In the diploma thesis, a small signal model for a grid inverter is developed using a modified method of bipolar pulse width modulation. Based on theoretical analysis, controllers are designed to improve dynamics and ensure system stability. High power factor and low harmonics are desired in a transmission system to improve power line efficiency.
The current control of the single-phase grid-connected inverter has sinusoidal reference to achieve power factor correction (PFC) and harmonic suppression. Designed current loop using the PI controller (a) Bode plot of the current loop (b) Polar diagram Fig.
Introduction
Then, with the help of a resonant term, the PR controller can track the sinusoid without the steady-state error [14]-[18]. Although the PR controller is commonly used in single-phase inverter systems due to the aforementioned merits, it introduces steady-state error when the reference signal has a step change. The dc link may contain ripples including ripples of double the line frequency because the dc link voltage is the rectified utility line.
This thesis takes advantage of the PI controller to track the constant reference of the voltage regulator. Controlling the DC link voltage and synchronizing the inductor current with the mains voltage are control purposes of the bidirectional AC-DC converter.
Operation Principle and Small-Signal Modeling
Operation Principle
In case of bipolar PWM method, diagonal switches turn on or off simultaneously as shown in fig. 2(b), has lower coupling loss and EMI than bipolar PWM method, but it is not negligible. Compared with conventional switching methodology, modified bipolar PWM method can reduce the switching loss by using only two switches.
2(c) shows the modified bipolar PWM method applied to the inverter for the control. The capacitor current is equal to the sum of the negative DC side current and the negative inductor current.
Small-Signal Modeling
2, when the triangular waveform frequency is much higher than modulation waveform frequency, the switching function of switches can be averaged. In the process of the linearization, the non-linear relationship near the operating point is approximated with the linear relationship. All the state variables and input variables in (4) can be expressed as the sums of the dc component and the ac component.
The DC components represent operating point and AC components represent small variation near the operating point. Based on the small-signal model, stability and dynamics of the inverter can be analyzed mathematically. Considering the small-signal DC link voltage and inductor current as the output variables, the inverter has six power stage transfer functions.
There is the line voltage-to-dc link voltage transfer function 𝐺𝑣𝑠(𝑠), the line voltage-to-inductor current transfer function 𝐺𝑖𝑠(𝑠), the DC-side current-to-dc link voltage transfer function 𝑍𝑝,(𝑑), the DC-side current-to-inductor the current transfer function 𝑍𝑞(𝑠), the duty ratio-to-dc link voltage transfer function 𝐺𝑣𝑑(𝑠) and the duty ratio-to-inductor current transfer function 𝐺𝑖𝑑(𝑠). The derived open-loop small-signal transfer functions of the inverter are verified using PSIM simulation. Solid lines represent theoretical results obtained from the mathematical model, and dashed lines represent AC sweep results obtained from PSIM simulation.
Near or above half the switching frequency, the dynamics of the converter is affected by sampling effects and cannot be predicted using classical analysis.
Controller Design
Current Controller Design
- PI Controller
- PR Controller
- PIR Controller
- Simulation Results
For simplicity of analysis, one of the two is assumed to be constant while the other affects the frequency responses. Therefore, the magnitude responses of the current loop increase at the high frequency and the crossover frequency increases as shown in Fig. Therefore, the magnitude response of the current loop increases at the low frequency as shown in Fig.
The value of 𝑘𝑝𝑖 is chosen to limit the division frequency lower than the ten percent of the switching frequency. 13(a), the magnitude of the constructed current loop at the line frequency is 40 dB and the crossover frequency is Hz. Designed current loop using the PI controller (a) Bode plot of the current loop (b) Polar map.
First, 𝑘𝑟 and 𝜉 are assumed to be constant for simple analysis of the effect of 𝑘𝑝 on the frequency responses. If the crossover frequency of the current loop is far from the line frequency, 𝑘𝑟 has no effect on the crossover frequency. In common with 𝑘𝑟, 𝜉 does not affect the crossover frequency of the current loop if the crossover frequency is much higher than 𝜔0.
When the crossover frequency is the ten percent of the switching frequency, it is much higher than the resonance frequency and is determined only by 𝑘𝑝. As previously mentioned, the PR controller can have the sufficient gain at the line frequency regardless of the crossover frequency. To design the PIR controller, the effect of the integrated gain 𝑘𝑖 should be considered.
19 (a), as 𝑘𝑖 increases, the order of magnitude of the proposed controller corresponds to a single integrator and the phase margin decreases. In conclusion, in terms of the transient response speed and the steady-state error, the PIR controller has intermediate characteristics between the PI controller and the PR controller.
Voltage Controller Design
The output impedance describes the noise transmission characteristics from the dc side current to the dc link voltage and can be expressed as (36). From (35) and (36), the audio sensitivity and output impedance are affected by the overall gain of the loop. Because the reference value of the voltage controller is constant, the PI controller is adopted to control the dc link voltage.
In the same way as the current controller design, one of the two is assumed to be constant while the other affects the frequency response. Thus, the audio sensitivity and output impedance characteristics are improved at the low frequency. The overall loop gain and the outer loop gain provide the same absolute stability information, but different relative stability information.
The total loop gain which is defined as the sum of the loop voltage and current loop has additional gain margin and phase difference. Also, the gain of the outer loop provides information on the stability of the voltage loop when the current loop is prefixed. Because there are double line frequency ripples in the dc link voltage, the crossover frequency of the outer loop gain is limited to twenty percent of the line frequency.
Similarly, the PI voltage controller is designed for cases where the PI controller and the PR controller are used as current controllers. 28(a), at low frequency, the overall loop gain has the largest magnitude when the PI controller is adopted to control the current. Thus, similar dc link voltage responses are expected in the presence of changes in dc side current for three current controllers.
Time domain expression of the DC link voltage has the relationship with the output impedance as follows. The solid line shows the exact equation of the output impedance and the dashed line shows the asymptotic approximation.
Experimental Results
Experimental results under steady state condition (a) PI current controller (b) PR current controller (c) PIR current controller. 33 shows the overshoot waveforms that occur when the side dc current varies from 20% to 70% load and Fig. 34 shows the overshoot waveforms that occur when the dc side current changes from 70% to 20% of the load.
Because the output impedances are the same regardless of current control, similar responses are shown for the DC link voltage as expected. If it is taken into account that asymptotic approximation is used and the ideal step loading cannot be implemented in actual experimental condition, the error result from the small-signal model is negligible. Experimental results of step load response from 20% to 70% load (a) PI current controller (b) PR current controller (c) PIR current controller.
Experimental results of step load response from 70 % to 20 % load (a) PI current controller (b) PR current controller (c) PIR current controller. Although three cases meet IEEE Std 519-2014 harmonic current limits, either the PR current controller or the PIR current controller has lower harmonic content than the PI current controller. Experimental results of harmonics under full load condition (a) PI current controller (b) PR current controller (c) PIR current controller.
When the PR controller and the PIR controller are used for current control, the inverter is slightly more efficient than when the PI current controller is adopted. So, even if the system meets the regulation for harmonics, it is necessary to reduce harmonics to improve the efficiency.
Conclusion
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