller in response to a negative step change, as shown in Figure 4(b) which is as follows:

As seen in Figure 4, the proposed controller ensures that iL and vCf arrive at the desired value by precisely following the projected best trajectory while simultaneously reducing the settling time and voltage fluctuations that occur throughout the transient operation. Figure 5 indicates the operation diagram of the proposed predictive control algorithm in conjunction with the linear regulator. A digital control chip makes it easier to combine linear and nonlinear control algorithms to generate the PWM signal. All the transient switching state sequence and ON/OFF state time duration under the different load change conditions are calculated using on-line calculation or prestored off-line results as a look-up table in the on-chip flash. Furthermore, the linear controller parameters can be efficiently designed with relatively high bandwidth to handle the slow load change condition. Once the abrupt change in load current is noticed, the nonlinear controller quickly bypasses the linear regulator and controls the power devices in one switching period.

4. Mathematical Derivation and Controller Design

The optimal trajectory control technique is implemented in four steps based on comprehensive theoretical research and mathematical derivation.

Step 1. Detection of load current step change at θ₀.
After detecting an abrupt change in load current, the on-line trajectory control module halts the linear controller. The power devices are instantly switched. S_ap = S_bn = ON and S_an = S_bp = OFF are fixed for step up, and S_an = S_bp = ON and S_ap = S_bn = OFF are fixed for step down.

Step 2. Calculate θ₁ and capacitor discharge portion S₁.
As shown in Figure 4(a), i_L goes up from I_o1sin (θ₀) to I_o2sin (θ₁), and C_f discharges between θ₀ and θ₁.
As demonstrated in Figure 4(a), i_L is rising from I_o1sin (θ₀) to I_o2sin(θ₁), and C_f is discharging during θ₀ and θ₁. Based on the current increasing slope coefficient k₁ expressed in (1), i_L meetsThe red waveform in Figure 4(a) shows that the load current does not significantly alter after the rapid step shift between θ₀ and θ₁, so I_o2sinθ₁≈I_o2sinθ₀. Equation (3) can be simplified, and θ₁ is calculated as follows:The green-shaded region S₁, representing the charge discharged by C_f, may be estimated using elementary geometric theory.

Step 3. Calculate θ₂, θ₃, and capacitor charging portion S₂.
At θ₂, as presented in Figure 4(a), S_ap = S_bn = OFF and S_an = S_bp = ON are set. i_L begins to decline at the slope of k₂, and C_f starts to increase till θ₃. Throughout the transient θ₀ and θ₃, i_L encountersBased on the green-shaded portion S₂, the charge absorbed by C_f can be calculated as follows:To ensure that both i_L and approach the region of the new quasi-steady state, the C_f released charge, represented by the green-shaded area S₁ during θ₀ and θ₁, should be equal to the absorbed charge, represented by S₂ during θ₁ and θ₃ and S₁ = S₂ as follows:Apparently, equations (6) and (8) are nonlinear equations that include two unknown variables, θ₂ and θ₃. It is easy to get the numerical solution of θ₂ and θ₃ using computer-assisted software such as Matlab or Mathcad.
Even though the accurate time instants θ₂ and θ₃ can be obtained using off-line numerical calculation, the on-line calculation is more suitable for practical engineering applications using a digital signal processor (DSP). Theoretical research and simulation demonstrate that the transient regulation time interval (θ₃–θ₀) is too short, so the load current can seem like a constant value. To simplify the calculation and apply the aforementioned optimal trajectory control algorithm in the C2000 DSP, some reasonable assumptions (I_o2sinθ₃≈I_o2sinθ₂≈I_o2sinθ₁) are made to simplify the analysis. Equations (6) and (8) are rewritten as follows:By solving (9), θ₂ and θ₃ are as follows:in which , ,, and .

Step 4. Trajectory control module deactivation (θ₃)
When i_L and enter the vicinity of a new quasi-steady state at θ₃, the trajectory control module is deactivated.
The entire settling period θ_settling for positive/negative step change described as θ₃–θ₀ is derived in (11).The suggested on-line optimal trajectory control approach equations were found for a step-up change in load current. The operating principles and equations for adapting the approach to a negative load current step change are almost identical to those described above.
Table 1 lists the theoretical value of θ₁, θ₂, and θ₃ by the numerical calculation and the analytical solution from the simplified equations (4) and (10) under the different load current step-change amplitude at π/3 and 2π/3. The corresponding simplification error δ, defined as (12), is shown in Figure 6.The regulation time intervals of theoretical and simplified calculation results are approximately equivalent to each other, enabling the real-time on-line application and preserving enough control precision. δ grows as the step-change amplitude and filter inductance L increase. However, the slight simplification difference does not significantly impact the control performance.
The settling time of the on-line trajectory control algorithm during step change of load current is independent of the linear regulator parameters and mainly determined by the increasing and decreasing slopes of i_L, step-change initial angle θ_0, and amplitude. This is different from the dynamic behaviour of the single-phase VSI with the conventional controller because it does not use the output voltage as a feedback mechanism during the transient response from θ₀ to θ₃.
Figure 7 shows the program flow chart of the on-line trajectory control algorithm. After detecting the rapid step change, the transient regulation module is activated. In the end, the control system returns to using the linear regulator. Figure 8 illustrates the total settling time for the single-phase VSI with different inductance L_f, θ₀, and amplitude under the operation conditions = 200 V, = 154 V, and I_o1 = 5 A.
From the previous analysis, the on-line trajectory control algorithm is more suitable for the dramatic load change. It does not work well under relatively small or slow load change conditions in the vicinity of 0 and π, as shown in Figure 9. A linear regulator can work very well in these operating conditions and provide a good dynamic performance.

(a)

(b)

5. Implementation Techniques for the Detection of Critical Points for Transferring

The proposed implementation techniques that can be used to detect these critical points and provide a seamless transition between controllers are given as follows:

5.1. Detection of Load Step Changes

The proposed load step detection approach uses analog and digital signal processing techniques. A delayed output voltage signal and a comparator are used to detect load step changes in a power supply or load. The delayed signal is generated by a simple RC filter circuit, and the comparator generates a pulse signal if their difference exceeds a certain threshold. The microcontroller processes the comparison results to realize the step detection of load. Figure 10 shows a load change detector circuit with an adjustable delay.

5.2. Smooth Transferring Back to Linear Control

To transition smoothly to the linear controller, this article suggests increasing sampling frequency, maximizing system bandwidth, and running the linear controller at max speed. After the transient, the control system reverts to the PI controller for steady-state regulation. Before reverting to the linear controller, the optimal approach computes new steady-state values denoted as i_Lnew and D_rew for inductor current and duty cycle to avoid oscillations.

6. Simulation and Experiment Results

Numerical simulations in MATLAB/Simulink were used to evaluate the suggested on-line trajectory control approach and theoretical analysis. For comparison, a well-designed single-phase VSI with a linear controller is constructed. The main circuit parameters are = 200 V, = 154 V, C₁ = C₂ = 5000 uF, L_f = 1 mH, C_f = 20 uF, R_L1 = 20 Ω, and R_L2 = 50 Ω. The switching frequency f_s is 100 kHz, and the output line frequency f_line is 50 Hz. The main parameters of the PI controller are k_vol(z) = 0.5 + 0.005 z/(z − 1) and k_cur(z) = 4.2 + 0.025 z/(z − 1).

The transient response of DC-AC with a traditional 7.8 A to 10.5 A and back to 7.8 A is presented in Figure 11, and Figure 12 depicts the transient response utilizing the on-line trajectory method. The measured waveforms comprise the identified load step-change signals, , i_L, and .

(a)

(b)

(a)

(b)

To control i_L and from the one steady state i_{L_int} = 7.8 A and = 133.36 V to the final steady state i_{L_final} = 10.5 A and = 136.36 V, the optimal trajectory of a transient is predicted and implemented. According to (4) and (10), the on-state time interval of S_a 116 μs and off-state time interval 10 μs calculated in the theory for positive step change are identical to the simulation results displayed in Figure 12(a). The recovery period is decreased from 1 ms to 126 μs, which is a more than 80% decrease compared to a traditional dual-loop linear PI controller. Additionally, voltage undershoot is significantly decreased.

It is demonstrated in Figure 12(b) that with load current ranging from 10.5 A to 7.7 A, the transient settling time is reduced from 1 ms with the conventional dual-loop controller to 60 us using the on-line trajectory control algorithm.

As illustrated in Figure 13, a laboratory prototype of the 800 W single-phase VSI is constructed to validate the theoretical calculations, and the corresponding system diagram is shown in Figure 14. The six-step on-line trajectory control method can be readily created and executed using the EPWM and high-speed ADC capabilities of the TMS320F28377D. Each switching instant’s duration is computed on-line and saved on a chip. AC load step change is executed via a bidirectional switch with two MOSFETs linked in reverse. In addition, the external interrupt is employed to identify the abrupt change in load current i_L, and is measured using two operational amplifiers with excellent bandwidth and isolation. The voltage and current loop’s PI controller guarantees that tracks the reference with zero steady-state error. The PWM module then creates four driving signals for power devices. As soon as a step-change is recognized, the linear controller shifts to a nonlinear optimal controller.

(a)

(b)

The on/off time module generates a control signal according to the calculated time instants. PI parameters are reset at the completion of the transient, and the control system is switched back to the linear regulator. The controller undergoes a smooth transition with negligible switchover effects because , i_L, and Dnew are at their new quasi-steady-state values. A “slow” PI controller does not need to maintain stability during mode switchovers. Consequently, the maximum speed of the linear control may be attained. Table 2 lists the specifications of the experimental prototype, and key parameters of dual-loop PI controllers are k_vol(z) = 0.2 + 0.0034·z/(z − 1) and k_cur(z) = 1.2 + 0.017 z/(z − 1). Two step-change points, π/3 and 4π/3, are chosen for the hardware demonstration of the on-line trajectory control algorithm. Figure 15 illustrates the transient experimental waveforms of a single-phase VSI with the linear controller during a rapid step change at π/3. And Figure 16 depicts the similar transient response of the suggested nonlinear optimal trajectory control approach during the identical step change in load current. Both and i_L recover rapidly. The settling time using the on-line trajectory control method is approximately consistent with the theoretical calculation, coming in at 113 microseconds for positive step changes in load current and 36 microseconds for negative step changes in load current.

(a)

(b)

(a)

(b)

The voltage/current overshoot and oscillation amplitude throughout the transient process are smaller than the simulation value due to the hardware test platform’s relatively large system damping coefficient. Figures 17 and 18 show the measured experimental results of the single-phase VSI with the existing linear controller and the on-line trajectory control algorithm during step change at 4π/3, respectively.

(a)

(b)

(a)

(b)

7. Conclusion

This article presents a hardware-efficient, low cost, and simple on-line trajectory controller with the following merits: The controller can work conveniently (bandwidth-free) with a linear-nonlinear combination without sacrificing stability or having any steady-state errors. The controller is far superior to conventional methods, in addition to being less complicated, less expensive, and having fewer limitations than the control methods mentioned above. The voltage deviation may be lowered by 74%, and the settling time can be reduced by 80% in response to the load current step change in a positive direction. On the other hand, if the load current is altered in a negative direction, the converter overshoot will be reduced by 70%, and the settling time will be cut down by 75%. At the same time, a set of nonlinear optimization equations are constructed, which can change the output voltage according to the application scene and realize the real-time on-line control. Simulation and experimental platforms are built to verify the proposed scheme’s correctness. The results show that the proposed control scheme can effectively shorten the stabilization time, reduce the voltage spike in the transient state process, and improve the reliability of the equipment.

Nomenclature

K₁:	Upward slope
K₂:	Downward slope
S₁:	Charge portion
S₂:	Discharge portion of
i_L:	Inductor current
i_load:	Load current
i_ref:	Reference current
i_L:	Sensed current
:	Reference voltage
I_o1:	Initial load current
I_o2:	Final load current
:	Input dc voltage
:	Output voltage
:	Capacitor voltage
L_f:	Filter inductor
C_f:	Filter capacitor
S_d:	SSR relay switch
R_L1:	Initial load
R_L2:	Second load
D_new:	New duty ratio
f_line:	Line frequency.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The study was supported by University Enterprise Fund.

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Copyright © 2023 Shah Zaman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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