Instantaneous Power Theory with Fourier and Optimal Predictive Controller Design for Shunt Active Power Filter

This paper presents a novel harmonic identification algorithm of shunt active power filter for balanced and unbalanced three-phase systems based on the instantaneous power theory called instantaneous power theory with Fourier. Moreover, the optimal design of predictive current controller using an artificial intelligence technique called adaptive Tabu search is also proposed in the paper. These enhancements of the identification and current control parts are the aim of the good performance for shunt active power filter. The good results for harmonic mitigation using the proposed ideas in the paper are confirmed by the intensive simulation using SPS in SIMULINK.The simulation results show that the enhanced shunt active power filter can provide theminimum%THD (Total Harmonic Distortion) of source currents and unity power factor after compensation. In addition, the %THD also follows the IEEE Std.519-1992.


Introduction
Power systems connected nonlinear loads can generate the harmonics into the systems.These harmonics cause a lot of disadvantages such as loss in transmission lines and electric devices, protective device failures, and short-life electronic equipment in the system [1][2][3].Therefore, it is very important to reduce or eliminate the harmonics in the system.It is well known that the harmonic elimination via a shunt active power filter (SAPF) [4] provides higher efficiency and more flexibility compared with a passive power filter.There are four main parts (part A-D) for using the SAPF to mitigate the harmonics in the system as shown in Figure 1.
The Part A is the harmonic identification method to calculate the reference currents for SAPF.There are many methods for harmonic identification such as an instantaneous power theory (PQ) [5,6], a synchronous reference frame (SRF) [7], a synchronous detection (SD) [8], a sliding window Fourier analysis (SWFA) [9], an a-b-c reference frame [10], and a DQ-axis with Fourier (DQF) [11].
The Part B is the SAPF structure.There are two types of SAPF topology such as the voltage source inverter (VSI) [12,13] and the current source inverter (CSI) [13,14] with six IGBTs.The VSI topology is used in the paper because this topology is simple and provides the good performance for harmonic elimination.
The Part C is the control technique to control the compensating current of SAPF.There are several techniques to control the compensating current injection such as a hysteresis control [15,16], a PWM technique with PI controller [15,16], a sliding mode control [17,18], a predictive control [19][20][21], a fuzzy logic control [22,23], and a neural network control [16].
The Part D is the last part for harmonic elimination using SAPF.This part is the DC bus voltage control of SAPF.There are various types of the voltage control to regulate the DC bus voltage such as PI controller [24,25], fuzzy logic controller [26], and RST controller [24].In the paper, the PI controller is used to control the DC bus voltage.
The aim of this paper is the minimum %THD of source currents after compensation via SAPF.The Part A and Part C are the significant parts to achieve the minimum %THD.Therefore, the performances of Part A and Part C must be improved.In Part A, the PQ method is selected for improvement because this algorithm is simple and because of unity power factor confirmation after compensation.The  conventional PQ method uses the analog filter to draw the harmonic component of the instantaneous active power from fundamental component.This approach has an error to calculate the harmonic component.Therefore, the SWFA technique is applied to draw the harmonic component for harmonic identification improvement.The PQ with SWFA method called an instantaneous power theory with Fourier (PQF) algorithm is presented in the paper.The details of the PQF algorithm and the performance comparison between the PQ and PQF for balanced and unbalanced systems are explained in Section 2.
There are many advantages for minimum %THD of source currents such as minimum loss in transmission lines and electric devices, more accuracy of protective devices, and long-life electronic equipments.Therefore, the minimum %THD of source currents is necessary.Normally, many research works [13,14,19,21,24,27,28] focus on how to reduce %THD of the system to follow the IEEE Std.519-1992 but do not care about the minimum %THD.The improvement of harmonic identification part (Part A) is not sufficient to achieve the minimum %THD nearly global solution.Therefore, the development of the compensating current controller (Part C) is the additional approach to present in the paper.The predictive current control is selected to improvement in Part C because this controller compensates the delay incurred through digital control implementation and provides good static and dynamic performances.The conventional predictive current control uses the first-order Lagrange equation to approximate the predicted reference currents.Presently, it is well known that there are many artificial intelligence (AI) techniques to apply for the optimization problems in the engineering researches such as the multiobjective harmony search (MOHS) [29], artificial bee colony (ABC) [30,31], competition particle swarm optimization (CPSO) [32], genetic algorithm (GA) [33], and adaptive Tabu search (ATS) [34][35][36][37][38][39][40][41][42][43][44][45][46][47].The ATS method is developed by Puangdownreong et al. in 2002 [34].In order to perform its effectiveness, the ATS has tested against several wellknown benchmark functions, that is, Bohachevsky, Rastrigin, Shekel's foxholds, Shubert, and Schwefel functions [42][43][44][45][46].Moreover, the convergence property of the ATS has been proved to assure that it can reach the optimal solution within finite search time [42][43][44][45][46][47].Thus, the ATS is selected to design the predictive current controller in the paper.The ATS approach can provide the good performance to control the compensating currents injection and guarantees the optimal solution for searching.The review of the conventional predictive current control on dq-axis is described in Section 3. The ATS method is briefly explained in Section 4. In Section 5, the optimal design of the predictive current controller using the ATS method is fully shown.Finally, Section 6 concludes and discusses the advantages of the proposed ideas to enhance the performance of SAPF.In the paper, the improvement of the harmonic identification and current controller design parts of SAPF is called the enhanced shunt active power filter (ESAPF).

Instantaneous Power Theory with Fourier
The harmonic identification algorithm for reference current calculations is very important for the harmonic mitigation with SAPF.The perfect reference currents are necessary for an enhanced shunt active power filter or ESAPF.Therefore, a novel algorithm to calculate the reference currents of ESAPF is presented in this section.This algorithm is called the instantaneous power theory with Fourier algorithm or PQF.The PQF algorithm is developed from the instantaneous power theory (PQ).The PQ algorithm is firstly public in 1983 by Akagi et al. [5].The performance comparison between the PQ and PQF algorithm is discussed in this section.The performance indices for comparison are %THD of source currents and power factor after compensation.The harmonic mitigation systems with the ideal shunt active power filter for balanced and unbalanced systems as shown in Figures 2 and  7, respectively.In Figure 2, the three-phase bridge rectifier feeding resistive and inductive loads (R = 130 Ω and L = 4 H) behaves as a nonlinear load into the balanced three-phase system.In Figure 7, the three single-phase bridge rectifiers with different RL loads are the nonlinear load for an unbalanced three-phase system.The ideal current source is used to represent the ideal shunt active power filter to perfectly inject the compensating currents (  ,   ,   ) into the power system because of using the ideal current source model for SAPF.The block diagram to calculate the reference currents using PQ and PQF algorithm for balanced and unbalanced three-phase systems is depicted in Figure 3. Figure 3 shows that there are six steps to calculate the reference currents.
Step 3. Calculate the instantaneous active power (  ) and reactive power (  ) on the 0 frame in the block number 3.
The   from the block number 3 consists of two components, the fundamental component (  ) and the harmonic component ( p ).
Step 4. Draw the p from the   .For PQ algorithm, the separation of the fundamental and harmonic components uses the analog filter (high-pass filter: HPF).In this paper, the cutoff frequencies of HPF for balanced and unbalanced systems are equal to 280 Hz and 50 Hz, respectively.On the other hand, the sliding window Fourier analysis (SWFA) is used to separate these components for PQF algorithm.In this step, the method to separate the fundamental and harmonic components is the different point between the PQ and PQF algorithm.After to draw the p from   , the reference active power (  ) can be obtained from subtracting between p and   (output of the PI controller in the DC bus voltage control part).In the paper, the reference reactive power is set equal to   because of the unity power factor after compensation.
Step From Figure 3, it can be seen that the zero sequence calculations are necessary for unbalanced three-phase system.For the balanced system, the zero sequence quantities are equal to zero.
The SWFA technique for PQF algorithm uses the Fourier series of active power as shown in (1).From this equation,  0 ,  ℎ , and  ℎ are the Fourier series coefficients,   is the sampling interval,  is time index, ℎ is the harmonic order, and  is the angular fundamental frequency of the system.The fundamental component (or DC component) of active power is represented by  0 coefficient as shown in (2).The  ℎ coefficient in (1) can be calculated by (3).The  0 coefficient or DC component can be calculated by substitute ℎ = 0 in (3) as shown in (4).The  0 and  in (3) and ( 4) are the starting point for computing and the total number of sampled point in one cycle, respectively.The calculation of  0 in the first period can be calculated using (4) so as to achieve the initial value for the PQF algorithm.For the next period, the  0 can be calculated by (5) in which this approach is called SWFA [9].The SWFA approach can be summarized in Figure 4: Figure 4: The flow chart of the SWFA approach.
The simulation results of the performance comparison between the PQ and PQF algorithms for the balanced system in Figure 2 with   = 10 mH are addressed in Table 1.The cutoff frequency of HPF for PQ method is set to 280 Hz.The average %THD of source currents (%THD ,av ) and the power factor after compensation (pf) are the performance indices for the comparison.The %THD av and pf can be calculated by ( 6) and ( 8), respectively.The %THD of source currents in each phase (%THD , ) can be calculated by (7).The fundamental and harmonic (order n) values in (7) are denoted by subscript 1 and n, respectively.The pf disp and pf dist in (8) are the displacement and distortion power factors in which these values can be calculated by ( 9) and (10), respectively: The results from Table 1 show that the PQF algorithm can provide the best performance in term of %THD ,av .From Table 1, the %THD ,av of the source currents before compensation is equal to 24    distorted as shown in Figure 5.These waveforms are equal to the load currents (  ,   ,   ) before compensation because the SAPF is not connected to the system.From Figure 5, the compensating currents (  ,   ,   ) from SAPF are injected into the system at t = 0.04 s.For t = 0.04-0.06s, the compensation is nonperfect because this interval is used for initial of SWFA algorithm.The SWFA algorithm is the main approach for PQF method.After t = 0.06 s, the SAPF generates the perfectly compensating currents into the system (reactive power and harmonic compensations).
From Figure 5 ( ≥ 0.06 s), it can be seen that the source currents after compensation are nearly sinusoidal waveforms.The %THD ,av of these currents is equal to 0.95 and 0.04 for PQ and PQF, respectively as shown in Table 1.These values are satisfied under IEEE std.519-1992.Moreover, the power factor after compensation is unity, while before compensation the power factor is equal to 0.95.From Figure 3, the different point between the PQ and PQF algorithm is the method to separate the fundamental and harmonic components.Therefore, the accurate instantaneous active power for harmonic component ( p ) is the main objective to identify the harmonic currents of the system.The spectrum comparison of the p values calculated by PQF and PQ algorithms is shown in Figure 6.The p,act is the spectrum of the instantaneous harmonic active power calculated by FFT approach from MATLAB programming.The p,PQF and p,PQ are calculated by PQF and PQ algorithms, respectively.From Figure 6, it can be seen that the p,PQF value calculated by PQF algorithm is nearly the same as the p,act value.The errors between the p values calculated by PQF and PQ algorithms compared with the p,act value are shown in Table 2.In the paper, the authors focus on the total error ( tot ) for the performance comparison between the PQ and PQF algorithms.From Table 2, the  tot from PQF algorithm (0.56%) is less than the PQ algorithm (1.56%).Therefore, the PQF algorithm is the perfect method to calculate the reference currents for ESAPF.
The simulation results of the performance comparison between the PQ and PQF algorithms for the unbalanced system in Figure 7 are addressed in Table 3.The results from Table 3 show that the PQF algorithm can provide the best performance in term of %THD ,av and %unbalance after compensation.The %unbalance in this table can be calculated by (11).From Table 3, the %THD ,av and %unbalance of source currents before compensation are equal to 31.52% and 15.43%, respectively.The waveforms of source current (  ,   ,   ) before compensation ( = 0-0.04s) are extremely distorted and unbalanced as depicted in Figure 8.For  = 0.04-0.06s, this interval is the initial calculation for PQF algorithm using a SWFA technique.For  ≥ 0.06 s, the PQF algorithm can completely eliminate the harmonic currents and balance the amplitude and phase of source currents after compensation.The %THD ,av of these currents are equal to 0.60 and 0.01 for PQ and PQF, respectively, as given in Table 3.The %unbalance after compensation using PQ and PQF algorithms is equal to 0.43 and 0, respectively.It means that the source currents after compensation are perfectly balanced using the PQF algorithm compared with the %unbalance before compensation (15.43%).From the simulation results of the balanced and unbalanced system, the PQF algorithm is the perfect method to calculate the reference currents for ESAPF.In the future works, the positive sequence detection is added to the PQF algorithm for the harmonic current elimination in the distorted and unbalanced voltage systems: %unbalance =     maximum current deviation from average rms current     average rms current

Predictive Current Control on dq-Axis
In this section, the predictive current control for SAPF with balanced three-phase system is proposed.The predictive current control technique is applied to control the injection of compensating currents with SAPF as shown in Figure 9.
The voltage source inverter with six IGBTs is the SAPF topology in the paper.The PQF algorithm described in the previous section is used to identify the harmonic currents in the system.The three-phase bridge rectifier feeding resistive and inductive loads behaves as a nonlinear load into the power system.The predictive current control is the suitable technique for a digital control [21].The equivalent circuit in Figure 10 is used to derive the relationship equation between the SAPF output voltages (u () ) and the voltages at PCC point (u PCC() ) as given in (12).The compensating currents or active filter currents are represented by i () .The discrete form of ( 12) can be represented by ( 13) and  sc is the sampling time of the controller: The concept of the reference currents prediction is shown in Figure 11.From this figure, the three-phase reference current at time instants () and ( + 1) is denoted by i (),ref () and i (),ref ( + 1), respectively.The predicted three-phase reference currents (i (),ref ( + 1)) for the next sampling period are calculated by (14)    coefficients of the first-order in Lagrange equation ( 0 = 2,  1 = −1).The Lagrange equation is used to approximate the reference currents one sampling instant ahead by using known values from a few previous sampling instant.The output voltages of SAPF are assumed to be constant during the one sampling time: Equations ( 12)-( 14) are used for three-phase values.In the paper, the predictive current control is applied on dqaxis.Therefore, the equations to calculate the output voltages of SAPF and the predicted reference currents on dq-axis are shown in ( 15) and ( 16), respectively.The Park's transformation is used to transform the three-phase quantities to dq-axis quantities.The overall procedure to calculate the output voltages of SAPF using predictive current control is depicted in Figure 12.The output voltages of SAPF are used to generate the six-pulse of IGBTs ( 1 −  6 ) via the PWM technique: The simulation results of the system with   = 0.01 mH and   = 10 mH in Figure 9 are shown in Table 4.The inductor (  ), capacitor ( dc ), and the DC bus reference voltage ( dc,ref ) of SAPF are equal to 39 mH, 250 F, and 750 V, respectively.The PI controller is applied to regulate the DC bus voltage (  = 3,   = 24).The %THD ,av of source currents (  ,  sb ,   ) before compensation is equal to 24.91%, Figure 11: The concept of predictive current control.
3    while %THD ,av after compensation with predictive current control technique using first-order Lagrange equation is 1.40%.The current and voltage waveforms of the system in Figure 9 are depicted in Figure 13.
In Figure 13, the compensating currents (  ,   ,   ) from SAPF are injected into the system.The source currents before compensation are highly distorted waveform (%THD ,av = 24.91%).After compensation, the source currents are nearly sinusoidal waveform (%THD ,av = 1.40%).Moreover, the PI controller can regulate the DC bus voltage to 750 V.The design of the predictive current control using the adaptive Tabu search (ATS) method without the first-order Lagrange equation is explained in Section 5.

Review of ATS Algorithm
The adaptive Tabu search or ATS method [34][35][36][37][38][39][40][41][42][43][44][45][46][47] is used to design the predictive current controller to minimize %THD ,av of source currents after compensation.The review of the ATS algorithm is described in this section.The ATS algorithm is improved from the Tabu Search (TS) method by adding two mechanisms, namely, back-tracking and adaptive search radius.The modified version of the TS method has been named the adaptive tabu search of ATS.The ATS algorithm can be outlined as follows.
Step 1. Initialize the tabu list TL and Count (a number of search round) = 0.
Step 2. Randomly select the initial solution  0 from the search space. 0 is set as a local minimum and  0 = best neighbor as shown in Figure 14.
Step 3. Update Count then randomly select  new solutions from the search space of a radius .Let  1 () be a set containing  solutions as shown in Figure 15.
Step 4. Compute the cost value of each member of  1 ().Then, choose the best solution and assign it as best neighbor1 (see Figure 15).(see Figure 16), and set  0 = best neighbor (see Figure 17).Otherwise, put best neighbor1 in the TL instead.
Step 6. Evaluate the termination criteria (TC) and the aspiration criteria (AC).If Count MAX Count (the maximum number allowance of search round), stop the searching process.The current best solution is the overall best solution.Otherwise, go back to Step 2 and start the searching process again until all criteria is satisfied (see Figure 18).The back-tracking process allows the system to go back and look up the previous solutions in TL.The better solution is then chosen among the current and the previous solutions.Figure 19 illustrates details of the back-tracking process.
Given this new search space to explore, the search process is likely to have more chances of escaping from the local optimum.The back-tracking mechanism can be added into Step 5 to improve the searching performance.
The adaptive radius process as depicted in Figure 20 decreases the search area during the searching process.The adaptive radius mechanism has been developed to adjust the radius (R) by using the cost of the solution.The criterion for adapting the search radius is given as follows: where DF is a decreasing factor.The adaptive search radius mechanism can be added into the end of Step 6 to improve the searching performance.The more details of ATS algorithm can be found in [34][35][36][37][38][39][40][41][42][43][44][45][46][47].

Optimal Design of Predictive Current Controller
In Section 3, the predicted currents are calculated by the firstorder Lagrange equation in (14) with  0 = 2,  1 = −1.
In this section, the ATS algorithm is applied to determine the appropriate coefficients ( 0 and  1 ) of ( 14) for %THD ,av minimization.The block diagram to explain how to search the  0 and  1 coefficients using the ATS algorithm is depicted in Figure 21.As can be seen in Figure 21, the ATS will try to search the best coefficients of ( 14) to achieve the minimum %THD ,av .The cost value of the ATS searching is %THD ,av of source currents.In each searching round, the %THD ,av value can be calculated by M-file programming, while the actual three-phase source currents are obtained from Simulink as shown in Figure 21.
In the ATS process, the  0 and  1 coefficients are adjusted to achieve the best solution; here it is the minimum %THD ,av .The convergence of the %THD ,av value is shown in Figure 22.It can be seen that %THD ,av can converge to the minimum point.The %THD ,av in Figure 22 can escape the local point to get the better solution because of the back tracking approach in the ATS algorithm.Moreover, the convergences of  0 and  1 coefficient values are shown in Figures 23 and 24   respectively.In the paper, the maximum of searching iteration for ATS is set to 300 rounds, number of initial solution = 400, number of N neighborhood = 40, initial radius of search space = 0.4, and decreasing factor value (DF) = 1.2.From the ATS searching results,  0 and  1 coefficients are equal to 2.85 and −1.86, respectively.The simulation results of the system in Figure 9 with the predictive current controller designed by ATS algorithm are shown in Figure 25.The source currents after compensation are nearly sinusoidal waveform and %THD ,av of these currents are equal to 0.96 as shown in Table 4. From the results, the predictive current controller designed by ATS algorithm can provide the smaller %THD ,av compared with the current controller using firstorder Lagrange equation.The results show that the ATS approach is very useful and more convenient for the optimal design of predictive current control in SAPF system.The   simulation results for harmonic currents elimination with dynamic load changing are shown in Figure 26.From this figure, the load of three-phase bridge rectifier is suddenly changed at  = 1s.After load changing, the SAPF can also mitigate the harmonic currents and the DC bus voltage controller can also regulate the DC voltage equal to 750 V.

Conclusion
The instantaneous power theory with Fourier or PQF algorithm is proposed in the paper.The performance comparison between the PQ and PQF is also presented by the simulation via the software package.The simulation results show that the PQF algorithm can provide the accurate reference currents for a shunt active power filter.Moreover, the optimal design of predictive current controller by ATS method is shown in the paper.This controller can provide the best performance of harmonic elimination compared with the conventional predictive current control.The shunt active power filter using the PQF algorithm to identify the harmonic and using the compensating current controller designed by ATS method is called the enhanced shunt active power filter (ESAPF).The results from simulation confirm that the ESAPF provides the minimum %THD and unity power factor of power supply at PCC point.

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Modelling and Simulation in Engineering

Figure 1 :
Figure1: The harmonic elimination system via shunt active power filter.

Figure 3 :
Figure 3: The block diagram of PQ and PQF algorithms.

Figure 5 :
Figure 5: The simulation results using PQF algorithm for ideal shunt active power filter with balanced system.

Figure 7 :
Figure 7: The unbalanced power system with ideal shunt active power filter.

Figure 8 :
Figure8: The simulation results using PQF algorithm for ideal shunt active power filter with unbalanced system.

Figure 9 :Figure 10 :
Figure 9: The balanced power system with the predictive current control of SAPF.

Figure 12 :
Figure 12: The overall procedure of the predictive current control for SAPF.

Figure 13 :
Figure13: The simulation results using first-order Lagrange equation.

Step 5 .
If best neighbor1 < best neighbor, then keep best neighbor in the TL, set best neighbor = best neighbor1

Figure 16 :
Figure 16: Assign a new best neighbor.

Figure 21 :
Figure 21: The design of predictive current controller using ATS algorithm.
( + 1): the predicted three-phase reference currents  (),ref () and  (),ref ( + 1): the three-phase reference current at time instants () and ( + 1)  0 ,  1 : the coefficients of the first-order in Lagrange  dc,ref : the DC bus reference voltage of SAPF  dc : t h e D C b u s v o l t a g e o f S A P F : a n u m b e r o f s e a r c h r o u n d  : t h em a x i m u mn u m b e ra l l o w a n c e of search round DF: a decreasing factor.

Table 1 :
The performance comparison between the PQ and PQF algorithms for balanced system.
.48% in which this value is extremely greater than the IEEE std.519-1992.The source current waveforms before compensation (  ,   ,   ) are highly

Table 2 :
The error of instantaneous active power for harmonic component calculation.

Table 3 :
The performance comparison between the PQ and PQF algorithms for unbalanced system.

Table 4 :
The simulation results.
PCC ,  PCC ,  PCC0 : the voltages at PCC point on 0 frame   ,   ,   : the three-phase load currents   ,   ,  0 : the load currents on 0 frame   and   : the instantaneous active power and reactive power   : the fundamental component of instantaneous active power p : the harmonic component of instantaneous active power   : the reference active power  ,ref ,  ,ref ,  0,ref : the reference currents on 0 frame  ,ref ,  ,ref ,  ,ref : the three-phase reference currents  0 ,  ℎ ,  ℎ : the Fourier series coefficients