PIL Implementation of Feedback Linearisation-SVM Control of 3-Phase Multifunctional Grid-Tied Solar PV Integrated UPQC

Figure 1 depicts the basic operation of the suggested control approach for the PVG-UPQC associated with a nonlinear load. A two-level Space Vector Modulator generates the switch control signals. The SVM technique has been opted in this paper to achieve several common objectives: reduction of the total harmonic distortion (THD) and therefore less switching losses and giving more possibility of controlling the system with a constant frequency. In addition, the SVM control technique has taken the focusing of many works in controlling their system inverters [15-17].

The instantaneous PQ theory for currents and PQ-PLL for voltages are used to establish the harmonic references [18, 19]. Compensation goals include voltage and current harmonics mitigation, reactive power compensation and DC bus regulation during the bidirectional active power exchange between the PV generator, two active filters and power system grid. A detailed analysis of the different components of PVG-UPQC is presented hereafter.

1.png

Figure 1. Feedback linearisation- SVM control scheme of the PVG-UPQC

2.1 Mathematical model of GPV-UPQC

The PVG-UPQC’s dynamic model is described by a differential equation defined in α-β stationary frame. The latter is presented in Eqns. (1) and (3).

The parallel filter model is governed by the following equation [9]:

$\begin{align}  & \frac{d{{i}_{fp\alpha }}}{dt}=-\frac{{{R}_{fp}}}{{{L}_{fp}}}{{i}_{fp\alpha }}-\frac{{{v}_{s\alpha }}}{{{L}_{fp}}}+\frac{{{v}_{fp\alpha }}}{{{L}_{fp}}} \\  & \frac{d{{i}_{fp\beta }}}{dt}=-\frac{{{R}_{fp}}}{{{L}_{fp}}}{{i}_{fp\beta }}-\frac{{{v}_{s\beta }}}{{{L}_{fp}}}+\frac{{{v}_{fp\beta }}}{{{L}_{fp}}} \\ \end{align}$                (1)

It can also be written in form (2).

${{\dot{x}}_{p}}={{f}_{p}}({{x}_{p}})+{{g}_{p}}({{x}_{p}}){{u}_{p}}$                 (2)

where:

$f_{p}\left(x_{p}\right)=\left[\begin{array}{cc}-\frac{R_{f p}}{L_{f p}} i_{f p \alpha}-\frac{v_{s \alpha}}{L_{f p}} \\ -\frac{R_{f p}}{L_{f p}} i_{f p \beta}-\frac{v_{s \beta}}{L_{f p}}\end{array}\right], g_{p}\left(x_{p}\right)=\left[\begin{array}{cc}\frac{1}{L_{f p}} & 0 \\ 0 & \frac{1}{L_{f p}}\end{array}\right]$, $x_{p}=\left[\begin{array}{c}i_{f p \alpha} \\ i_{f p \beta}\end{array}\right], u_{p}=\left[\begin{array}{c}v_{f p \alpha} \\ v_{f p \beta}\end{array}\right]$

                 $y_{p}=\left[\begin{array}{l}y_{p 1} \\ y_{p 2}\end{array}\right]=\left[\begin{array}{l}h_{p 1} \\ h_{p 2}\end{array}\right]$                   .

v_sαβ are source voltages in α-β frame, i_fpαβ and v_fpαβ are the α-β axis currents and voltages of the shunt filter respectively.

The series filter model is described by [12]:

$\begin{align}  & \frac{d{{i}_{fs\alpha }}}{dt}=-\frac{{{R}_{fs}}}{{{L}_{fs}}}{{i}_{fs\alpha }}-\frac{{{v}_{inj\alpha }}}{{{L}_{fs}}}+\frac{{{v}_{fs\alpha }}}{{{L}_{fs}}} \\  & \frac{d{{i}_{fs\beta }}}{dt}=-\frac{{{R}_{fp}}}{{{L}_{fs}}}{{i}_{fs\beta }}-\frac{{{v}_{inj\beta }}}{{{L}_{fs}}}+\frac{{{v}_{fs\beta }}}{{{L}_{fs}}} \\ \end{align}$                  (3)

These equations can also be expressed as in (4).

${{\dot{x}}_{s}}={{f}_{s}}({{x}_{s}})+{{g}_{s}}({{x}_{s}}){{u}_{s}}$                 (4)

where:

$f_{s}\left(x_{s}\right)=\left[\begin{array}{cc}-\frac{R_{f s}}{L_{f s}} i_{f s \alpha}-\frac{v_{i n j \alpha}}{L_{f s}} \\ -\frac{R_{f s}}{L_{f s}} i_{f s \beta}-\frac{v_{i n j \beta}}{L_{f s}}\end{array}\right]$,

$g_{s}\left(x_{s}\right)=\left[\begin{array}{ll}\frac{1}{L_{f s}} & 0 \\ 0 & \frac{1}{L_{f s}}\end{array}\right], x_{s}=\left[\begin{array}{l}i_{f s \alpha} \\ i_{f s \beta}\end{array}\right], u_{s}=\left[\begin{array}{l}v_{f s \alpha} \\ v_{f s \beta}\end{array}\right]$

$y_{s}=\left[\begin{array}{l}y_{s 1} \\ y_{s 2}\end{array}\right]=\left[\begin{array}{l}h_{s 1} \\ h_{s 2}\end{array}\right]$

v_injαβ are the injected voltages of the series filter in the α-β coordinates, i_fsαβ are the α-β axis currents of the series filter, and v_fsαβ are series filter’s voltages.

The DC-Bus is modeled as follows:

$\frac{dv_{dc}^{2}}{dt}=\frac{2{{P}_{c}}}{{{C}_{dc}}}$                         (5)

It can be also expressed in the form of (6).

${{\dot{x}}_{dc}}={{f}_{dc}}({{x}_{dc}})+{{g}_{dc}}({{x}_{dc}}){{u}_{dc}}$                   (6)

where: $f_{d c}\left(x_{d c}\right)=0, g_{d c}\left(x_{d c}\right)=\frac{2}{C_{d c}} \cdot$ and $\cdot u_{d c}\left(x_{d c}\right)=P_{c}$.

The output of the photovoltaic generator (PVG) is connected to a boost converter, as represented in Figure 2.

2.png

Figure 2. Boost converter of the PV generator

The state space model for this converter is represented by the dynamic equations below [14]:

$\begin{align}  & \frac{d{{v}_{pv}}}{dt}=\frac{1}{{{C}_{pv}}}{{i}_{pv}}-\frac{1}{{{C}_{pv}}}{{i}_{Lpv}} \\  & \frac{d{{i}_{Lpv}}}{dt}=\frac{1}{{{L}_{pv}}}{{v}_{pv}}-\frac{1}{{{L}_{pv}}}{{v}_{dc}} \\ \end{align}$                  (7)

with D is defined as duty cycle, the boost’s average model becomes as follows:

$\begin{align}  & \frac{d{{i}_{Lpv}}}{dt}=\frac{1}{{{L}_{pv}}}{{v}_{pv}}-\frac{1}{{{L}_{pv}}}(1-D){{v}_{dc}} \\  & \frac{d{{v}_{pv}}}{dt}=\frac{1}{{{C}_{pv}}}{{i}_{pv}}-\frac{1}{{{C}_{pv}}}(1-D){{i}_{Lpv}} \\ \end{align}$                 (8)

The equations can be put in the following form:

${{\dot{x}}_{b}}={{f}_{b}}({{x}_{b}})+{{g}_{b}}({{x}_{b}}){{u}_{b}}$             (9)

where:

$f_{b}\left(x_{b}\right)=\left[\begin{array}{c}\frac{1}{L_{p v}} v_{p v} \\ \frac{1}{C_{p v}} i_{p v}\end{array}\right], g_{b}\left(x_{b}\right)=\left[\begin{array}{cc}-\frac{1}{L_{f p}} & 0 \\ 0 & -\frac{1}{C_{p v}}\end{array}\right]$,

$x_{b}=\left[\begin{array}{c}i_{L p v} \\ v_{p v}\end{array}\right], u_{b}=\left[\begin{array}{c}v_{d c} \\ i_{L p v}\end{array}\right]$

$y_{b}=\left[\begin{array}{l}y_{b 1} \\ y_{b 2}\end{array}\right]=\left[\begin{array}{l}h_{b 1} \\ h_{b 2}\end{array}\right]$

2.2 MPPT detection algorithm

The DC-DC boost converter is commonly utilised in solar PV units, amplifying and regulating the PV panel voltage at a given level while extracting maximum power. The boost converter is controlled by a Maximum Power Point Tracking (MPPT) controller [5].

In this study, the P&O detection algorithm is employed because of its simplicity of use and low-cost implementation [7]. The flowchart of P&O detection algorithm is presented in Figure 3.

The Perturb & Observe algorithm consists of changing the operating point of the PV generator by raising or reducing the duty cycle of the boost converter for the aim of measuring the output power before and after the perturbation. The algorithm perturbs the structure in the same direction as the power increases; otherwise, it perturbs the structure in the reverse direction. As shown in Figure 3, there are four possible options that are presented during the tracking of the MPPT.

2.3 Harmonic identification

The identification strategy used to remove harmonics from perturbed waveforms has a big impact on the performance of the active filter [10, 19]. The methods employed for extracting harmonics are detailed in the following parts.

2.3.1 Harmonic currents identification using PQ-theory

The instantaneous power theory technique (PQ theory) is used in this study as illustrated in Figure 4.

The load's instantaneous powers are computed as follows [18]:

$\left[\begin{array}{c}P_{l} \\ Q_{l}\end{array}\right]=\left[\begin{array}{cc}v_{s \alpha} & v_{s \beta} \\ v_{s \beta} & -v_{s \alpha}\end{array}\right]\left[\begin{array}{l}i_{l \alpha} \\ i_{l \beta}\end{array}\right]$             (10)

3.png

Figure 3. P&O detection algorithm

4.png

Figure 4. Harmonic currents extraction scheme using PQ theory

5.png

Figure 5. Harmonic voltages extraction scheme using PQ-PLL theory

The powers could be written in the following way:

$\left\{\begin{array}{l}P_{l}=\bar{P}_{l}+\tilde{P}_{l} \\ Q_{l}=\bar{Q}_{l}+\tilde{Q}_{l}\end{array}\right.$                  (11)

For compensating reactive power and mitigating harmonics, the total reactive power ($\bar{Q}_{l}$ and $\widetilde{Q}_{l}$ components) with the oscillatory component of active power are chosen as compensatory power references, then, the compensating currents reference are calculated as in (12).

$\left[\begin{array}{c}i_{f p \alpha}^{*} \\ i_{f p \beta}^{*}\end{array}\right]=\frac{1}{v_{s \alpha}^{2}+v_{s \beta}^{2}}\left[\begin{array}{cc}v_{s \alpha} & v_{s \beta} \\ v_{s \beta} & -v_{s \alpha}\end{array}\right]\left[\begin{array}{c}\tilde{P}_{l} \\ Q_{l}\end{array}\right]$             (12)

2.3.2 Harmonic voltages identification using PQ-PLL theory

It consists of two parts in which the first is to extract voltage harmonics, and it is similar to PQ theory for currents (13).

$\left[\frac{\overline{{\mathcal{V}}_{s \alpha}}}{{\mathcal{V}}_{s \beta}}\right]=\frac{1}{i_{l \alpha}^{2}+i_{l \beta}^{2}}\left[\begin{array}{cc}i_{l \alpha} & i_{l \beta} \\ -i_{l \beta} & i_{l \alpha}\end{array}\right]\left[\begin{array}{l}\overline{P}_{l} \\ \overline{Q_{l}}\end{array}\right]$                 (13)

The harmonic voltages are calculated by:

$\left[\begin{array}{l}\mathcal{V}_{h \alpha} \\ \mathcal{V}_{h \beta}\end{array}\right]=\left[\begin{array}{l}\mathcal{V}_{s \alpha} \\ \mathcal{V}_{s \beta}\end{array}\right]-\left[\begin{array}{l}\overline{\mathcal{V}_{s \alpha}} \\\overline{\mathcal{V}_{s \beta}}\end{array}\right]$            (14)

By adding the voltage drop across the load, the reference voltages are given by:

$\left[\begin{array}{c}v_{f s \alpha}^{*} \\ v_{f_{s} \beta}^{*}\end{array}\right]=\left[\begin{array}{c}v_{h \alpha} \\ v_{h \beta}\end{array}\right]+\left[\begin{array}{c}\Delta v_{l \alpha} \\ \Delta v_{l \beta}\end{array}\right]$               (15)

Figure 5 represents the schematic diagram of the PQ-PLL theory.

2.4 Feedback linearisation controller (FLC) synthesis

Consider the nonlinear system represented in:

$\dot{x}=f(x)+\sum_{i=1}^{p} g_{i}(x) u_{i} \quad i=1,2, \ldots, p$

                             $y_{i}=h_{i}(x)$                                                                     (16)

g(x), f(x) and h(x) represent a scalar functions.

The well-known approach for forming the previous system's feedback linearisation law is shown in Figure 6 [20]. The problem of determining the system's vector relative degree (13) necessitates differentiation of each output signal until one of the input signals is explicitly included in the differentiation. We define r_j as the minimal integer for each output signal that has one input in $y_{j}^{\left(r_{j}\right)}$ [9]:

$y_{j}^{\left(r_{j}\right)}=L_{f}^{r_{j}} h(x)+\sum_{i=1}^{p} L_{g i}\left(L_{f}^{r_{j}-1} h_{j}(x)\right) u_{i} \quad j=1,2, \ldots, ., p$                        (17)

where: $L_{f} h_{j}$ and $L_{g i}\left(L_{f}^{r_{j}-1} h_{j}\right)$ represent the Lie derivatives of h with regard to f and g.

The global relative degree equals to the total of all relative degrees calculated using (13) [21]. It should be less than or equal to the system's order.: $r=\sum_{j=1}^{p} r_{j} \leq n$_.

The formula (17) can be represented in its matrix form to obtain the expression of linearising law u that permits making the relationship between inputs and outputs linear [9]:

$\left[\begin{array}{cccc}y_{1}^{r_{j}} & \ldots & \ldots & y_{p}^{r_{p}}\end{array}\right]^{t}=\zeta(x)+D(x) u$              (18)

where:

$\zeta (x)=\left( \begin{align}  & {{L}_{f}}^{{{r}_{1}}}{{h}_{1}}(x) \\  & {{L}_{f}}^{{{r}_{2}}}{{h}_{2}}(x) \\  & \text{       }. \\  & \text{       }. \\  & \text{       }. \\  & {{L}_{f}}^{{{r}_{p}}}{{h}_{p}}(x) \\ \end{align} \right)$

And

$D(x)=\left( \begin{align}  & {{L}_{{{g}_{1}}}}{{L}_{f}}^{{{r}_{1}}-1}{{h}_{1}}\,\,\,\,\,\,\,{{L}_{{{g}_{2}}}}{{L}_{f}}^{{{r}_{1}}-1}{{h}_{1}}\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,{{L}_{{{g}_{p}}}}{{L}_{f}}^{{{r}_{1}}-1}{{h}_{1}} \\  & {{L}_{{{g}_{1}}}}{{L}_{f}}^{{{r}_{2}}-1}{{h}_{2}}\,\,\,\,\,\,{{L}_{{{g}_{2}}}}{{L}_{f}}^{{{r}_{2}}-1}{{h}_{2}}\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,{{L}_{{{g}_{p}}}}{{L}_{f}}^{{{r}_{2}}-1}{{h}_{2}} \\  & \,\,\,\,\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,\,\,\,\, \\  & {{L}_{{{g}_{1}}}}{{L}_{f}}^{{{r}_{p}}-1}{{h}_{p}}\,\,\,\,{{L}_{{{g}_{2}}}}{{L}_{f}}^{{{r}_{p}}-1}{{h}_{p}}\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,\,.\,\,\,\,\,\,{{L}_{{{g}_{p}}}}{{L}_{f}}^{{{r}_{p}}-1}{{h}_{p}} \\ \end{align} \right)$

D(x) is called the decoupling matrix system.

The linearizing control law has the following form [9]:

$u=D{{(x)}^{-1}}(-\zeta (x)+v)$             (19)

It is important to note that linearisation is only possible if the decoupling matrix D(x) is reversible. Figure 6 shows the linearised system's block diagram.

2.4.1 DC-link voltage FLC design

The design of the DC link voltage controller is based on (5). The derivative of the output y=h = v_dc² is computed as:

$\overset{\centerdot }{\mathop{y}}\,={{L}_{f}}h(x)+{{L}_{g}}h(x)u=\frac{2}{{{C}_{dc}}}P_{dc}^{{}}$              (20)

The control P_dc is figured in (20); hence, its relative degree r =1. The relative degree of this output is equal to the order of the system which clearly corresponds to an exact linearisation [10].

The control is then determined by:

$P_{dc}^{*}=\frac{{{C}_{dc}}}{2}v$               (21)

where: $\dot{y}=v$.

For the problem of trajectory tracking defined by v_dc*(t), the linearizing control law v is defined by:

$v={{k}_{dc}}(v_{dc}^{{{*}^{2}}}-v_{dc}^{2})+\frac{dv_{dc}^{*2}}{dt}$                   (22)

where, k_dcis a positive constant.

2.4.2 Parallel currents FLC design

Each output derivative is given by:

${{\dot{y}}_{j}}={{L}_{f}}^{{}}h(x)+\sum\limits_{i=1}^{2}{{{L}_{gi}}({{L}_{f}}^{{}}{{h}_{j}}(x))}{{u}_{i}}\,\,\,j=1,2\,$                 (23)

Then, the matrix form of (23) can be expressed as:

$\left[\begin{array}{c}\dot{y}_{p 1} \\ \dot{y}_{p 2}\end{array}\right]=\left[\begin{array}{c}-\frac{R_{f p}}{L_{f p}} x_{p 1} \\ -\frac{R_{f p}}{L_{f p}} x_{p 2}\end{array}\right]+\left[\begin{array}{cc}\frac{1}{L_{f p}} & 0 \\ 0 & \frac{1}{L_{f p}}\end{array}\right]\left[\begin{array}{l}u_{p 1} \\ u_{p 2}\end{array}\right]$                   (24)

The decoupling matrix is reversible (det(D_p(x)) ≠0), then the linearizing control law can be given as:

$u_{p}=\left[\begin{array}{l}u_{p 1} \\ u_{p 2}\end{array}\right]=D_{p}\left(x_{p}\right)^{-1}\left[-\zeta_{p}\left(x_{p}\right)+\left[\begin{array}{l}v_{p 1} \\ v_{p 2}\end{array}\right]\right]$                   (25)

By applicating the linearisation law, we obtain the following decoupled linear system:

$\left[\begin{array}{l}\dot{y}_{p 1} \\ \dot{y}_{p 2}\end{array}\right]=\left[\begin{array}{c}v_{p 1} \\ v_{p 2}\end{array}\right]$              (26)

6.png

Figure 6. Block diagram of linearised MIMO system

The control law utilised for tracking the current reference is:

$v_{p 1}=k_{p 1}\left(i_{f p \alpha}^{*}-i_{f p \alpha}\right)+\frac{d i_{f p \alpha}^{*}}{d t}$

                              $v_{p 2}=k_{p 2}\left(i_{f p \beta}^{*}-i_{f p \beta}\right)+\frac{d i_{f p \beta}^{*}}{d t}$                  (27)

where, k_p1 and k_p2are positive constants.

From (25) and (27), the control law is given by:

$v_{f p \alpha}^{*}=R_{f p} i_{f p \alpha}+v_{s \alpha}+L_{f p}\left(k_{p 1}\left(i_{f p \alpha}^{*}-i_{f p \alpha}\right)+\frac{d i_{f p \alpha}^{*}}{d t}\right)$

                                      $v_{f p \beta}^{*}=R_{f p} i_{f p \beta}+v_{s \beta}+L_{f p}\left(k_{p 2}\left(i_{f p \beta}^{*}-i_{f p \beta}\right)+\frac{d i_{f p \beta}^{*}}{d t}\right)$                     (28)

2.4.3 Series currents FLC design

Each output derivative is given by:

$\dot{y}_{j}=L_{f} h(x)+\sum_{i=1}^{2} L_{g i}\left(L_{f} h_{j}(x)\right) u_{i} \quad j=1,2$                 (29)

Then, (29) can be written in the matrix form as:

$\left[\begin{array}{l}\dot{y}_{s 1} \\ \dot{y}_{s 2}\end{array}\right]=\left[\begin{array}{c}-\frac{R_{f s}}{L_{f s}} x_{s 1} \\ -\frac{R_{f s}}{L_{f s}} x_{s 2}\end{array}\right]+\left[\begin{array}{cc}\frac{1}{L_{f s}} & 0 \\ 0 & \frac{1}{L_{f s}}\end{array}\right]\left[\begin{array}{l}u_{s 1} \\ u_{s 2}\end{array}\right]$                   (30)

The decoupling matrix is reversible (det(D_s(x)) ≠0), then the linearizing control law can be given as:

$u_{s}=\left[\begin{array}{l}u_{s 1} \\ u_{s 2}\end{array}\right]=D_{s}(x)^{-1}\left[-\zeta_{s}(x)+\left[\begin{array}{l}v_{s 1} \\ v_{s 2}\end{array}\right]\right]$                    (31)

By applying the linearisation law, we obtain the following decoupled linear system:

$\left[\begin{array}{l}\dot{y}_{s 1} \\ \dot{y}_{s 2}\end{array}\right]=\left[\begin{array}{l}v_{s 1} \\ v_{s 2}\end{array}\right]$              (32)

The control law utilized for tracking the voltage reference is:

$v_{s 1}=k_{s 1}\left(i_{f s \alpha}^{*}-i_{f s \alpha}\right)+\frac{d i_{f s \alpha}^{*}}{d t}$

                        $v_{s 2}=k_{s 2}\left(i_{f s \beta}^{*}-i_{f s \beta}\right)+\frac{d i_{f s \beta}^{*}}{d t}$               (33)

where, k_s1 and k_s2are positive constants.

From (31) and (33), the control law is given by:

$v_{f s \alpha}^{*}=R_{f s} i_{f s \alpha}+v_{i n j \alpha}+L_{f s}\left(k_{s 1}\left(i_{f s \alpha}^{*}-i_{f s \alpha}\right)+\frac{d i_{f s \alpha}^{*}}{d t}\right)$

                          $v_{f s \beta}^{*}=R_{f s} i_{f s \beta}+v_{i n j \beta}+L_{f s}\left(k_{s 2}\left(i_{f s \beta}^{*}-i_{f s \beta}\right)+\frac{d i_{f s \beta}^{*}}{d t}\right)$               (34)

2.4.4 Boost FLC design

Each output derivative is given by:

$\dot{y}_{j}=L_{f} h(x)+\sum_{i=1}^{2} L_{g i}\left(L_{f} h_{j}(x)\right) u_{i} \quad j=1,2$                (35)

Then, the matrix form of (35) can be expressed as:

$\left[\begin{array}{l}\dot{y}_{b 1} \\ \dot{y}_{b 2}\end{array}\right]=\left[\begin{array}{l}\frac{1}{L_{p v}} x_{b 1} \\ \frac{1}{C_{p v}} x_{b 2}\end{array}\right]+\left[\begin{array}{cc}-\frac{1}{L_{p v}} & 0 \\ 0 & -\frac{1}{C_{p v}}\end{array}\right]\left[\begin{array}{l}u_{b 1} \\ u_{b 2}\end{array}\right]$               (36)

The decoupling matrix is reversible (det(D_b(x)) ≠0), then the linearizing control law can be given as:

$u_{b}=\left[\begin{array}{l}u_{b 1} \\ u_{b 2}\end{array}\right]=D_{b}\left(x_{b}\right)^{-1}\left[-\zeta_{b}\left(x_{b}\right)+\left[\begin{array}{l}v_{b 1} \\ v_{b 2}\end{array}\right]\right]$                     (37)

By applying the linearisation law, we obtain the following decoupled linear system:

$\left[\begin{array}{l}\dot{y}_{b 1} \\ \dot{y}_{b 2}\end{array}\right]=\left[\begin{array}{l}v_{b 1} \\ v_{b 2}\end{array}\right]$                 (38)

The control law utilized for tracking the reference is:

$v_{b 1}=k_{b 1}\left(i_{L p v}^{*}-i_{L p v}\right)+\frac{d i_{L p v}^{*}}{d t}$

                               $v_{b 2}=k_{b 2}\left(v_{p v}^{*}-v_{p v}\right)+\frac{d v_{p v}^{*}}{d t}$                   (39)

where, k_b1 and k_b2are positive constants.

From (37) and (38), the control law is given by:

$v_{d c}^{*}=v_{p v}+L_{p v}\left(k_{b 1}\left(i_{L p v}^{*}-i_{L p v}\right)+\frac{d i_{L p v}^{*}}{d t}\right)$

$i_{L p v}^{*}=i_{p v}+C_{p v}\left(k_{b 2}\left(v_{p v}^{*}-v_{p v}\right)+\frac{d v_{p v}^{*}}{d t}\right)$              (40)

2.5 Space vector modulation

In this part, the space vector modulation (SVM) algorithm is described to produce the switches PWM control signals (s_a, s_b and s_c) [22]. The SVM schematic of a 3-phase IGBT-based voltage source inverter is a hexagon (Figure 7), constituted of six sectors. Each sector is an equilateral triangle of a height h =√3/2 [23]. The sector of operation to every given reference vector is obtained using Eq. (41).

$S_{i}=\operatorname{int}\left(\frac{\theta_{s}}{60}\right)+1$                (41)

7.png

Figure 7. SVM diagram

All sectors use the same on-time calculation. The following is the volt-second equation [24]:

$\overrightarrow{v_{f}} T_{s}=t_{i} \vec{v}_{i}+t_{i+1} \vec{v}_{i+1}+t_{0} \vec{v}_{0}$              (42)

where, f_s: switching frequency, with T_s=1/f_s.

In sector 1,

$\left\{\begin{array}{l}\vec{v}_{1}=\sqrt{\frac{2}{3}} v_{d c} \\ \vec{v}_{2}=\sqrt{\frac{2}{3}} v_{d c}\left(\frac{1}{2}+j \frac{\sqrt{3}}{2}\right) \\ \vec{v}_{0}=0\end{array}\right.$                (43)

The reference to be regenerated could also be expressed as follows,

$\overrightarrow{v_{f}}=v_{f \alpha}+j v_{f \beta}$            (44)

Eqns. (42)-(44) can be used to get:

$\left\{\begin{array}{l}v_{f \alpha}=\sqrt{\frac{2}{3}} v_{d c} \frac{t_{1}}{T_{s}}+\frac{1}{\sqrt{6}} v_{d c} \frac{t_{2}}{T_{s}} \\ v_{f \beta}=\frac{1}{\sqrt{2}} v_{d c} \frac{t_{2}}{T_{s}}\end{array}\right.$         (45)

Finally, the ON-times are calculated using Eq. (45) [25].

$\left\{\begin{array}{l}t_{1}=\frac{\sqrt{6} v_{f \alpha}-\sqrt{2} v_{f \beta}}{v_{d c}} T_{s} \\ t_{2}=\frac{\sqrt{2} v_{f \beta}}{v_{d c}} T_{s} \\ t_{0}=T_{s}-\left(t_{1}+t_{2}\right)\end{array}\right.$          (46)

The SVM scheme is determined by which null vector is used.

There are three options: only use the null vector v₀, only use the null vector v₇, or use a combination of null vectors. Altering the null vector of every cycle and reversing the sequence after every null vector is a popular SVM approach as shown in Figure 8. This is what we'll call it, the symmetric 7-segment approach [24].

8.png

Figure 8. 7-segment switching sequence for V_ref in sector

2.6 Comparative PI controller design

The suggested FL-SVM's performance is compared with that of the PI controller to illustrate its efficiency. As shown below, the design of the PI controller is detailed.

The role of the DC PI controller is to allow tracking the reference by controlling the active power flow between the PCC and the DC bus [26].

From Eq. (5), the following transfer function is calculated:

$\frac{v_{d c}^{2}(s)}{P_{c}(s)}=\frac{2}{C_{d c} s}$          (47)

Figure 9 shows a closed - loop schematic of DC voltage regulation.

9.png

Figure 9. DC voltage regulation

The closed loop transfer function is given in:

$G(s)=\frac{v_{d c}^{2}(s)}{v_{d c}^{* 2}(s)}=\frac{\frac{2 k_{p d c}}{C_{d c}} s+\frac{2 k_{i d c}}{C_{d c}}}{s^{2}+\frac{2 k_{p d c}}{C_{d c}} s+\frac{2 k_{i d c}}{C_{d c}}}$           (48)

By the identification of (48) to a second order canonic transfer function (49):

$H(s)=\frac{2 \zeta \omega_{n} s+\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}}$             (49)

We can find:

$k_{i d c}=C_{d c} \omega_{n}^{2} / 2$

                      $k_{p d c}=\xi \omega_{n} C_{d c}$          (50)

$\omega_{n}=2 \pi f_{n}$ is the Natural pulsatance of the controller, and 0<ξ<1 is the damping factor for a good dynamic and acceptable oscillations, we choose $f_{n}=25 H z \xi=0.7$.

By the same method, we can also obtain the PI coefficients of currents, voltages and Boost controllers as summarized in Table 1.

Table 1. PI parameters

2.7 Principle of PIL co-simulation

Prototyping "Processor in the Loop" has the advantage of validating the digital implementation of control algorithms in the control part while the computer emulates the power part. It is therefore possible to evaluate the control part in a virtual environment where adjustments in the control algorithms are easily achieved by reprogramming without costly hardware iteration. The latter results in a minimisation in development’s time, as well as cost of the project.

Controlling the suggested PVG-UPQC system in the Simulink.

Platform using the generated control algorithm embedded in STM32F429I-DISC DSP board, as illustrated in Figure 10.

2.8 The PIL implementation technique steps of the control algorithms

In PIL co-simulation, an embedded platform running the control algorithm is connected to a host computer on which the physical system model is executed. Then, an evaluation of the execution circumstances of the developed algorithm can be performed in order to optimize some important factors such as memory footprint, code size and execution of the algorithm according to the required time.

During PIL prototyping, at each simulation step, the power part of the electrical system is simulated in the Matlab / Simulink platform and the output signals are sent to the STM32F429i-Discovery DSP board. When it receives the signals from the standard PC, it executes the implemented control algorithms. The DSP board then sends back to the PC the control commands to the electrical system simulated in the Matlab / Simulink platform. At this stage, a PIL simulation cycle is done. The exchange of data between the standard PC and the DSP board is synchronized using an FTDI232 interface (serial communication) to connect the DSP board to the standard PC as indicated in Figure 10. To do so, we need to achieve the following steps:

10.png

Figure 10. The structure of the PIL co-simulation

Phase I:

The objective of this step is to generate the C code for all the control algorithms of the system.

Step 01:

Simulation of the system using the Embedded Matlab Function.

Phase II:

The PIL co-simulation is performed between the DSP board and the Simulink MATLAB platform. The implementation of the PIL procedure with existing resources is as follows:

Step 01:

-Grouping of all the control algorithms of the proposed system in a single simulation subsystem,

Step 02:

-Define the reference of the DSP board in the Simulink Matlab platform and define the appropriate inputs and outputs of the control system. After that, the proposed control algorithm will be built and integrated into the DSP board,

Step 03:

-Select a communication interface between the DSP board and the Simulink platform,

Step 04:

-Set the appropriate time step in Simulink according to the application,

Step 05:

-Download the compiled algorithm control model of the entire subsystem to be incorporated into the DSP board,

Step 06:

-Configure and control the proposed system on the Simulink Matlab platform using the control algorithm embedded in the DSP board.

This paper has attained a control design of the PVG-UPQC in addition to its mathematical model and harmonic extraction methods for both currents and voltages. Additionally, Feedback linearisation combined with SVM controller technique is derived to suppress currents and voltages harmonics in the grid. A two-level SVM technique were used due to its benefits in terms of a fixed frequency implementation. Co-simulation results based on the PIL technique has been fulfilled in all stages the system operation under nonlinear load changes and voltage sag/swell as well. The results shown in this paper has clearly proves the superiority and the robustness of the proposed controller of the PVG-UPQC, and offers much better performance compared to the traditional PI controller. Otherwise, in the future, to suppress the limitations of the injected power, the system will be implemented with more kilo watts delivered to the grid in addition to involve real practical implantations instead of PIL technique.