Robust switched fractional controller for performance improvement of single phase active power filter under unbalanced conditions

Abstract

A novel controller is proposed to regulate the DC-link voltage of a single phase active power filter (SPAPF). The proposed switched fractional controller (SFC) consists of a conventional PI controller, a fractional order PI (FO-PI) controller and a decision maker that switches between them. Commonly, the conventional PI controller is used in regulation loops due to its advantages in steady-state but it is limited in transient state. On the other hand, the FO-PI controller overcomes these drawbacks but it causes dramatic degradation in control performances in steady-state because of the fractional calculus theory and the approximation method used to implement this kind of controller. Thus, the purpose of this paper is to switch to the PI controller in steady-state to obtain the best power quality and to switch to the FO-PI controller when external disturbances are detected to guarantee a fast transient state. To investigate the efficiency and accuracy of the SFC considering all robustness tests, an experimental setup has been established. The results of the SFC fulfill the requirements, confirm its high performances in steady and transient states and demonstrate its feasibility and effectiveness. The experiment results have satisfied the limit specified by the IEEE harmonic standard 519.

Keywords

conventional PI controller / fractional calculus (FC) / total harmonic distortion (THD) / Oustaloup continuous approximation (OCA) / single phase active power filter (SPAPF)

Introduction

In fact, current harmonics have become a serious problem in power distribution systems caused by connecting nonlinear loads to the electrical grid, especially power electronic converters used in embedded devices in household, commercial and electronic-based appliances [ 1− 5]. To cope with these harmonics, traditional solutions based on passive filters (RLC circuits) have been designed regarding their advantages presented in simplicity and low cost. Moreover, these kinds of passive filters especially eradicate low order frequencies (3th, 5th, 7th, 11th…) but they are susceptible to originating series and parallel resonance phenomena with the grid [ 6, 7]. Unfortunately, the traditional solutions for power quality improvement are ineffective [ 1]. Recently, active power filters have been extensively used for reactive power compensation by reducing current and voltage harmonics [ 8]. Single phase active power filters (SPAPFs) are preferred because they have a lower cost, higher efficiency and a possibility to connect several SPAPFs in parallel to increase the range of power compensation [ 8] compared to series topologies which used transformers to be connected between the power supply and the loads.

A number of new control techniques have been proposed to control SPAPFs with regard to the integration of control and sensorless methods [ 1, 8]. The majority of these approaches, based on complicated calculus of instantaneous active and reactive power, lead to slow computing time and powerful calculation units (such as DSP). For these reasons, the direct current control (DCC) strategy which is generally composed of two control loops has been implemented. An outer voltage loop is used to set DC-link voltage while an inner current loop is used for regulating the filter phase currents [ 1]. The accuracy of the SPAPF depends essentially on the performance of the control loops [ 1].

In industrial fields, the PID controllers are known as a good solution in regulation loops, because of their simplicity, their low cost and their easy implementation. These controllers are good in steady-state but the critical point is how fast the system reaches the equilibrium state when external conditions have been introduced [ 1]. Thus, another kind of these controllers has been proposed in the literature named fractional order PID (FO-PID) controllers which can overcome the drawbacks of the conventional ones by giving them extra degree of freedom, offering fast response time in transient state [ 9− 11] and being flexible to parameter variations because of their flat phase margin around the gain crossover frequency [ 10, 11], but FO-PID controllers have bad effects on the power quality in steady-state, especially the need of an approximation method when building these kinds of controllers [ 12]. However, in active power filtering voltage regulation loops, a conventional PI controller [ 1] or a fractional order PI controller [ 2] is necessary to maintain the capacitor voltage at a desired level.

This paper proposes a robust switched fractional controller (SFC) integrated in the DCC algorithm to control a SPAPF. In the improved DCC approach, the DC-link voltage is regulated by the SFC which consists of a conventional PI controller, a FO-PI and a decision maker which switches between them regarding the working conditions. In steady-state, the conventional PI controller is selected by the decision maker to regulate the capacitor voltage at a suitable level to achieve the compensation objectives. On the other hand, the FO-PI controller is selected when severe load changes have been detected to deal with the abnormal conditions. The FO-PI controller with its adjusted order of integration ensures fast response time and low overshoot. Indeed, the FO-PI controller is insensitive to external load disturbances and parameter variations which have been proved by Podlubny in Refs. [ 12− 14]. To design the FO-PI controller, the theory of fractional calculus (FC) and the Oustaloup continuous approximation (OCA) method are necessary. The benefits of this approximation are demonstrated in Refs. [ 10, 11]. The validity of the enhanced DCC approach has been investigated through real time bench implementation. This improved approach is termed as SFC-DCC. The steady-state and dynamic behavior of the SFC-DCC technique have been presented with the robustness tests in practice.

Power system and control algorithm

Power system configuration

Consider a single phase compensation system as shown in Fig. 1. The electrical grid supplies nonlinear load consisting of the full bridge rectifier with an RL load which deteriorates the power quality of the mains. SPAPF is a voltage source inverter (VSI) with 4 insulated-gate bipolar transistors (IGBTs), including a capacitor C_dc in the DC side and an inductance L_f in the other VSI side. The SPAPF is connected through filtering inductance (L_f) to compensate the reactive power by eliminating the current harmonics and correct the power factor.

DCC technique

The technique presented in this paper to control the SPAPF is DCC that is chosen due to its simplicity and easiness of implementation as exhibited in Fig. 2. The proposed SFC is designed to make the capacitor voltage (V_dc) achieve its reference ( V_{\mathrm{d}\mathrm{c}}^{*} ) in fast settling time without being influenced by external disturbances. In more details, the SFC is integrated in the outer voltage regulation loop to generate the proper maximum source current amplitude (I_smax) which is multiplied by the source voltage that is passed through a phase locked loop (PLL) to build the reference current ( I_{\mathrm{s}}^{*} ) which is compared with the measured current (I_s) and passed through an hysteresis band that provides the VSI control signals.

DC-link voltage regulation loop

In regulation loops, the PID controllers were looked as a good solution in steady-state but were limited in transients. To deal with that, fractional order PID controllers have been proposed to offer better transients but they have bad effects on power quality in steady-state which is caused by the approximation methods used to implement this kind of controllers. The main idea of this paper is to take the advantages of both controllers to build a new robust controller by giving satisfying performances in steady and transient states. However, in active power filtering regulation loops, a simple conventional PI controller is required to maintain the capacitor voltage oscillating around its reference. Thus, the proposed SFC integrates a conventional PI controller and an FO-PI controller and a decision maker that switches between them depending on the ε value (the error between the capacitor voltage V_dc and its reference V_{\mathrm{d}\mathrm{c}}^{*} ) as depicted in Fig. 3.

Steps to design the proposed SFC

Observing Fig. 3, the steps to design the proposed controller are:

Step 1 Calculate the proportional and integrator gains of the conventional PI controller (k_pc, k_ic), (selected in steady-state).

Step 2 Calculate the proportional and integrator gains (k_i, k_p) and the order of integration α of the FO-PI controller (selected in dynamic state).

Step 3 Approximate the term s^{-\alpha } using OCA to transfer function (to be implemented in practical tests).

Step 4 Build the bode diagram to make a comparison between the conventional and fractional order controller (amplitude, phase margin).

Step 5 Give the decision maker intervals to select the best controller.

Conventional PI controller

In steady-state, the conventional PI controller of Fig. 4 is selected to meet the requirement of the power quality by providing proper maximum source current amplitude (I_smax).

The closed loop of the DC-link voltage regulation loop of the SPAPF is demonstrated in Fig. 5.

The transfer function (TF) of the system is given by Eq. (1) and the TF of the conventional PI controller is given by Eq. (2).

(1) P(s)={\displaystyle \frac{\mathrm{1}}{C_{\mathrm{dc}}s}},

(2) C_c(s)=k_{\mathrm{pc}}+{\displaystyle \frac{k_{\mathrm{ic}}}{s}}.

The TF of the regulation loop (Fig. 5) is calculated and given by Eq. (3).

(3) G_c(s)={\displaystyle \frac{V_{\mathrm{dc}}}{V_{\mathrm{dc}}^{*}}}={\displaystyle \frac{k_{\mathrm{pc}}s/C_{\mathrm{dc}}+k_{\mathrm{ic}}/C_{\mathrm{dc}}}{s^{\mathrm{2}}+k_{\mathrm{pc}}s/C_{\mathrm{dc}}+k_{\mathrm{ic}}/C_{\mathrm{dc}}}}.

Equation (3) is a second order equation which is similar to Eq. (4).

(4) G_c(s)={\displaystyle \frac{V_{\mathrm{dc}}}{V_{\mathrm{dc}}^{*}}}={\displaystyle \frac{\mathrm{2}\xi {\omega }_{n}s+{\omega }_n^2}{s^{\mathrm{2}}+\mathrm{2}\xi {\omega }_{n}s+{\omega }_n^{\mathrm{2}}}}.

By identification of Eq. (3) and Eq. (4), Eq. (5) is obtained.

(5) \{\begin{array}{l}{k}_{\mathrm{ic}}=C_{\mathrm{dc}}{\omega }_n^2\hfill \\ k_{\mathrm{pc}}=2\xi {\omega }_n{C}_{\mathrm{dc}}.\hfill \end{array}

FO-PI controller

The scheme of the FO-PI controller is the same as the one displayed in Fig. 4, but the integration order of (1/s) is adjusted from integer to real value ( \mathrm{1}/s^{\alpha } ) as described in Eq. (6).

(6) \{\begin{array}{ll}\alpha =\mathrm{1}\hfill & Conventional\; PI,\hfill \\ \alpha \mathrm{\in }[\mathrm{0},\mathrm{1}]\hfill & FO-PI.\hfill \end{array}

In the next section, a method to calculate the parameters k_i, k_p and α is clearly explained (being well elaborated in Ref. [ 12]).

Parameters calculus of FO-PI controller

The FO-PI controller has the following transfer function:

(7) C(s)=k_p+{\displaystyle \frac{k_i}{s^{\alpha }}}.

Assume that the gain crossover frequency ω_c and phase margin φ_m are given. From the basic definition of gain crossover frequency and phase margin, the following specifications can be obtained [ 14]:

1)Phase margin specification

(8) \arg [G({\omega }_c)]=\arg [C(j{\omega }_c)P(j{\omega }_c)]=-\pi +{\phi }_m.

2)Robustness to gain variations of the plant

(9) {\left({\displaystyle \frac{d\; arg(C(j\omega )P(j\omega ))}{d\omega }}\right)}_{\omega ={\omega }_c}=\mathrm{0.}

With the condition that the phase derivative at the ω_c frequency is zero, i.e., the phase bode plot is flat at the gain crossover frequency (Fig. 6), the system is more robust to gain changes since the response overshoot remain unchanged.

3)Amplitude specification:

(10) \left|G(j{\omega }_c)\right|=\left|C(j{\omega }_c)P(j{\omega }_c)\right|=\mathrm{1.}

The phase and amplitude of the plant in frequency domain can be given from Eq. (1) by

(11) \arg [P(\omega )]=-{\displaystyle \frac{\pi }{\mathrm{2}}},

(12) \left|P(j\omega )\right|={\displaystyle \frac{\mathrm{1}}{C_{\mathrm{dc}}\omega }}.

From the FO-PI controller transfer function (Eq. (7)), its frequency response can be obtained.

(13) C(j\omega )=k_p(\mathrm{1}+k_i{(j\omega )}^{-\alpha })=k_p\left[\left(\mathrm{1}+k_i{\omega }^{-\alpha }\cos {\displaystyle \frac{\alpha \pi }{\mathrm{2}}}\right)-j{k}_i{\omega }^{-\alpha }\sin {\displaystyle \frac{\alpha \pi }{\mathrm{2}}}\right].

The phase and amplitude are

(14) \arg [C(j\omega )]=-{\tan }^{-\mathrm{1}}\left({\displaystyle \frac{k_i{\omega }^{-\alpha }\sin (\alpha \pi /\mathrm{2})}{\mathrm{1}+k_i{\omega }^{-\alpha }\cos (\alpha \pi /\mathrm{2})}}\right),

(15) \left|C(j\omega )\right|=k_p\sqrt{{\left(\mathrm{1}+k_i{\omega }^{-\alpha }\cos {\displaystyle \frac{\alpha \pi }{\mathrm{2}}}\right)}^{\mathrm{2}}+{\left(k_i{\omega }^{-\alpha }\sin {\displaystyle \frac{\alpha \pi }{\mathrm{2}}}\right)}^{\mathrm{2}}}.

The open-loop TF G(s) is such that

(16) G(s)=C(s)P(s)

From Eq. (11) and (14), and specification Eq. (1), the phase of G(jω) can be expressed as

(17) \arg [G(j{\omega }_c)]=-\pi +{\phi }_m=-{\tan }^{-\mathrm{1}}\left({\displaystyle \frac{k_i{\omega }_c^{-\alpha }\sin (\alpha \pi /\mathrm{2})}{\mathrm{1}+k_i{\omega }_c^{-\alpha }\cos (\alpha \pi /\mathrm{2})}}\right)-{\displaystyle \frac{\pi }{\mathrm{2}}}.

From Eq. (17), the relationship between k_i and α can be established as

(18) k_i={\displaystyle \frac{\tan (-\pi /\mathrm{2}+{\phi }_m)}{{\omega }_c^{-\alpha }\sin (\alpha \pi /\mathrm{2})-{\omega }_c^{-\alpha }\cos (\alpha \pi /\mathrm{2})\tan (-\pi /\mathrm{2}+{\phi }_m)}}.

According to specification Eq. (2), Eq. (19) can be obtained.

(19) {\left[{\displaystyle \frac{d(\arg (G(j\omega )))}{d\omega }}\right]}_{\omega ={\omega }_c}={\left[{\displaystyle \frac{d\left(-{\tan }^{-\mathrm{1}}\left({\displaystyle \frac{k_i{\omega }_c^{-\alpha }\sin (\alpha \pi /\mathrm{2})}{\mathrm{1}+k_i{\omega }_c^{-\alpha }\cos (\alpha \pi /\mathrm{2})}}\right)-{\displaystyle \frac{\pi }{\mathrm{2}}}\right)}{d\omega }}\right]}_{\omega ={\omega }_c}=\mathrm{0.}

From Eq. (18) and (19), the parameters k_i and α can be calculated.

According to specification Eq. (3), an equation about k_p as follows can be established.

(20) k_p={\displaystyle \frac{C_{\mathrm{dc}}\omega }{\sqrt{{[\mathrm{1}+k_i{\omega }^{-\alpha }\cos (\alpha \pi /\mathrm{2})]}^{\mathrm{2}}+{[k_i{\omega }^{-\alpha }\sin (\alpha \pi /\mathrm{2})]}^{\mathrm{2}}}}}.

Clearly, Eq. (18)−(20) can be solved to get k_i, α and k_p. So, the parameters of the proposed controllers have been calculated but in simulation and practical test, the integration term s^−α should be approximated and given as a TF.

Oustaloup continuous approximation (OCA)

To approximate the term s^−α, several methods have been used, like the predictor-corrector approach [ 15], analytical and numerical calculation of the inverse Laplace transform [ 16], state-space representation [ 17] and OCA [ 10, 11]. Of these methods, the latter seems to be the most suitable in addition to providing the fractional order PI controller as a transfer function.

The fractional {\mathrm{PI}}^{\lambda }{D}^{\mu } controller proposed by Podlubny [ 13] has been presented with its transfer function and described as.

(21) C(s)=k_p+k_i{s}^{-\lambda }+k_d{s}^{\mu }\hspace{1em}(\lambda ,\mu >\mathrm{0}).

The relationship between the error input e(t) and control output u(t) of the FO-PI controller is given by Eq. (22).

(22) u(t)=k_{p}e(t)+k_{i\mathrm{0}}{D}_t^{-\alpha }e(t),

where the operator o_{\mathrm{0}}{D}_t^{-\alpha } denotes the αth order integrator with the fixed lower terminal (initial time) 0 and the moving upper terminal t [ 18]. Based on the Riemann−Liouville, the definition of fractional integration can be rewritten as Eq. (23) [ 18].

(23) u(t)=k_{p}e(t)+k_i\underset{\mathrm{0}}{\overset{t}{{\displaystyle \int }}}{\displaystyle \frac{{(t-\tau )}^{\alpha -\mathrm{1}}}{Г(\alpha )}}e(\tau )d\tau .

A comparison of the second terms of Eqs. (21) and (23) reveals that, in the fractional FO-PI controller, the weighted error is integrated instead of the error value. In this weighted integration, at time t, the function {(t-\tau )}^{\alpha -\mathrm{1}}/\mathrm{\Gamma }(\alpha ) plays the role of weight function for integrating the error history e(\tau ) where \tau \in [\mathrm{0},t] .

The existence of the weight function {(t-\tau )}^{\alpha -\mathrm{1}}/\mathrm{\Gamma }(\alpha ) in the structure of the FO-PI controller can improve the capabilities of this controller and help to overcome practical limitations of a traditional PI controller [ 12, 14].

The Oustaloup presented the approximation algorithm is used when a frequency band of interest is given by [ω_b, ω_h]; the term s^α can be substituted with Eq. (24) [ 18].

(24) s^{\alpha }=K{\displaystyle \prod }_{k=-N}^N{\displaystyle \frac{s+{\omega }_k^{\prime }}{s+{\omega }_k}}.

The Oustaloup’s approximation model of the term s^α is given in Ref. [ 18], where s is Laplace transform variable and α is a real number in the range of (‒1, 1). s^a is called a fractional order differentiator if 0<α<1 and a fractional order integrator if ‒1<α<0. The TF of the term s^a is given by Eq. (25).

(25) H(s)=K{\left({\displaystyle \frac{{\omega }_b}{{\omega }_h}}\right)}^{\alpha }{\displaystyle \prod }_{k=-N}^N{\displaystyle \frac{\mathrm{1}+s/{\omega }_k^{\prime }}{\mathrm{1}+s/{\omega }_k}},

where

{\omega }_k^{\prime }={\omega }_b{\left({\displaystyle \frac{{\omega }_h}{{\omega }_b}}\right)}^{\frac{k+N+(\mathrm{1}-\alpha )/\mathrm{2}}{\mathrm{2}N+\mathrm{1}}},{\omega }_k={\omega }_b{\left({\displaystyle \frac{{\omega }_h}{{\omega }_b}}\right)}^{\frac{k+N+(\mathrm{1}+\alpha )/\mathrm{2}}{\mathrm{2}N+\mathrm{1}}},

and K={\omega }_h^{\alpha } . {\omega }_k^{\prime }(k=\mathrm{0},\pm 1,\pm 2,\cdots \pm N) are the zeros, {\omega }_k(k=\mathrm{0},\pm 1,\pm 2,\cdots \pm N) are the poles of rang k, and 2N+ 1 is the order of approximation function [ 18].

The approximated controller (C_TF(s)) has the following transform:

(26) C_{\mathrm{TF}}\left(s\right)=k_p+k_{i}K{\left({\displaystyle \frac{{\omega }_b}{{\omega }_h}}\right)}^{\alpha }{\displaystyle \prod }_{k=-N}^N{\displaystyle \frac{1+s/{\omega }_k^{\prime }}{1+s/{\omega }_k}}.

The parameters of OCA’s to optimize the term s^−α are N = 3, ω_b=10⁻² rad/s, ω_h=10⁻⁶ rad/s, with α = 0.8.

Bode diagram comparison

Figure 6 shows a bode diagram of the conventional and fractional controllers. On the one hand, it presents the open loop diagram of the conventional PI controller that is used when the SPAPF is working in steady-state. In more details, the capacitor voltage is close to its reference. On the other hand, when perturbations are introduced into the SPAPF, the decision maker switches to the fractional controller to deal with the abnormal variations because of its flat phase until a frequency of 10⁶ rad/s.

Decision maker

The decision maker is a block that has the error ε as input and the command signal u(ε) as output to switch S for the PI controller or FO-PI controller according to Eq. (27).

(27) u\left(\epsilon \right)=\{\begin{array}{ll}0\hfill & \left|\epsilon \right|\in \left[-a,a\right],\hfill \\ 1\hfill & \mathrm{otherwise}.\hfill \end{array}

Figure 7 illustrates the values of command signal u(ε) versus the error ε. When the error is in the interval, the conventional PI controller is chosen by switching S to 0 (initial value); when external disturbances are detected, the decision maker switches to the FO-PI controller (1).

After building the SFC, practical tests have been conducted to validate the performance of the SFC-DCC.

Experiments results

Figure 8 presents the experimental setup of 1.5 kW and reproduces exactly the system shown in Fig. 1 which is composed of a single phase inverter based on four IGBTs in an H-bridge structure with a capacitor (C_dc = 1100 µF) on the DC side and a filtering inductance (L_f = 4 mH) on the AC side. The load used in the following experiment is a full bridge rectifier with an RL(L_L = 20 mH, R_L = 22 Ω) load. Besides these power elements, voltage and current sensors, measurement equipments (oscilloscope, quality of energy analyzer and multimeter), a real-time dSPACE 1104 card integrated in PC Pentium working in Matlab/Simulink environment is used to implement the proposed technique (SFC-DCC) with a sampling time equal to T_s = 50 µs.

Figure 9 represents the voltage and current waveforms using the SFC-DCC algorithm in steady-state under normal conditions. It shows that the load current is rich with harmonics. As seen in Fig. 9, the waveform of the source current after filtering become sinusoidal and in phase with the source voltage. Thus, the SPAPF is properly compensating for the grid harmonic currents.

Figures 10 and 11 represent the main results obtained from the power quality analyzer. In steady-state, the decision maker selects the conventional PI controller to offer proper power quality. Before filtering (Fig. 10), the source current has the same waveform as the load current with a total harmonic distortion (THD) equal to 31.4% (Fig. 10 a), the reactive power is equal to 170 var and the power factor is equal to 0.886. After filtering (Fig. 11), the THD is reduced to 2.5% and the reactive power to 67 var. Thus, the power factor is corrected from 0.886 to 0.984.

To verify the robustness of the proposed algorithm in transient state when selecting the FO-PI controller, an external variation of the nonlinear load has been introduced. As the load changes, the capacitor voltages (V_dc) of Fig. 12 smoothly follow their references (200 V) and behave as expected. The comparison of both parts of Fig. 12 shows that the DC-link voltage is more suitable when using the proposed SFC controller over the conventional PI controller with a short response time and low overshoot. In more details, when the decision maker detects a variation, it selects the FO-PI controller which is fast in achieving its reference in less than one period of settling time (tr<20 ms) and very low overshoot (1%) over the conventional one with a slow response time (tr>60 ms) and high overshoot (5%).

The comparisons of Table 1 are based on obtained practical results with previous published papers in single phase active power filtering. THD is used in steady-state comparison while the response time (t_r) and the overshoot (D) are used in transient state by increasing and decreasing the load to test the robustness of these techniques.

The advantages of the proposed method over the conventional one have been proved previously in practice in both steady and transient states and summarized in Table 1. To confirm the validity of the results, a comparison of the results with those in other works has been made (Ref. [ 19] and Ref. [ 20]) which also integrate the conventional PI controller and sliding mode respectively in regulation loops of SPAPF. This paper proves the superiority of the switched fractional controller over the previous controllers. All experimental results confirm the robustness of the SFC-DCC technique in both dynamic and steady-states. The efficiency of reactive power compensation and THD reduction to less than 3% makes the proposed algorithm an interesting solution for active power filtering. Thus, developing a simple algorithm with powerful performances, it means a reduction of the size, cost, and space of the overall power installation. It also means reducing the power losses in the equipment.

Conclusions

A novel controller named SFC has been proposed by integrating the DC-link voltage regulation loop of the SPAPF. The SFC consists of a conventional PI controller selected in steady-state to obtain the best power quality compensation, a FO-PI controller selected in transient state to deal with external disturbances and a decision maker which switches between them according to a simple algorithm. The different SFC design steps are presented by computing the gains of the conventional PI controller, then building the FO-PI controller based on the FC theory and OCA, and finally determining the band limits of the decision maker. To verify the effectiveness of the direct current control algorithm integrating the proposed controller, an experimental bench has been implemented in the laboratory. The obtained total harmonic distortion (THD_i = 2.5%) satisfies the IEEE harmonic standard 519 (THD_i<5%). All robustness tests prove the superiority of the improved algorithm with a settling time of less than 20 ms and a low overshoot. Thanks to its higher performances in steady and transient states, the proposed algorithm could be an interesting technique in regulation loops in general and especially in active power filtering. In addition to its feasibility, the low computational cost of this method can be considered as an attractive solution to be integrated in the existing inverters for renewable energy. In this way, valuable features can be offered to the customers without the need for an additional investment in equipment.

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