A Double Update PWM Method to Improve Robustness for the Deadbeat Current Controller in Three-Phase Grid-Connected System

In the grid-connected inverter based on the deadbeat current control, the filter inductance variation and single update PWM affect the distortion of the grid current, stability, and dynamic of the system. For this, a double update PWM method for the deadbeat current controller in three-phase grid-connected system is proposed, which not only effectively decreases the grid current distortion and control delay, but also improves the system stability and dynamic response speed due to reducing the characteristic root equation order of the closed-loop transfer function. The influence of the filter inductance deviation coefficient on the system performance is analyzed. As a conclusion, the corresponding filter inductance deviation coefficient in the system critical stability increases with increase in the parasitic resistance of the filter inductance and line equivalent resistance and decreases with increase in the sampling frequency. Considering the system stability and dynamic response, the optimal range of the control parameters is acquired. Simulation and experimental results verify the effectiveness of the proposed method.


Introduction
With the increasingly serious energy crisis and environmental pollution, renewable energy distributed generation technology has been widely concerned and researched in [1][2][3][4].The grid-connected inverter is the core of the distributed generation system in [5][6][7][8].Its role is to convert the DC power generated by renewable energy to the AC power accepted by the grid.The deadbeat current control is based on the mathematical model of the grid-connected inverter and depends on the actual electrical parameters of the main circuit.Theoretically, no static-state error can be achieved in [9][10][11][12].Because of its fast current transient response, accurate current tracking characteristic, and all digital control, the deadbeat current control has been applied more often.
However, reliance on the accurate electrical model and control delay are the main constraints in the deadbeat control.
On the one hand, the filter inductance value cannot be accurately detected.Even with the increasing of the grid current, the magnetic flux of the filter inductance tends to be saturated, resulting in the decrease of the filter inductance.This will lead to a certain deviation between the model filter inductance and actual filter inductance, affect the control accuracy of the deadbeat control, and cause the grid current distortion in [13,14].On the other hand, the inherent delay of the sampling and calculation limits the maximum duty cycle of the grid-connected inverter.For this, the delay caused by the proposed single update PWM in [15] may increase poles of the open-loop pulse transfer function, which affects the stability and dynamic response speed of the system.Therefore, how to improve the stability and dynamic response speed of the system and decrease the distortion of the grid current has become the research focus and goal of the gridconnected inverter.
In [16], a current prediction control method for the inductance online identification is proposed, which can accurately identify the inductance value in static and dynamic processes.An online estimation method of the filter inductance parameter is proposed in [17], which can prevent the current phase difference and system instability when the filter inductance parameter is not matched by the traditional prediction deadbeat current control method.However, the delay caused by the single update PWM has not been considered in the above methods.
The predictive current control technology is proposed to compensate the delay in [15,[18][19][20].A supply current predictive controller that adopts the variable step-size adaptive algorithm is proposed in [15], which solves the problem that the maximum duty is limited by the delay caused by the sampling and calculation.Because the weighted coefficient is regulated by the error calculated in each sampling period, this algorithm shows well predictive accuracy.But a single voltage vector is only used in a control cycle, so it needs very high sampling frequency to get good performance [18].In [19,20], the linear prediction method is proposed, which can estimate the information of the next beat based on the control object model and the information of the current and past period.Although it eliminates the control delay, this method depends on the accuracy of the model, which may exist a certain estimation error.
At the same time, the current observer is used to predict the next beat current in [10,21,22], and the current reference is predicted in advance to compensate for the delay.In [10,21], the current predictive algorithm based on repetitive control observer is proposed, which solves the problem of the current instability caused by the control delay and improves the current predictive precision.However, this method does not involve the dynamic analysis.In [22], an improved deadbeat current control scheme with a novel adaptive self-tuning load model is proposed.An improved deadbeat current controller with delay compensation is used to achieve high bandwidth current control characteristic, which compensates for the delay of the total system.However, a single prediction algorithm is adopted in the current reference variation, which may cause a large current overshoot or phase lag and deteriorate the system performance.
Digital PWM in double update mode is used to reduce the modulation delays and achieve performance.In [23], the effects of the measurement sampling and PWM updating methods on PI-based current control performance have been studied for a converter system.Reference [24] proposed proportional resonant controller implementation with double update mode digital PWM for single-phase grid-connected inverter.Digitally controlled grid-connected inverters with converter current control scheme and converter current plus grid current control scheme have been studied in [25].A combination of two samples' time displacement and the line current PWM ripple was proposed to cancel that error and boost the performance of such drives in [26].Reference [27] demonstrated the improved performance of a threephase voltage source inverter when digital multi-sampled space vector modulation was used.The bandwidth expansion strategy was proposed to achieve the stator current double sampling and PWM duty cycle double update in a carrier period in [28].However, the deadbeat current control has not been considered in the above methods.A fast robust PWM method for photovoltaic grid-connected inverter is proposed in [29], which effectively solves the delay of the one-stepdelay control and improves the system stability.
In this paper, the double update PWM method to improve robustness for the deadbeat current controller in three-phase grid-connected system is proposed.The paper is organized as follows.Section 2 presents the structure and control method for three-phase grid-connected system.The double update PWM method for the deadbeat current controller is proposed in Section 3. Finally, simulations and experiments are illustrated and discussed in Sections 4 and 5. Some conclusions are given in Section 6.

Control Method for Three-Phase Grid-Connected System
Structure of photovoltaic grid-connected system is shown in Figure 1, including the photovoltaic array, inverter circuit, and LC filter. dc is the DC side storage capacitance, which is used to stabilize the DC voltage  dc .The power transistors  1 - 6 constitute a three-phase full bridge inverter circuit that converts the DC voltage  dc into the AC output voltage, which is the same as the amplitude and phase of the grid voltage.LC filter is formed by the inductance L and capacitance C. In the grid-connected mode, LC filter is equivalent to a single filter inductance.The resistance r is the sum of the parasitic resistance of the filter inductance L and line equivalent resistance. dc is the DC current. inv (j=a,b,c) is the inverter output current. g is the grid current.Since the current flowing through the filter capacitance C is small,  inv is approximately equal to  g .Diagram of the double update PWM method for the deadbeat current controller is shown in Figure 2, including the double closed-loop control and double update PWM.The outer voltage loop adopts the PI control to stabilize  dc .The inner current loop adopts the deadbeat control, which decreases the distortion of the grid current caused by the filter inductance variation.The double update PWM effectively solves the delay caused by the single update PWM and improves the stability and dynamic response speed of the system. dcr is the DC voltage command,  g is the grid voltage,  1 is the fundamental angular frequency of the grid,  gr is the input amplitude command of the inner current loop,  gr is the grid current command, and D  is the equivalent duty cycle.

The Double Update PWM Method for the Deadbeat Current Controller
3.1.The Single Update PWM Method.In digital control, the single update PWM is shown in Figure 3, where  ms is the single update PWM wave.The sampling is carried out at the peak of the (k-1)th triangular carrier.The calculation is based on the sampling value.At the peak of the kth triangular carrier, the single update PWM wave  ms (k-1) of the (k-1)th  carrier cycle is loaded.So the equivalent duty cycle D(k) of the kth carrier cycle can be expressed as where d(k-1) is the duty cycle calculated by the sampling values at the peak of the (k-1)th triangular carrier.
Obviously, the loading time of the single update PWM wave is lagged behind a sampling period at the beginning of the sampling.So the one-step delay is caused by the single update PWM.

The Double Update PWM Method.
The double update PWM method is shown in Figure 4, where  md is the double update PWM wave.The sampling is carried out at the peak of the triangular carrier.The loading is carried out at the peak and valley of the triangular carrier.That is to say, one sampling and double loading are taken in each carrier cycle.The modulation process is as follows: the sampling is carried out at the peak of the kth triangular carrier.Meanwhile, the double update PWM wave corresponding to the d(k-1) is loaded, so that the conduction time of the first half of the carrier cycle is 0.5 s d(k-1).At the valley of the kth triangular carrier, the double update PWM wave corresponding to the difference calculated by subtracting the d(k-1) from the 2d(k) is loaded, so that the conduction time of the second half of the carrier cycle is  s (d(k)-0.5d(k-1)).So the equivalent duty cycle D(k) of the kth carrier cycle can be expressed as  where d(k) is the duty cycle calculated by the sampling values at the peak of the kth triangular carrier.Therefore, the proposed double update PWM method eliminates the control delay.Meanwhile, the time margin of the sampling and calculation decreases from  s to  s /2.
Diagram of the deadbeat current control for the gridconnected inverter is shown in Figure 5, where  gr (s) is the input quantity,  g (s) is the disturbance input quantity,  g (s) is the output quantity,  s is the sampling cycle, and S is the synchronous sampling switch.With the increasing of the grid current, the magnetic flux of the filter inductance tends to be saturated, resulting in the decrease of the filter inductance.There is the deviation between the filter inductance L 1 and actual filter inductance L.  at is the filter inductance deviation coefficient. at =L 1 /L. h (s) is the continuous domain transfer function of the zero order holder (ZOH).
d (s) is the continuous domain transfer function of the single update PWM, which can be expressed as The delay comparison between the single update PWM and proposed method is shown in Table 1.Considering the parasitic resistance of the filter inductance and line equivalent resistance, the influence of the control delay on the stability and dynamic of the system is analyzed, and the appropriate control parameter range is given.

Control System Stability Analysis.
In Figure 5, using the single update PWM,  d (s)= −s . * g (s) is Laplace transform of  g (s), which can be expressed as where  L (s) is the transfer function of the filter,  L (s)=sL+r, , and  * gr (s) is Laplace transform of the input quantity  gr (s).By substituting z= −s into (5), the closed-loop pulse transfer function in the z domain is obtained using the single update PWM.
By substituting z=(+1)/(-1) into ( 6), the characteristic root equation of the closed-loop transfer function of the system can be obtained as where  is transform operator from the  domain to the  domain.
Based on the Routh criterion [30], the range of the filter inductance deviation coefficient  at can be expressed as In Figure 5, using the proposed method,  d (s)=1.The closed-loop pulse transfer function in the z domain can be expressed as In ( 6) and ( 9), compared to the single update PWM, the characteristic root equation order of the closed-loop transfer function of the system is reduced when the proposed method is used.
When || <1, the system is stable and the range of the filter inductance deviation coefficient  at can be expressed as The corresponding filter inductance deviation coefficient in the system critical stability  at critical can be expressed as Relationship among the resistance r, sampling frequency f s , and  at critical by the single update PWM and proposed method is shown in Figure 6.Figures 6(a) and 6(b) correspond to the single update PWM and proposed method, respectively.When f s is constant,  at critical increases with increase in r.When r is constant,  at critical decreases with increase in f s .When the proposed method is used, the range of  at critical is 0<  at critical ≤2.1, while the range of  at critical is 0<  at critical ≤1.15 by using the single update PWM.So the range of  at critical by using the proposed method is obviously larger than the range of  at critical by using the single update PWM.
When  at critical =0.1, 0.5, 1.15, 1.5, 2.1, and 2.5, the root locus analysis with the single update PWM and proposed method is shown in Figure 7. Figures 7(a) and 7(b) correspond to the single update PWM and proposed method, respectively.In Figure 7(a), when  at critical =0.1 and 0.5, the pole is inside the unit circle, and the system is in a stable state.When  at critical =1.15, the pole is on the unit circle, and the system is in a critical stable state.When  at critical =2.1 and 2.5, the pole is outside the unit circle, and the system is in an unstable state.
In Figure 7(b), when  at critical =0.1, 0.5, 1.15, and 1.5, the pole is inside the unit circle, and the system is in a stable state.When  at critical =2.1, the pole is on the unit circle, and the system is in a critical stable state.When  at critical =2.5, the pole is outside the unit circle, and the system is in an unstable state.So the range of  at critical by using the proposed method is obviously larger than the range of  at critical by using the single update PWM.

Control System Dynamic Analysis.
When the single update PWM is used, system response to unit step change with  at critical changing is shown in Figure 8, while the sampling frequency f s =10kHz and resistance r=0.01Ω are constant.Figures 8(a) and 8(b) correspond to 0<  at critical ≤1 and 1<  at critical ≤1.15, respectively.When 0<  at critical ≤1, the dynamic response of the system is convergent.When 1<  at critical ≤1.15, the dynamic response of the system is divergent.Therefore, the range of  at critical is 0<  at critical ≤1.
In Figure 8(c), the curves A, B, C, D, E correspond to  at critical =0.7, 0.5, 0.3, 0.2, 0.1, respectively.In curve C, the system has the best dynamic response and no overshoot when  at critical is equal to  at critical0 ( at critical = at critical0 =0.3).In curves A and B, when  at critical >  at critical0 , the closed-loop poles are conjugate complex poles located in the right half unit circle of the z plane, and the dynamic response of the system is the oscillation convergence pulse sequence.The smaller  at critical is, the closer poles are to the coordinate origin, the smaller the overshoot is, and the faster the response speed is.In curves D and E, when  at critical <  at critical0 , the closed-loop poles are located in the real axis of the right half unit circle of the z plane.The dynamic response of the system is unidirectional positive convergence pulse sequence and has no overshoot.The larger  at critical is, the closer poles are to the coordinate origin, and the faster the response speed is.When the single update PWM is used, system response to unit step change with the resistance r changing is shown in Figure 9, while the sampling frequency f s =10kHz and  at critical =0.5 are constant.When the resistance r changes, the peak time and the adjustment time of the system do not change much.
When the proposed method is used, system response to unit step change with  at critical changing is shown in Figure 10, while the sampling frequency f s =10kHz and resistance r=0.01Ω are constant.Figures 10(a) and 10(b) correspond to 0<  at critical ≤2 and 2<  at critical ≤2.1, respectively.When 0<  at critical ≤2, the dynamic response of the system is convergent.When 2<  at critical ≤2.1, the dynamic response of the system is divergent.Therefore, the range of  at critical is 0<  at critical ≤2.
Contrast analysis of system response to unit step change between the single update PWM and proposed method is shown in Figure 11.Curves A and B correspond to   at critical =0.5 and 0.3 in the single update PWM, and curves C and D correspond to  at critical =1 and 0.5 in the proposed method.Compared to curve D, the system has optimal response to unit step change, fast response speed, and no overshoot in curve C. Comparing curve C with curve B, the optimal response to unit step change of the system in curve C using the proposed method is superior.Comparing curve D with curve A, in curve A using the single update PWM, the unit step response of the system has overshoot, and the oscillation and adjustment time become longer, resulting in the poor dynamic of the system.In curve D using the proposed method, the unit step response of the system has no overshoot and the dynamic becomes better.

Project Implementation Design Method.
The implementation principle of the proposed modulation method based on the DSP2812 controller is shown in Figure 12.DSP adopts TI's TMS320F2812 chip, which is a 32-bit fixed point micro control unit (MCU) with a main frequency of up to 150MHz.And the selected switching frequency is up to 10kHz.The general-purpose timer T1 is set to operate in continuous up/down counting mode.When the value of the count register T1CNT is equal to the period register T1PR, the timer T1 has a cycle interrupt, which starts the AD sampling unit, and ( 1-d(k-1))T1PR is assigned to the compare register T1CMPR.When the value of the count register T1CNT is 0x0000, the timer T1 has an underflow interrupt.The time margin from the start of the sampling to the end of the calculation is 0.5 s .The duty cycle d(k) is obtained, and (1-2d(k)+d(k-1))T1PR is assigned to the compare register T1CMPR.When the value of the count register T1CNT is equal to the compare register T1CMPR, the compare match event occurs, and the level of the pin T1PWM will jump.to the buffer for isolation and transformation.And then its level is reversed by the phase inverter.The final voltage signal is assigned to the corresponding IPM module.
Method 2. The compare output pin T1PWM of the timer is set to active low in the DSP2812 controller.The DSP pin output driver signal is shown in Figure 12(c).The drive signal is sent to the buffer for isolation and transformation.The final voltage signal is assigned to the corresponding IPM module.

Simulation Verification
In order to verify the effectiveness of proposed method, the simulation model of three-phase grid-connected inverter is built by using PSIM 9.0 based on Figure 1.The maximum output power of photovoltaic array is 50kW.System parameters are shown in Table 2.
In order to verify the implementation process of the proposed method, the simulation waveforms of the modulation wave and triangular carrier wave are shown in Figure 13.When the single update PWM and proposed method are used, the modulation processes are similar to the description of Figures 3 and 4. The correctness of the theoretical analysis is verified.
Steady state simulation waveforms of grid currents with  at critical =0.5 and  at critical =2 are shown in Figures 14 and 15, respectively.And the steady state simulation results of grid currents are shown in Table 3.In Figure 14(a), when the single update PWM is used with  at critical =0.5, THD of the grid current is 2.48%.Meanwhile, in Figure 14(b), when the proposed method is used with  at critical =0.5, THD of the grid current is 2.35%.Under the two modulation methods, the system is in a stable state.
In Figure 15(a), when the single update PWM is used with  at critical =2, the system is in an unstable state.However, in Figure 15(b), when the proposed method is used with  at critical =2, THD of the grid current is 2.95%, and the system   is in a stable state.Therefore, if the value of  at critical exceeds its critical range with using the single update PWM, the instability will occur.But when this parameter is applied to the proposed method, the system is still in a stable state.So the range of  at critical by using the proposed method is obviously larger than the range of  at critical by using the single update PWM.
The dynamic simulation waveforms of grid currents using the single update PWM and proposed method are shown in Figure 16.And the dynamic simulation results of grid currents are shown in Table 4.In Figure 16(a), the grid inrush current is 50A and THD is 2.7% during the start-up process when the single update PWM is used.However, the grid inrush current can be decreased to 25A and THD can be reduced to 1.8% during the start-up process by using the proposed method, as shown in Figure 16(b).

Experimentation Verification
In order to verify the validity of simulation results, the experimental platform of three-phase grid-connected system is built shown in Figure 17, which includes the main circuit, the control board, and the LC filter circuit.DSP adopts TI's TMS320F2812 chip, and IGBT selects Infineon  FF300R12ME4 module.The system parameters are shown as in Table 2.
By real-time debugging and setting breakpoints for the proposed method in CCStudio V3.3, the time spent on the sampling is 11.30s, the time consumed by the outer voltage loop is 11.80s, and the time required for the inner current loop is 7.60s.The total time spent is 30.70s.The selected switching frequency is up to 10kHz, and the switching period is 100.00s.Therefore, the time consumed by the sampling Journal of Electrical and Computer Engineering Figure 19: The experimental waveforms and THD of grid currents using the proposed method ( at critical =0.5).and calculation is only 30.70% of the switching period, which is less than 0.5 times the switching period.The sampling and calculation can be completed within 0.5 times the switching period.
The steady state experimental results of grid currents are shown in Table 5.And the experimental waveforms and THD of grid currents using the single update PWM and proposed method are shown in Figures 18 and 19, respectively.The theoretical root mean square (RMS) values of the grid currents are 50kW/(3×220V)≈75.76A.Considering some losses, the actual RMS values of grid currents are less than 75.76A.There is a measurement error among three current clamps of FLUKE, so three display values are not equal.Under the same  at critical =0.5 and control parameters, THD of the grid current using the single update PWM is 2.8%, while THD of the grid current using the proposed method is only 2.4%.
The experimental waveforms and THD of grid currents using the proposed method with  at critical =1, 1.5 are shown in Figures 20 and 21, respectively.In Figures 19, 20, and 21, THD of the grid current changes from 2.4% to 2.1%, and the next  22.In Figure 22(a), when  at critical changes from 0.5 to 2 with the single update PWM, the system becomes unstable.However, when  at critical changes from 0.5 to 2 with the proposed method, the system can still be in a stable state, as shown in Figure 22(b).
Dynamic experimental waveforms between the single update PWM and proposed method are compared as shown in Figure 23.Figures 23(a) and 23(b) correspond to  at critical =0.5 and 0.3 in the single update PWM.Figures 23(c) and 23(d) correspond to  at critical =1 and 0.5 in the proposed method.And the dynamic experimental results of grid currents are as shown in Table 6.The adjustment time is 75ms, 55ms, 15ms, and 35ms, respectively.When the above two individual methods are adopted with  at critical =0.5, in   , it can be obtained that the optimal dynamic of the system is superior by using the proposed method.The experimental results prove the correctness of theoretical analysis for Figure 11.

Conclusion
In this paper, the proposed double update PWM method for the deadbeat current controller not only effectively decreases the grid current distortion and control delay, but also improves the system stability and dynamic response speed due to reducing the characteristic root equation order of the closed-loop transfer function.The influence of the filter inductance deviation coefficient on the system performance is analyzed.As a conclusion, the corresponding filter inductance deviation coefficient in the system critical stability increases with increase in the parasitic resistance of the filter inductance and line equivalent resistance and decreases with increase in the sampling frequency.

Figure 4 :
Figure 4: The double update PWM method.

Figure 5 :
Figure 5: Diagram of the deadbeat current control for the grid-connected inverter.

Figure 6 :Figure 7 :
Figure 6: Relationship among the resistance r, sampling frequency f s , and  at critical .

1 Figure 8 :Figure 9 :
Figure 8: Single update PWM system response to unit step change with the constant sampling frequency f s =10kHz and resistance r=0.01 when  at critical changes.

Figure 10 :Figure 11 :
Figure 10: Proposed control system response to unit step change with the constant sampling frequency f s =10kHz and resistance r=0.01 when  at critical changes.

Figure 12 :
Figure 12: The implementation principle of the proposed modulation method based on the DSP2812 controller.

Method 1 .Figure 13 :Figure 14 :
Figure 13: The simulation waveforms of the modulation wave and triangular carrier wave.

Figure 16 :
Figure 16: The dynamic simulation waveforms of grid currents.

Figure 17 :
Figure 17: Experimental platform of three-phase grid-connected system.

Figure 18 :
Figure 18: The experimental waveforms and THD of grid currents using the single update PWM ( at critical =0.5).

Figure 20 :
Figure 20: The experimental waveforms and THD of grid currents using the proposed method ( at critical =1).

Figure 21 :Figure 22 :
Figure 21: The experimental waveforms and THD of grid currents using the proposed method ( at critical =1.5).

Figure 23 (
Figure23(a), the system has overshoot, and the adjustment time becomes longer, resulting in the poor dynamic.However, in Figure23(d), the system has no overshoot and the dynamic becomes better with using the proposed method.Comparing Figure23(c) with Figure23(d), the adjustment time of the former is only 15ms, while the latter is 35ms.Thus, the system has faster response speed when the proposed method is used with  at critical =1.Comparing Figure23(a)with Figure23(b), the system has the better dynamic with  at critical =0.3 rather than  at critical =0.5 when the single update PWM is used.From Figures23(c) and 23(b), it can be obtained that the optimal dynamic of the system is superior by using the proposed method.The experimental results prove the correctness of theoretical analysis for Figure11.

Figure 23 :
Figure 23: Contrast analysis of dynamic experimental waveforms between the single update PWM and proposed method.

Table 1 :
The delay comparison between the single update PWM and proposed method.

Table 3 :
The steady state simulation results of grid currents.

Table 4 :
The dynamic simulation results of grid currents.

Table 5 :
The steady state experimental results of grid currents.

Table 6 :
The dynamic experimental results of grid currents.