Anti-Windup FOPID-Based DPC for SAPF Interconnected to a PV System Tuned Using PSO Algorithm

So far, most of the world's energy is generated from fossil fuels (coal, oil, and gas), where these energy sources contribute to harmful gas emissions which are heavily involved in the global warming, as well as inducing pollution of the earth and organisms [1]. Furthermore, the excessive consumption of energy by these resources systematically leads to reduction of reserves of this kind of energy potential. Moreover, energy production is still a challenge of a great importance for the coming years since it is continually employed almost everywhere, i.e., in residential, commercial, and industrial areas. A good alternative is the power generation from renewable energies such as hydroelectric, geothermal, biomass, wind, and photovoltaic (PV), which operate without pollution effects on the atmosphere after use [2]. The availability of solar energy as an environment-friendly, unlimited, and free energy on the entire globe surface [3] has prompted researchers to select it among other existing sources of renewable energy for study and investigation. Meanwhile, the rapid growth of nonlinear loads integration causes problems in the electrical grids, such as reactive power and harmonic currents [4, 5]. Active power filters (APF) are the most resorted to solution for reducing the negative repercussions of such loads in the electrical grids [5]. Within the APF family, the shunt active power filter (SAPF), which is paralleled to the grid to inject a current that is opposing both the current harmonics and reactive power emitted by the load, to eventually make the current supplied by the electrical power system sinusoidal and in phase with its voltage, is commonly opted [6, 7]. An interesting combination would be to integrate a SAPF in a PV system to harvest their granular features mentioned above, all together [8, 9]. In the literature, many techniques have been presented to control the APF. Direct power control (DPC) is invented by Noguchi et al. in 1998 whose idea is inspired from the direct torque control (DTC) intended for electrical machines drives [5, 10]. The DPC method does not require current control loops or pulse-width-modulator (PWM) block. The switching table based on the correction of the reactive and active powers, as well as based on the sector indicating the angular position of the source voltage vector, is intended to select the switching states of the converter [11, 12]. In this context, researchers gave great importance to this table which was treated by Boukezata et al. [8] to achieve good performance of the DPC control. In most cases, the DPC is fed by a reference of zero reactive power and active one produced via the Proportional-Integral (PI) regulator of the converter dc-link voltage [5, 8]. Various control techniques are employed to maintain this latter at its reference value optimally, regardless of the operating conditions. Among those, the conventional PI controller is simple to implement and presents a good response in steady state [13, 14]. This controller, however, exhibits poor performance during dynamic conditions, which are in the case of SAPF integrated in a PV system start-up, changing solar irradiance, and changing load.

To overcome this situation, the proposed control in this paper is carried out by an anti windup fractional order proportional-integral differentiator AW-FO($\mathrm{P} \mathrm{I}^{\mathscr{E}} \mathrm{D}^{\eta}$) controller, replacing the standard PI regulator that keeps the DC bus voltage at its desired value. The benefits offered by this AW-FO($\mathrm{P} \mathrm{I}^{\mathscr{E}} \mathrm{D}^{\eta}$) regulator with two extra freedom degrees ℰ and η allows having better dynamic response and shorter response time compared to the conventional PI controller [15-17]. Moreover, the output of the proposed AW-FOPID controller effectively participates in the delivery of the active power compared to the standard PI regulator employed in DPC that suffers from weak responses in dynamic mode. Indeed, this is the first time that the AW-FOPID controller requiring the determination of five optimized parameters has been incorporated into the DPC. Regarding the adjustment of this AW-FOPID parameters, Particle Swarm Optimization (PSO) algorithm is used to minimize the Integral Time Absolute Error (ITAE) [18].

As the solar insolation varies, the voltage corresponding the maximum power point (MPPT) varies. Therefore, several algorithms of MPPT such as incremental conductance (IC), perturb and observe (P&O), and hill climbing (HC) have been proposed in the literature [19, 20]. The advantages offered by these aforementioned algorithms reside in the ease of calculation and implementation. Due to the deficiencies of the aforementioned algorithms especially during dynamically changing weather conditions, intelligent controllers like fuzzy logic has been used in tracking effectively the maximum power point (MPPT) in PV systems, whatever abrupt changes affecting solar irradiance and temperature [9, 21]. On the other hand, the design of fuzzy MPPT proposed in this paper is not subject to well-defined criteria but is mainly based on experience.

This paper is sectioned in the following way: Description of the operating principle of the SAPF is explained in Section 2; The principle of the DPC applied to the SAPF together with PSO tuned AW-FO($\mathrm{P} \mathrm{I}^{\mathscr{E}} \mathrm{D}^{\eta}$) controller design to regulates the DC-link voltage, and fuzzy logic controller (FLC) for tracking the maximum power point, are presented in Section 3. Simulation results are given and discussed in Section 4. Finally, the presented work is concluded in Section 5.

3.1 Direct power control

The basic principle of DPC was proposed by Noguchi [11], while it was initially inspired from the DTC of electric motors control [23]. In the DPC strategy, reactive and active powers imitate, respectively, the electromagnetic torque and the amplitude of the stator flux of the DTC. This non-linear method is known as a direct power control technique because it chooses the optimal voltage vector without need for any modulation technique or coordinates transformation. The basic concept of the DPC is to select the appropriate states from the switching table based on localisation of the source voltage vector and errors [22-24].

In the proposed DPC method, the continuous bus voltage is maintained at the desired level through an AW-FOPID regulator-based voltage regulation loop.

The instantaneous active and reactive powers are calculated starting from the following equations:

${{P}_{s}}={{V}_{sa}}{{I}_{sa}}+{{V}_{sb}}{{I}_{sb}}+{{V}_{sc}}{{I}_{sc}}$         (2)

${{Q}_{s}}=\frac{1}{\sqrt{3}}\left[ \left( {{V}_{sb}}-{{V}_{sc}} \right){{I}_{sa}}+\left( {{V}_{sc}}-{{V}_{sa}} \right){{I}_{sb}}+\left( {{V}_{sa}}-{{V}_{sb}} \right){{I}_{sc}} \right]$      (3)

${{S}_{s}}={{P}_{s}}+j{{Q}_{s}}$          (4)

The reference of the reactive power is set to zero value to ensure a unity power factor. Whereas, the reference of the active power is developed with the multiplication of the peak value of the current source generated by the AW-FOPID regulator and the optimal value of the PV generator voltage. Then, the powers are compared and the obtained errors are applied to the hysteresis regulators [9, 22, 24]. The used hysteresis regulators allow restricting the errors of the instantaneous reactive and active powers in the desired band, as shown in Figure 2. The output of the controller switches between 1 and 0, where it is 1 if the error is positive and 0 otherwise. The influence of each control vector applied to the APF on the reactive and active powers is dependent on the actual position of the source voltage vector. Thus, the switching Table 1 has as inputs the signals from both hysteresis comparators and the information on the source voltage vector.

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Figure 2. General structure of SAPF controlled by the proposed DPC approach in presence of the PV system

Table 1. Switching table of DPC strategy

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Figure 3. Sectors on stationary coordinates

According to the angle between the inverter output voltage reference and the α axis, the sector will be selected as shown in Figure 3. Consequently, the angle is determined by an inverse trigonometric function based on the vector components of the voltage in the fixed reference space (α, β):

${{\theta }_{p}}={{\tan }^{-1}}\left( \frac{{{v}_{s\beta }}}{{{v}_{sa}}} \right)$       (5)

p is the number of the sector. Accordingly the inverter output voltage reference is selected based on the desired reactive and active power values.

3.2 Particle swarm optimization (PSO) based fractional order $\mathrm{P} \mathrm{I}^{\mathscr{E}} \mathrm{D}^{\eta}$ controller design

3.2.1 AW-FOPID controller

The conventional PI controller suffers from some weakness in the dynamic state. To overcome this drawback, the proposed control is carried out by an AW-FOPID, replacing the standard PI regulator to keep the DC bus voltage at its desired value with shorter response time during dynamic conditions, while the overshoots and undershoots are maintained at minimum levels. The AW-FOPI^εD^η has been introduced in 1999 with its general form in which the integral and derivative action orders, ε and η respectively, are not integers [25], as shown in Figure 4. The AW-FOPID controller has good convergence and conservation of the adjusted variable to its desired value. Moreover, due to the better dynamic response, flexibility and low sensitivity to eventual variations of the system parameters, AW-FOPID controllers belong to the dominating industrial controllers which have attracted the attention of several researches in different fields, such as: aerospace control systems [26], hypersonic flight vehicle, and automatic voltage regulation [27-30].

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Figure 4. AW-FOPID controller structure

From Figure 4, the transfer function G(s) of the AW-FOPID controller is calculated by the following equation:

$G\left( s \right)=\frac{U\left( s \right)}{E\left( s \right)}={{K}_{p}}+{{K}_{i}}{{s}^{-\varepsilon }}+{{K}_{d}}{{s}^{\eta }}$                 (6)

where K_p, K_i, K_d are the proportional, integral and derivative gain factors, respectively; ε, η are the integral and derivative orders respectively; R(s) is the input signal; C(s) represents the plant’s transfer function; E(s) is the error signal and Y(s) is the output signal.

It can be noticed that the selection of $\varepsilon$, $\eta$ gives the conventional controllers, i.e. PID controller ($\varepsilon$, $\eta$=1), PD controller ($\varepsilon$=0) and PI controller ($\eta$=0).

(1) Approximation method of fractional order operators

The method proposed by Oustaloup [31] is more elaborated and more convenient to approximate the fractional order (FO) to Laplace transfer functions (TFs). The term s^α of the Oustaloup’s approximation model is defined in the study [32], where s is the Laplace transform variable and α is a real number which ranges between -1 and 1. s^α is designated as an FO differentiator if (0<α<1) and as an FO integrator (−1<α<0). Moreover, this method is called recursive Oustaloup's filter and it distributed in a limited frequency band [wb wh]. So, the approximate of the fractional order (FO) to Laplace transfer functions (TFs) is assessed as follows:

${{s}^{\alpha }}=K\prod\limits_{k=-N}^{N}{\frac{s+{{{{\omega }'}}_{k}}}{s+{{\omega }_{k}}}}$           (7)

where:

${{{\omega }'}_{k}}={{\omega }_{b}}{{\left( \frac{{{\omega }_{h}}}{{{\omega }_{b}}} \right)}^{\left( \frac{k+N+0.5(1-\alpha )}{2N+1} \right)}}$            (8)

${{\omega }_{k}}={{\omega }_{b}}{{\left( \frac{{{\omega }_{h}}}{{{\omega }_{b}}} \right)}^{\left( \frac{k+N+0.5(1+\alpha )}{2N+1} \right)}}$         (9)

$K=\omega _{h}^{\alpha }$           (10)

With ω_k′ and ω_k are respectively the zeros and poles of interval k; K represents the adjustment gain; ω_b and ω_h are respectively the low and high frequencies; N is the number of poles and zeros, and (2N+1) is the approximation function order.

(2) Particle swarm optimization (PSO)

According to (6), the AW-FOPID controller has five parameters to be adjusted (K_d, K_p, K_i, ℰ and η). To tune these parameters, the PSO algorithm is used by minimizing the objective function (F). PSO algorithm is a stochastic population-based computer algorithm to find an optimal solution to a problem. This technique was initially invented by Russel James Kennedy and Eberhart in 1995. It was developed based on the behaviour of some animals, such as fish and birds [28, 29]. In PSO algorithm, each individual is named “particle” which is considered as a candidate solution to find the optimal solution to the problem. A particle’s position is affected by its own best found position, as well as the position of the best particle in its neighbourhood. For the local best PSO, a smaller neighbourhoods are determined for each particle. Whereas in the global best PSO, the neighborhood for each particle is all particle’s of swarm (entire swarm).

The fitness function evaluates and quantifies the performance of a particle [28, 30]. The personal best position of the particle i is estimated as

${{y}_{i}}(t+1)=\left\{ \begin{align}  & {{y}_{i}}(t)\text{  if F(}{{x}_{i}}(t+1)\ge F({{y}_{i}}(t)) \\  & {{x}_{i}}(t+1)\text{ if F(}{{x}_{i}}(t+1)\langle F({{y}_{i}}(t)) \\ \end{align} \right.$       (11)

With x_i is the particle's current position; F is the objective function. This best position of the particle i is updated at time step t. For the global best position, y is defined as:

$y=\text{ min}\left\{ F \right.\text{(}{{\text{y}}_{0}}(t)),........,F({{y}_{s}}(t)\left. ) \right\}$             (12)

With s is the swarm’s size. The standard equations of PSO for each dimension j: jϵ {1,..., N_d } are given as:

${{v}_{i}}_{,j}\text{=}\omega \text{.}{{v}_{i}}_{,j}(t)+{{c}_{1}}.{{\Delta }_{1}}+{{c}{2}}.{{\Delta }_{2}}$           (13)

$\text{  }{{x}_{i}}(t+1)={{x}_{i}}(t)+{{V}_{i}}(t+1)$         (14)

where:

${{\Delta }_{1}}={{r}_{1,j}}.({{y}_{i,j}}(t)-{{x}_{i,j}}(t))\text{ }$             (15)

${{\Delta }_{2}}={{r}_{2,j}}.(y_{j}^{n}(t)-{{x}_{i,j}}(t))\text{ }$              (16)

v_{i, j}  is the jth element of the velocity vector of the ith particle, ω is the inertia weight, c₁ and c₂ are the acceleration constants, and r_1,j , r_2,j are random coefficients in the range [0, 1]. When the velocity updates tend to zero, this process is stopped.

Various optimization criteria are used for adjusting the process controllers. Among those, the minimization of Integral Time Absolute Error (ITAE):

$Min\text{ F}=Min[\int\limits_{0}^{\infty }{t\left| e(t) \right|}dt]$          (17)

The PSO algorithm used to design the AW-FOPID controller parameters is represented in Figure 5.

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Figure 5. PSO-based AW-FOPID controller parameters design

In every iteration, each particle needs to update its own best individual fitness. The individual fitness of each particle is measured by using the Integral Time Absolute Error (ITAE) which can be expressed as:

$MSE=\frac{\sum\limits_{r=0}^{N}{{{e}^{2}}}}{N}$             (18)

where, N denotes the total number of points for which the optimization is carried out, t_s is the time rang of simulation. In this study, PSO with the spreading factor (PSO-SF) [32-38] is used and modified so that the algorithm can be used to tune the AW-FOPID controller parameters. By using the spreading factor approach, the values of the acceleration coefficient and inertia weight can be linearly converged to the predefined values over time as illustrated by the following equations:

${{w}_{e}}={{e}^{(-current\text{ epoch/(SF}\times total\text{ epoch))}}}$            (19)

where: SF = 0.5(spread + deviation).

${{c}_{1}}={{c}_{2}}=2\times (1-(current\text{ epoch/total epoch))}$          (20)

The instructions to be followed by the algorithm of tuning process of AW-FOPID controller by PSO are given as follows:

1- Initialize the parameters and specify the lower and upper bounds of the five controller parameters: inertia weight (we) from 0 to 1, acceleration coefficients range (c₁ and c₂) from 0.05 to 2, position range from 0.01 to 15 and velocity range from -0.001 to 0.5;

2- Randomly distribute particles within specified ranges;

3- While current number of epochs is less than 1000;

4- Evaluate the fitness of each particle using Eq. (18) with MSE tends to 0;

5- If the current individual fitness is better than the previous individual fitness, then update new individual fitness;

6- Identify the best particle fitness among the swarm;

7- If the current population fitness is better than the previous population fitness, then update new population fitness;

8- Calculate the velocity and update the position using Eqns. (13) and (14) respectively;

9- Calculate the new values of the acceleration coefficients and the inertia weight using Eqns. (20) and (19) respectively;

10- End.

3.3 Fuzzy logic MPPT controller

3.3.1 Photovoltaic array model

The photovoltaic array employed in the developed system is a KC200GT, characterized by a single diode model [33, 34]. From Figure 6, the equivalent circuit of the PV cell is illustrated by a current source in parallel with a diode and series/parallel resistors.

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Figure 6. Electrical equivalent circuit of a PV cell

The current relation of the PV cell is represented by the next equation:

$I={{I}_{pv,cell}}-{{I}_{0,cell}}\left[ {{e}^{\left( \frac{qV}{aKT} \right)}}-1 \right]$             (21)

where:

${{I}_{pv}}=\left( {{I}_{pv,n}}+{{K}_{I}}{{\Delta }_{T}} \right)\frac{G}{{{G}_{n}}}$              (22)

I_pv,cell is the generated current by the incident light; I_0,cell is the leakage current of the diode; K is Boltzmann’s constant, all measured in ampere [J.K^-1]; T is the effective cell’s temperature, measured in Kelvin [K]; q is the electron’s charge, measured in Coulomb [C]; a is the non-ideal junction factor, measured in ampere; I_pv;n is the solar generated current at the nominal condition; Δ_T=T-T_n (T_n and T is the nominal and actual temperatures, measured in Kelvin [K]; respectively); G_n(W/m²) is the nominal irradiation; G is the irradiation and K_I is the short circuit current/temperature coefficient.

Considering the PV panel series and shunt resistors gives the following:

$I={{I}_{pv}}-{{I}_{0}}\left[ {{e}^{\left( \frac{q\left( V+I{{R}_{s}} \right)}{a{{N}_{s}}KT} \right)}}-1 \right]-\frac{V+I{{R}_{s}}}{{{R}_{p}}}$          (23)

where: N_s is the number of cells mounted in series. A practical PV array consists of many PV cells connected in series and in parallel. This is achieved by adding some the series and parallel coefficients as follows [35]:

$\begin{align}  & I={{N}_{pp}}{{I}_{pv}}-{{N}_{pp}}{{I}_{0}}\left( {{e}^{\left( \frac{{{N}_{ss}}V+I{{R}_{s}}\left( {\scriptstyle{}^{Nss}/{}_{{{N}_{pp}}}} \right)}{{{N}_{ss}}\alpha {{V}_{t}}} \right)}}-1 \right) \,\,\,\,\,\,\,\,-\frac{{{N}_{ss}}V+I{{R}_{s}}\left( {\scriptstyle{}^{Nss}/{}_{{{N}_{pp}}}} \right)}{{{R}_{p}}\left( {\scriptstyle{}^{{{N}_{ss}}}/{}_{{{N}_{pp}}}} \right)} \\ \end{align}$             (24)

where: N_ss is the number of PV panels mounted in series and N_pp is the number of PV panels mounted in parallel.

3.3.2 Maximum power point tracking (MPPT)

(1) Fuzzy logic controller

Fuzzy logic method is employed for tracking the MPP of PV module to achieve good efficiency under any weather conditions. The fuzzy logic approach is very efficient for both linear and nonlinear controlled systems, while the mathematical model is not needed [9, 21].

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Figure 7. Fuzzy logic controller block diagram

Generally, the FLC is executed in three essential steps: Fuzzification, Rules inference, and Defuzzificationas shown in Figure 7 [21, 36]. The Fuzzification step is the process of changing the digital input variables into linguistic equivalent, which are achieved by using membership functions. The Rules inference step determines the output of the fuzzy logic controller by Mamdani method with a max-min technique depending on the set belonging to the rule base. The Defuzzification step converts the linguistic variables into a crisp value which calculates the output control variable. Since at the MPP of the PV array, ΔP(k) and ΔV(k) are null, the proposed algorithm, therefore, has two input variables, which are based on Eqns. (25) and (26) [37]:

$\text{ }\!\!\Delta\!\!\text{ P}=\text{P}\left( \text{k} \right)-\text{P}\left( \text{k}-1 \right)$         (25)

$\text{ }\!\!\Delta\!\!\text{ V}=\text{V}\left( \text{k} \right)-\text{V}\left( \text{k}-1 \right)$         (26)

where: P(k), P(k-1), V(k), and V(k-1) are respectively, the PV power and voltage at the sampling times k and (k-1), respectively. These inputs of the fuzzy MPPT are represented by the error E and its variation ∆E [36, 37]:

$\left\{ \begin{align}  & \text{E(k)}=\frac{\text{ }\!\!\Delta\!\!\text{ P}}{\text{ }\!\!\Delta\!\!\text{ V}} \\  & \text{ }\!\!\Delta\!\!\text{ E(k)}=\text{E(k)}-\text{E(k}-\text{1)} \\ \end{align} \right.$            (27)

The input variables E(k) and ΔE(k) are divided into five fuzzy sets which are denoted as: Negative Big (NB), Negative Small (NS), Zero (ZO), Positive Small (PS), and Positive Big (PB). The rule base connects the fuzzy inputs to the fuzzy output by the master rule of syntax: '' if X and Y, then Z'' [9, 37], as listed in Table 2.

Table 2. Fuzzy logic decision table

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Figure 8. Membership functions for inputs and output variables

For ease of calculation, equilateral triangle membership functions are chosen (Figure 8). The center of gravity method for defuzzification step is used to calculate the incremental duty cycle ΔD as [9, 37]:

$\Delta D=\frac{\sum\nolimits_{j=0}^{n}{{{w}_{j}}\Delta {{D}_{j}}}}{\sum\nolimits_{j=0}^{n}{{{w}_{j}}}}$       (28)

With: n is the maximum number of effective rules, w represents the weighting factor and ΔD_j is the value corresponding to ΔD. Finally, the duty cycle is obtained by adding this change to the previous value of the control duty cycle as expressed in the following [37]:

$D\left( K+1 \right)=D\left( K \right)+\Delta D\left( K \right)$              (29)