A Modified PI Control for Grid-tied Inverters to Improve Grid Injected Current Quality

The injection of good quality current into the grid is always prominent task for the grid-tied inverters. The usage of classical Proportional Integral (PI) controllers, though, dominates in all process control industry, they fail to track periodic signals like sine waves without DQ transformations. To eliminate the usage of DQ coordinates and also the steady state error in sinusoidal signals tracking, the sinusoidal error input feeding to the PI controller is divided into N-step shaped single. The value of N is decided by the switching frequency used in the controller loop, to eliminate the dead-zone effect of inverter switches. Mathematical analysis and extensive simulation results are proved that elimination of steady state error and injection of good quality of sinusoidal current when compared to Conventional PI controller. The THD analysis clearly approves the outperformance of the modified PI controller over the conventional one. Keyword Current control, feed-forward, grid-tied inverters, PI controllers, Steady state error, Steady-state error, Total Harmonic Distortion

Hence, the proposed work uses the most versatile structured PI controller with small modifications, widely used in 60% industrial applications [20],for tracking sinusoidal injected current tracking, by dividing error into N number of stepped wave form. The remaining paper is organized as follows: Section II, describes the working of the conventional PI controller with the inverter and also proves that it cannot eliminate the steady-state error for periodic signals. The modifications to the present PI controller are proposed in section III. The simulation steady has done for showing the out performance of the modified PI controller by tracking the grid reference current in section IV and finally, conclusions are drawn in section V.

II. GRID-TIED INVERTER WITH CONVENTIONAL PI CONTROLLER
The schematic diagram of the grid-tied inverter, with the current signal as feedback for the controller to generate the required pulse D to the inverter switches, is shown in Fig. 1. In the present work, closed loop current control uses the PI based controller along with the feedforward link to inject the required power demanded by loads. The Assume that DC link voltage at the input terminals of the inverter as V dc , and duty cycle to the inverter as D' and voltage at the output terminal of the inverter V o . Based on Equivalent Areas principle, the above three variables are related as below, To send the power always from source to grid, the inverter output voltage always greater than the grid voltage, Vg. The output voltage from the inverter is constructed form the two components :one component used to create the grid offset voltage, .the another part is useful to maintain the desired injected current. Hence, the duty cycle also having the two parts as shown in (2) To further simplify the model of the closed loop system, grid voltage can be excluded by incorporating the feedforward coefficient as 1/ Vdc . so that open loop transfer function of the system is free from the Vg. Hence the total system closed loop block diagram as shown in Fig. 2. From the Fig. 2., the open-loop transfer function of the system has been modified as follows, From (3), it reveals that the above system represents the type-1 system, hence there is always a steady-state error when the type-I system tracks a periodic signal.

III. GRID-TIED INVERTER WITH CONVENTIONAL PI CONTROLLER
The output of the conventional PI controller, which is also known as Control effort (U e ) in control terminology, is given as; Where, e(t)is the error signal generated from reference current signal and actual injected current signal. The inverter closed loop system with PI controller is type -1 system, Hence the idea, to avoid the steady state error from the periodic signals with this existed system, is to split the periodic signal into stepped waveform. Hence the error , which is a periodic signal divided into several samples of step signals and do proportional and integration of the same phase samples at each time. With this scheme, PI will view the sinusoidal current signal as a set of multiple step signals; which can easily makes the elimination of steady state error in the injected current signal. Moreover, the N stepped switching waves are collectively forms one full fundamental sine wave form, hence the duty cycle for the inverter switches is also fixed at each and every sample of the fundamental signal generated by the inverter. This may also give the freedom to the controller designer, to compare the same phase signal in the reference wave to the same phase signal in the actual current wave from, so that it excludes the effects caused by dead times of inverter switches. Thus, the final modified PI implemented in the simulation, will produces U e as shown below: The equation (5), clearly, shows that PI controller produces the control effort Ue by processing the errors at different phase angles whose instants are decided by the value of steps (N) considered for the dividing periodic waveform. Hence the duty cycle/ control effort U e obtained from the PI controller for the corresponding N-step Input error, without any steady error. Further any correction signal is required to eliminate further magnitude errors in each phase of the error signal, there is iterative integrating path with attenuation gain added in that path. Hence, the this arrangement will make the further production of corrective signal U ep ,until there is complete elimination of error corresponding each N-step reference and actual current signal. So, this further cumulative integration inside the current cycle completely eliminates the steady state error to zero, after few fundamental cycles. .is the error magnitude at the processed phase point p (p=1,2,…N) in the present fundamental cycle, ) are the error signals in previous cycles at the same phase point p. K P and K I represents the proportional and integral coefficients of same PI controller respectively.

IV. GRID-TIED INVERTER WITH CONVENTIONAL PI CONTROLLER
To check and realise the performance of the above proposed PI control method, a foxed-step MATLAB/SIMULINK environment is created. Inverter is realised with Fast switching IGBTs which can operate at 20 KHz. A low pass LC filter is attached to filter out the all switching frequency harmonics from the output voltage of the switched base inverter. Whole closed loop simulation is done using fixed step solver to replicate the real time closed loop simulations. And more over , controller also discretised using zero order hold blocks. The system parameters used for the simulation are listed in Table I. The proposed modified PI controller for inverter application is shown in Fig. 3. Here, authors are using sampling frequency of 36000 Hz for the controller, which results in 720 samples(=36000/50) for one cycle of sinusoidal waveform. Each 720 samples added with corresponding samples in each cycle to nullify the steady state error to zero. The K P and K I values are chosen as 15.6 and 12.0, respectively. The implementation of PI structure in the simulation is followed the well-known parallel-PID structure. The attenuation coefficient K is taken as 0.8. To synchronize the reference current command (i ref ) with the grid voltage (V g ), the Phase angel is retrieved from the gird the voltage signal. For generating the PWM Signal from the output of the proposed controller, a triangular of 20 KHz signal is generated.   Fig. 4. shows the grid voltage and grid collected current by the inverter operated with conventional PI controller for error tracking. Fig. 5. shows the grid voltage and grid collected current by the inverter with novel PI controller for error tracking. The Fig. 4. clearly shows the grid current ripple changing from the value 80 A to 100 A . But, in proposed Modified method, the grid injected current has almost zero ripple in the wave form signal which has clearly shown in zoomed view in Fig. 5. The THD analysis for both controlling methods are shown in Fig. 6., which proves that novel PI controller method improves the quality of injected current to the grid.

V. CONCLUSION
In this paper, modification of PI controller has been proposed to eliminate the steady state error while tracking the periodic signals. The modified PI has been tested with the grid-tied inverter application. Dividing the periodic signal into the N-Stepped wave signals and cumulative integration after the current cycle integration with the attenuation factor, makes the present proposal succeed in elimination of steady-state error in the sinusoidal signal. The extensive MATALB/ SIMULINK simulation results and THD analysis proves the quality of improved periodic signal tracking with the modified PI controller over the traditional PI controller. This method can be used for tracking the any type periodic signals without using DQ co-ordinates.