**4. Implementation and Robustness Analysis of SMC and LRC**

The intended outputs, or control objectives of the proposed controllers (each of SMC and LRC controller) is:

Y1 = VdC ≈ Vd ≈ 480 Volt

Y2 = VqC ≈ Vq ≈ (the lowest possible) Volt

Equation (22) gives the general system form affined within the control(s):

$$
\dot{\mathbf{x}} = f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\mathbf{u},
\tag{22}
$$

*4.1. Implementation and Robustness Analysis of Sliding Mode Controller against Parametric Uncertainties Including Uncertainties in Power of CPL*

Sliding mode control, or SMC, is an advanced non-linear control technique featuring prominent characteristics of accuracy, robustness, and ease of tuning. By using the discontinuous control signal that forces the output of the system to 'slide' along with sliding surface or a distinct cross-section of the minimal behavior of the system, it can adjust the dynamics of the system in a way [47]. The state feedback control law is a discontinuous time function here, and can shift from one structure to the next depending on the prevailing location in space in a continuous manner. Hence, sliding mode control can be described as a control technique with variable structures. As the system's certain operation mode slides along the predetermined control structure boundaries, it is called the sliding mode. The geometrical locus, which consists of the boundaries, is called the system's sliding surface. To implement the sliding mode controller, the state space model equation below can be rewritten as Equation (23). In this section, the robustness will be enhanced by considering the uncertainties in active power of CPL (*P*0) and reactive power of CPL (*Q*0). When *P*<sup>0</sup> is unknown in case of designing *u*1, we will also consider *x*<sup>3</sup> as unknown to avoid any complexity. Similarly, in case of *u*2, we will also consider *x*<sup>4</sup> as unknown.

$$
\begin{bmatrix}
\dot{\mathbf{x}}\_{1} \\
\dot{\mathbf{x}}\_{2} \\
\dot{\mathbf{x}}\_{3} \\
\dot{\mathbf{x}}\_{4} \\
\dot{\mathbf{x}}\_{5} \\
\dot{\mathbf{x}}\_{6}
\end{bmatrix} = \begin{bmatrix}
\omega \mathbf{x}\_{2} - \frac{R\_{1}}{L\_{1}} \mathbf{x}\_{1} - \frac{\mathbf{x}\_{3}}{L\_{1}} \\
\omega \mathbf{x}\_{4} + \frac{1}{\mathsf{C}} \mathbf{x}\_{1} - \frac{1}{\mathsf{C}} \frac{R\_{1}}{\mathsf{x}\_{3}} - \frac{1}{\mathsf{C}} \mathbf{x}\_{5} \\
\omega \mathbf{x}\_{6} + \frac{1}{\mathsf{C}} \mathbf{x}\_{3} - \frac{R\_{1}}{\mathsf{C}} \mathbf{x}\_{5} \\
\end{bmatrix} + \begin{bmatrix}
0 \\
0 \\
0 \\
0 \\
0
\end{bmatrix},
\tag{23}
$$

Although *P*<sup>0</sup> and *Q*<sup>0</sup> are unknown, they satisfy *P*<sup>0</sup> ≤ *δ<sup>P</sup>* and *Q*<sup>0</sup> ≤ *δ<sup>Q</sup>* for some known bounds *δ<sup>P</sup>* and *δQ*. The variation on CPL power can be summarized as:

$$d\_P = \Delta\_P / \,\Delta\mathfrak{x}\_{\mathfrak{Y}\_{\bullet}} \tag{24}$$

$$d\_Q = \Delta\_Q / \,\Delta\mathbf{x}\_{\mathbf{4}\prime} \tag{25}$$

where *dP* represents the uncertainties of *P*0, *dQ* represents the uncertainties of *Q*0, Δ*x*<sup>3</sup> is the uncertainties in *x*3, and Δ*x*<sup>4</sup> is the uncertainties in *x*4. As *x*<sup>3</sup> and *x*<sup>4</sup> are in the denominator, we need lower bounds of these parameters. Power uncertainty is expressed in term of current. We know that *x*<sup>3</sup> is the voltage of "d-axis" and it satisfies Δ*x*<sup>3</sup> ≤ *δx*<sup>3</sup> for some known, stringently positive bound *δx*3. Similarly, *x*<sup>4</sup> is the "q-axis" voltage. It satisfies Δ*x*<sup>4</sup> ≤ *δx*<sup>4</sup> for some known, stringently positive bound *δx*4. Overall, there are six unknowns with known bounds. The Sliding Mode Control input, *u*<sup>1</sup> will be designed first, with the similar method adopted to design *u*2, the other control input. Using the similar method as discussed in the previous section, let

$$x\_1 = \int (x\_3 - x\_{3d}) dt\_\prime \tag{26}$$

$$
\omega\_2 = \dot{\varkappa}\_1 = \varkappa\_3 - \varkappa\_{3d'} \tag{27}
$$

$$
\dot{\omega}\_2 = \dot{\mathbf{x}}\_3 - \dot{\mathbf{x}}\_{3d} \\
= f\_3(\mathbf{x}) + g\_3(\mathbf{x})u\_1 - \dot{\mathbf{x}}\_{3d} \tag{28}
$$

Expanding *f*3(*x*) and *g*3(*x*)

$$
\dot{\omega}\_2 = \omega \mathbf{x}\_4 + \frac{1}{c} \mathbf{x}\_1 - \frac{1}{c} \frac{P\_0}{\mathbf{x}\_3} - \frac{1}{c} \mathbf{x}\_5 - \frac{1}{c} u\_1 - \dot{\mathbf{x}}\_{3d} \tag{29}
$$

Let, the sliding surface be

$$s = \mathfrak{e}\_1 + \mathfrak{e}\_2,\tag{30}$$

After differentiating and considering the uncertainties:

$$
\dot{s} = \dot{e}\_1 + \dot{e}\_2.\tag{31}
$$

$$\dot{s} = \mathbf{e}\_2 + (\omega(\mathbf{\hat{x}}\_4 + \Delta \mathbf{x}\_4) + \frac{1}{c}(\mathbf{\hat{x}}\_1 + \Delta \mathbf{x}\_1) - \frac{1}{c}(\frac{P\_0}{\mathbf{x}\_3} + d\_P) - \frac{1}{c}\mathbf{x}\_5 - \frac{1}{c}\mu\_1 - \dot{\mathbf{x}}\_{3d}),\tag{32}$$

where *x*<sup>4</sup> = *x*ˆ4 + Δ*x*4. Then the total parametric uncertainty including uncertainty of CPL power can be represented as:

$$d = \frac{1}{c} \Delta \mathbf{x}\_1 + \omega \Delta \mathbf{x}\_4 - \frac{1}{c} d\_{P\prime} \, \|d\| \le d \max,\tag{33}$$

here *dmax* is the limit of the total disturbance d.

$$
\delta \, d\max = \frac{1}{\mathcal{c}} \delta\_{\mathbf{x1}} + \omega \delta\_{\mathbf{x4}} - \frac{1}{\mathcal{c}} \delta\_{\mathbf{P}} / \, \delta\_{\mathbf{x3} \prime} \tag{34}
$$

Then,

$$\dot{\mathbf{s}} = \mathbf{c}\_2 - \frac{1}{c} \mathbf{x}\_5 - \dot{\mathbf{x}}\_{3d} + \omega \mathbf{x}\_4 + \frac{1}{c} \mathbf{\hat{x}}\_1 - \frac{1}{c} \frac{P\_0}{\mathbf{x}\_3} - \frac{1}{c} u\_1 + d\_\prime \tag{35}$$

Let it be considered as the Lyapunov candidate function.

$$V = \frac{1}{2}s^2,\tag{36}$$

$$\dot{V} = s\dot{s} = s(e\_2 - \frac{1}{c}\mathbf{x}\_5 - \dot{\mathbf{x}}\_3 + \omega \mathbf{x}\_4 + \frac{1}{c}\mathbf{f}\_1 - \frac{1}{c}\frac{P\_0}{\mathbf{x}\_3} - \frac{1}{c}u\_1 + d),\tag{37}$$

*Energies* **2017**, *10*, 1959

We use *u*1.

$$u\_1 = -c \left[ -\mathbf{e}\_2 + \frac{1}{c} \mathbf{x}\_5 + \dot{\mathbf{x}}\_{3d} - \omega \mathbf{x}\_4 - \frac{1}{c} \mathbf{\hat{x}}\_1 + \frac{1}{c} \frac{P\_0}{\mathbf{x}\_3} + v \right],\tag{38}$$

Now, we can obtain: .

$$V = s(d+v),\tag{39}$$

*d* ≤ *dmax*, put into consideration, . *V* will be made negative by the subsequent discontinuous control, *v*. Consequently, it will guarantee stability.

$$
v = -d \max \ast \operatorname{sat} \left( \frac{s}{\varepsilon} \right); \ \varepsilon > 0,\tag{40}$$

In total, the control input is:

$$u\_1 = -c \left[ -\mathfrak{e}\_2 + \frac{1}{c} \mathfrak{x}\_5 + \dot{\mathfrak{x}}\_{3d} + \frac{1}{c} \frac{P\_0}{\mathfrak{x}\_3} - \omega \mathfrak{x}\_4 - \frac{1}{c} \mathfrak{x}\_1 - d \max \ast \operatorname{sat} \left( \frac{s}{\mathfrak{e}} \right) \right],\tag{41}$$

Such an analysis is also presented here for *u*2, let,

$$x\_3 = \int (x\_4 - x\_{4d}) dt,\tag{42}$$

$$
\varepsilon\_4 = \dot{\varepsilon}\_3 = \chi\_4 - \chi\_{4d\prime} \tag{43}
$$

$$
\dot{e}\_4 = \dot{\mathbf{x}}\_4 - \dot{\mathbf{x}}\_{4d} = f\_4(\mathbf{x}) + \mathbf{g}\_4(\mathbf{x})\mu\_2 - \dot{\mathbf{x}}\_{4d}.\tag{44}
$$

Taking the sliding surface as:

$$s = e\_3 + e\_4,\tag{45}$$

After differentiation and considering the uncertainties:

$$\dot{s} = \epsilon\_4 + \left( -\omega(\dot{\mathbf{x}}\_3 + \Delta \mathbf{x}\_3) + \frac{1}{c}(\dot{\mathbf{x}}\_2 + \Delta \mathbf{x}\_2) - \frac{1}{c}(\frac{Q\_0}{\mathbf{x}\_4} + d\_Q) - \frac{1}{c}\mathbf{x}\_6 - \frac{1}{c}\mu\_2 - \dot{\mathbf{x}}\_{4d} \right), \tag{46}$$

where *x*<sup>3</sup> = *x*ˆ3 + Δ*x*3. Then the total parametric uncertainty including uncertainty of CPL power can be represented as:

$$d = \frac{1}{\mathcal{c}} \Delta \mathbf{x}\_2 - \omega \Delta \mathbf{x}\_3 - \frac{1}{\mathcal{c}} d\_{Q'} \left\| d \right\| \le d \max,\tag{47}$$

where *dmax* is the limit for d, the total disturbance.

$$
gamma = \frac{1}{c} \delta\_{\rm x2} - \omega \delta\_{\rm x3} - \delta\_{\rm Q} / \delta\_{\rm x4} \tag{48}$$

Then,

$$\dot{s} = c\_3 - \frac{1}{c} \mathbf{x}\_6 - \dot{\mathbf{x}}\_{4d} + \omega \mathbf{x}\_3 + \frac{1}{c} \mathbf{\hat{x}}\_2 - \frac{1}{c} \frac{Q\_0}{\mathbf{x}\_4} - \frac{1}{c} \mathbf{u}\_2 + d,\tag{49}$$

Considering this as the Lyapunov candidate function:

$$V = \frac{1}{2}s^2,\tag{50}$$

$$\dot{V} = s\dot{s} = s(e\_3 - \frac{1}{c}\mathbf{x}\_6 - \dot{\mathbf{x}}\_{4d} + \omega\mathbf{x}\_3 + \frac{1}{c}\mathbf{x}\_2 - \frac{1}{c}\frac{Q\_0}{\mathbf{x}\_4} - \frac{1}{c}\mu\_2 + d),\tag{51}$$

We then use *u*2.

$$\mathbf{u}\_{2} = -\mathbf{c} \left[ -\mathbf{e}\_{3} + \frac{1}{c} \mathbf{x}\_{6} + \dot{\mathbf{x}}\_{4d} - \omega \hat{\mathbf{x}}\_{3} - \frac{1}{c} \hat{\mathbf{x}}\_{2} + \frac{1}{c} \frac{Q\_{0}}{\mathbf{x}\_{4}} + \upsilon \right],\tag{52}$$

Then, we can obtain: .

$$V = s(d+v),\tag{53}$$

Considering *d* ≤ *dmax*, . *V* will be made negative by the subsequent discontinuous control, *v*. Consequently, it will guarantee stability.

$$v = -d \max \ast \operatorname{sat} \left( \frac{s}{\varepsilon} \right); \ \varepsilon > 0,\tag{54}$$

In total, the control input is:

$$\mu\_2 = -c \left[ -\mathfrak{e}\_3 + \frac{1}{\mathfrak{c}} \mathfrak{x}\_6 + \dot{\mathfrak{x}}\_{4d} - \omega \mathfrak{x}\_3 - \frac{1}{\mathfrak{c}} \mathfrak{x}\_2 + \frac{1}{\mathfrak{c}} \frac{Q\_0}{\mathfrak{x}\_4} - d \max \ast \mathrm{sat} \left( \frac{\mathfrak{s}}{\mathfrak{c}} \right) \right], \tag{55}$$

*4.2. Implementation and Robustness Analysis of Lyapunov Redesign Controller against Parametric Uncertainties Including Uncertainties in Power of CPL*

The LRC is based only on Lyapunov function. Its nominal controller is designed to ensure the nominal system or disturbance-free system to be stable by forcing the Lyapunov function derivative of the nominal system to be negative. If there is disturbance in the system, the discontinuous control is used alone to handle the disturbance. The discontinuous controller is formulated by redesigning the Lyapunov function of the nominal system. In the redesigning process, the disturbance is introduced to the Lyapunov function of the nominal system and then solved for the discontinuous control to overcome that disturbance and force the new derivative Lyapunov function or be negative and consequently, the system to be globally stable. It has some chattering issues because of the discontinuous controller. The chattering magnitude is dependent on the magnitude of dmax. Having large dmax makes the system stable against large disturbance but it can cause larger chattering if it is set as a very large value. If the disturbance happens to be greater than the set dmax, the system can become unstable. But, the LRC has greater margin for stability because its nominal system is also ensured to be stable, thus provides better performance for large disturbance.

First of all, the Lyapunov Redesign Control input, *u*<sup>1</sup> will be designed, with the same approach followed next to design the other control input, *u*2. Using the similar method as discussed in last section, we introduce new state variables:

$$x\_1 = \int (x\_3 - x\_{3d}) dt\_\prime \tag{56}$$

$$
\omega\_2 = \dot{\mathbf{e}}\_1 = \mathbf{x}\_3 - \mathbf{x}\_{3d} \tag{57}
$$

$$
\dot{e}\_2 = \dot{\mathbf{x}}\_3 - \dot{\mathbf{x}}\_{3d} \\
= f\_3(\mathbf{x}) + g\_3(\mathbf{x})u\_1 - \dot{\mathbf{x}}\_{3d} \tag{58}
$$

Expanding *f*3(*x*) and *g*3(*x*):

.

$$
\dot{\mathbf{e}}\_2 = \omega \mathbf{x}\_4 + \frac{1}{c} \mathbf{x}\_1 - \frac{1}{c} \frac{P\_0}{\mathbf{x}\_3} - \frac{1}{c} \mathbf{x}\_5 - \frac{1}{c} u\_1 - \dot{\mathbf{x}}\_{3d} \tag{59}
$$

Considering the uncertainties:

$$\dot{e}\_2 = \omega(\dot{\mathbf{x}}\_4 + \Delta \mathbf{x}\_4) + \frac{1}{c}(\dot{\mathbf{x}}\_1 + \Delta \mathbf{x}\_1) - \frac{1}{c} \left(\frac{P\_0}{\mathbf{x}\_3} + d\_P\right) - \frac{1}{c}\mathbf{x}\_5 - \frac{1}{c}\mu\_1 - \dot{\mathbf{x}}\_{3d} \tag{60}$$

Then the total parametric uncertainty including uncertainty of CPL power can be represented as:

$$d = \frac{1}{c} \Delta \mathbf{x}\_1 + \omega \Delta \mathbf{x}\_4 - \frac{1}{c} d\_{P\prime} \colon \|d\| \le d \max,\tag{61}$$

here *dmax* is the limit of *d*, the total disturbance.

*Energies* **2017**, *10*, 1959

$$
gamma = \frac{1}{c} \delta\_{\text{x1}} + \omega \delta\_{\text{x4}} - \frac{1}{c} \delta\_{\text{P}} / \,\delta\_{\text{x3}} \tag{62}
$$

Following the methodology of Lyapunov redesign, the over-all input is *u*<sup>1</sup> = *u*<sup>0</sup> + *v*; where *u*<sup>0</sup> is the nominal stabilizing controller and *v* is to handle the disturbances. We get the linear state space of error as in Equation (63):

$$
\dot{e} = \begin{bmatrix} 0 & 1 \\ -k\_1 & -k\_2 \end{bmatrix} e,\tag{63}
$$

Now, we define the desired Eigen values for the linearized system. Desired Eigen values would be −10.

Let, Equation (63) be written as . *e* = *Ae* and *A* = 0 1 −*k*<sup>1</sup> −*k*<sup>2</sup> Generalized Eigen values of matrix "*A*":

$$sI - A = \begin{bmatrix} s & -1 \\ k\_1 & s + k\_2 \end{bmatrix}'\tag{64}$$

$$|sI - A| = s^2 + k\_2s + k\_{1\prime} \tag{65}$$

Characteristic polynomial (desired):

$$(s+10)(s+10) = s^2 + 20s + 100,\tag{66}$$

Comparing Equations (65) and (66):

$$k\_2 = 20, \ k\_1 = 100$$

So, the values of *k*<sup>1</sup> and *k*<sup>2</sup> will become +100 and +20 respectively.

$$
\dot{\mathbf{c}} = \begin{bmatrix} 0 & 1 \\ -100 & -20 \end{bmatrix} \mathbf{e}\_{\prime} \tag{67}
$$

$$A = \begin{bmatrix} 0 & 1 \\ -100 & -20 \end{bmatrix} \text{,} \tag{68}$$

$$P A + A^T P = -I,\tag{69}$$

$$P = \begin{bmatrix} \frac{21}{8} & \frac{1}{200} \\ \frac{1}{200} & \frac{101}{4000} \end{bmatrix} \text{\textsuperscript{0}} \tag{70}$$

$$V(\mathfrak{e}) = \mathfrak{e}^T \mathrm{Pe}\_{\prime} \tag{71}$$

$$w = 2e^T P G = 2\begin{bmatrix} e\_1 & e\_2 \end{bmatrix} \begin{bmatrix} \begin{array}{cc} 2 \\ \frac{1}{200} \end{array} & \begin{array}{cc} 200 \\ \frac{1}{4000} \end{array} \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \tag{72}$$

$$w = \frac{1}{100}\varepsilon\_1 + \frac{101}{2000}\varepsilon\_{2\prime} \tag{73}$$

Then, we can choose the Lyapunov function for the nominal system or disturbance-free system to be:

$$V = \frac{1}{2}c\_{2\prime}^2\tag{74}$$

$$\dot{V} = \mathfrak{e}\_2 \dot{\mathfrak{e}}\_2 = \mathfrak{e}\_2 \left( \omega \mathfrak{X}\_4 + \frac{1}{c} \mathfrak{x}\_1 - \frac{1}{c} \frac{P\_0}{\mathfrak{x}\_3} - \frac{1}{c} \mathfrak{x}\_5 - \frac{1}{c} u\_0 - \dot{\mathfrak{x}}\_{3d} \right),\tag{75}$$

*Energies* **2017**, *10*, 1959

If we choose,

$$\mu\_0 = -c \left[ \frac{1}{c} \frac{P\_0}{x\_3} - \omega \pounds\_4 + \frac{1}{c} x\_5 + \dot{x}\_{3d} - k\_1 \varepsilon\_1 - k\_2 \varepsilon\_2 \right] \tag{76}$$

then, . *V* < 0. The terms [−*k*1*e*<sup>1</sup> − *k*2*e*2] guarantee the global stability of the nominal system which is absent in SMC method. The overall system is stabilized using the discontinuous control in the presence of disturbances. Redesigning the Lyapunov function considering disturbances,

$$V = \frac{1}{2}c\_{2\prime}^2\tag{77}$$

$$\dot{V} = \varepsilon\_2 \dot{\varepsilon}\_2 = \varepsilon\_2 (\left(\omega \pounds\_4 + \frac{1}{c}\mathbf{x}\_1 - \frac{1}{c}\frac{P\_0}{\mathbf{x}\_3} - \frac{1}{c}\mathbf{x}\_5 - \frac{1}{c}\mu\_0 - \dot{\mathbf{x}}\_{3d}\right) + \left(\frac{1}{c}\mathbf{v} + d\right)),\tag{78}$$

 *ωx*ˆ4 + <sup>1</sup> *<sup>c</sup> <sup>x</sup>*<sup>1</sup> <sup>−</sup> <sup>1</sup> *c P*0 *<sup>x</sup>*<sup>3</sup> <sup>−</sup> <sup>1</sup> *<sup>c</sup> <sup>x</sup>*<sup>5</sup> <sup>−</sup> <sup>1</sup> *<sup>c</sup> <sup>u</sup>*<sup>0</sup> <sup>−</sup> . *x*3*<sup>d</sup>* is assured to be negative, then the discontinuous control can be designed as:

$$
\omega = -c \ast d \max \ast \operatorname{sat}(\frac{d \max \ast \omega}{\mu}),
\tag{79}
$$

Then, the overall input is:

$$u\_1 = -c \left[ \frac{1}{C} \frac{p\_0}{x\_3} - \omega \pounds\_4 + \frac{1}{C} x\_5 + \dot{x}\_{3d} - 100e\_1 - 20e\_2 - d \max \ast \text{sat} \left( \frac{d \max \left( \frac{1}{100} e\_1 + \frac{101}{2000} e\_2 \right)}{\mu} \right) \right], \tag{80}$$

Therefore, there is *μ* > 0 so that for *μ* < *μ*∗, the closed-loop system's origin is asymptotically stable globally according to absolute stability theorem. Similarly, we have Equation (81), when we design a controller for *u*<sup>2</sup> with same desired points.

$$u\_2 = -c \left[ \frac{1}{C} \frac{Q\_0}{x\_4} + \omega \epsilon \mathbf{\hat{x}}\_3 + \frac{1}{C} \mathbf{x}\_6 + \dot{\mathbf{x}}\_{4d} - 100 \mathbf{e}\_3 - 20 \mathbf{e}\_4 - d \max \ast \text{sat} \left( \frac{d \max \left( \frac{1}{100} \mathbf{e}\_3 + \frac{101}{2000} \mathbf{e}\_4 \right)}{\mu} \right) \right], \tag{81}$$

where,

$$
\sigma\_3 = \int (\mathbf{x}\_4 - \mathbf{x}\_{4d}) dt\_\prime \tag{82}
$$

$$
\dot{e}\_4 = \dot{e}\_3 = \mathbf{x}\_4 - \mathbf{x}\_{4d} \tag{83}
$$

$$d = \frac{1}{c} \Delta \mathbf{x}\_2 - \omega \Delta \mathbf{x}\_3 - \frac{1}{c} d\_Q; \ \|d\| \le d \max = \frac{1}{c} \delta\_{\mathbf{x}2} - \omega \delta\_{\mathbf{x}3} - \delta\_{\mathbf{Q}} / \,\delta\_{\mathbf{x}4\prime} \tag{84}$$

#### **5. Results**

Here, the parameters and the parametric values regarding the simulation done for comparative analysis by varying the CPL power have been shown in Table 1.


**Table 1.** Table of Parameters.

In Figure 9a,b, performance comparisons have been illustrated between SMC (colored in blue) and LRC (colored in green) for *d*-axis output voltage and *q*-axis output voltage respectively. The control objective for d-axis output voltage has been considered as 480 Volt, for normal conditions—where the variance value has been set as 10% to simulate noise. From Figure 9a, it is evident that the LRC controller shows considerably superior performance than that of the SMC controller, as its output stayed closer to the control objective. For *q*-axis output voltage, the control objective has been considered as low as possible and negligible in practice. In Figure 9b, the *q*-axis output voltage fluctuates more in the case of the SMC controller than that of the LRC controller. To determine the controller behaviors in a very noisy environment, more noise is added by setting the variance value as 100%, and noise rejection capabilities of SMC and LRC are tested. The results are shown in Figure 10a,b, which demonstrate LRC's superior capability to stick to the reference value with close proximity, whereas SMC has fluctuations of great magnitudes. For nonlinearity, LRC is capable of attaining the reference d-axis value with negligible time-delay, but SMC needs some time to reach that (Figure 11a). However, for *q*-axis, both controllers exhibit a similar performance (Figure 11b). In case of parametric uncertainties, LRC again proves to be the better suited one, displaying less fluctuations than SMC to maintain the control objective (Figure 12a,b). Hence, LRC offers appreciable stability considering CPL power variation and parametric uncertainties. In Figure 13a,b, performance comparisons have been presented between SMC (blue colored) and LRC (green colored) in the case of *d*-axis control input current, *Id* (*u*1) and *q*-axis control input current, *Iq* (*u*2) respectably considering CPL power variation and parametric uncertainties. Here, the more the fluctuation in control input current, the more the stress will be imposed on the storage system to compensate, consequently degrading the storage performance and overall life time. This situation, in practice, makes it harder to retain microgrid stability. The mean squared errors (MSE) obtained from these analyses for both SMC and LRC controllers are presented in Table 2. It is obvious from the presented values that LRC is the better controller, and in a noisy environment, SMC is no match for LRC, as the former displays error significantly greater than LRC. This observation also leads to believe that, for practical applications, LRC can provide better performance than LRC. Therefore, from the comparative analysis presented here, the Lyapunov Redesign Controller shows better performance to retain system stability in face of CPL power variation. Hence, the LRC controller is preferred to be adopted for storage-based load side compensation technique for microgrid stability improvement with dense CPL loads present.

**Figure 9.** Comparison of performance between SMC (blue colored) and LRC (green colored) for normal condition in case of (**a**) *d*-axis output voltage; (**b**) *q*-axis output voltage considering CPL power variation and parametric uncertainties. LRC controller shows considerably better performance than SMC by staying closer to the reference voltage.

**Figure 10.** Comparison of performance between SMC (blue colored) and LRC (green colored) for very noisy environment in case of (**a**) *d*-axis output voltage; (**b**) *q*-axis output voltage considering CPL power variation and parametric uncertainties. LRC controller shows far better performance than SMC, as the latter shows high fluctuations from the reference voltage, while LRC stays close to it.

**Figure 11.** Comparison of performance between SMC (colored in blue) and LRC (colored in green) for nonlinearity in case of (**a**) *d*-axis output voltage, (**b**) *q*-axis output voltage. Unlike SMC, LRC is capable of attaining the reference d-axis value with negligible time-delay.

**Figure 12.** Comparison of performance between SMC (colored in blue) and LRC (colored in green) in case of (**a**) *d*-axis output voltage, (**b**) *q*-axis output voltage considering parametric uncertainties. LRC controller shows considerably better performance than SMC by staying closer to the reference voltage.

**Figure 13.** Comparison of performance among SMC (colored in blue) and LRC (colored in green) in case of (**a**) *d*-axis control input current, *Id* (*u*1); (**b**) *q*-axis control input current, *Iq* (*u*2) considering CPL power variation and parametric uncertainties. LRC fluctuated less than SMC, causing less stress on the system and thus providing a longer lifetime.


**Table 2.** Mean squared error (MSE) values of SMC and LRC controllers for the different conditions.

#### **6. Reason behind Inferior SMC Performance and Solutions**

Sliding Mode Control presents many fascinating challenges to the mathematicians. It is also extensively used in engineering applications because of the comparatively easy implementation which does not require a deep understanding of the complex mathematical background. These two reasons put it in a unique position among control theories. There are three main stages of designing a Sliding Mode Controller: designing the sliding surface, selecting the control law that will hold the system trajectory on the sliding surface, and implementing in a chatter-free setup—which is the most important one of these three. Although in theory, Sliding Mode Control is a robust one, experiments show otherwise—SMC has some serious shortcomings. The most prominent one of them is chattering—the high frequency oscillation around the sliding surface. It reduces the control performance significantly. As an example, the following second-order system can be considered:

$$
\dot{x}\_1 = x\_{2\prime} \tag{85}
$$

$$
\dot{\mathbf{x}}\_2 = a\mathbf{x}\_1 + b\mathbf{x}\_2 + c\sin\mathbf{x}\_1 + d\mathbf{u}\_2\tag{86}
$$

*a* and *b* are negative constant values here whereas *c* and *d* are positive constants. For *c > |a|*, the system is known to be unstable. For the actuator, existence of fast dynamics is posited, and it is stable. These are not considered in the ideal model. The equations governing them are:

$$w\_1 = w\_\prime \tag{87}$$

$$
\dot{w}\_1 = w\_{2\prime} \tag{88}
$$

$$
\dot{w}\_2 = -\frac{1}{\mu^2} w\_1 - \frac{2}{\mu} w\_2 + \frac{1}{\mu^2} u\_\* \tag{89}
$$

*μ* is a constant, considered to have a positive, sufficiently small value. As demonstrated in Figure 14, with actuator unmodeled dynamics present, *w*(*t*) is the actual input of the system, not *u*(*t*) directly from the sliding mode controller. The sliding mode surface and the control input is chosen as:

$$u = -M \text{sign}(\sigma),\tag{90}$$

$$
\sigma = \lambda \mathbf{x}\_1 + \mathbf{x}\_2. \tag{91}
$$

where *λ* and *M* are positive constants, with *M* is required to be large enough to enforce sliding mode into the ideal model ( . *σσ* <sup>&</sup>gt; <sup>0</sup>). . *x* becomes a continuous time function in real system, thus making the expectation of sliding mode to occur invalid; and causes chatter.

**Figure 14.** An example system demonstrating sliding mode control for systems as described in Equations (85) and (86). There are actuator dynamics that are not included in the ideal system. Chattering is caused from the excitation of these unmodeled dynamics by the high frequency switching action.

According to theory, the unmodeled dynamics present in the system causes the chattering effect. A sliding mode control, which is "chattering free", is not attainable as the model used in designing the controller can never capture all the system dynamics. But, the chattering can be curtailed. The sliding mode is normally implemented with a relay—which represents the sign function. It creates a common problem with relative degree equal to one. An alternative to this approach is using approximations of the sign function, which is widely used. Sigmoids, saturation, and hysteresis functions are used often too, providing a continuous or smooth control signal, but also losing the invariance property of the sliding mode control along the way. Table 3 shows some methods to improve the effectiveness of SMC. Fuzzy Sliding Mode Control (FSMC)—which uses a low pass filter, and estimates the sliding variable through a disturbance estimator—is the one with the least effectiveness. Integral Sliding Mode Control (ISMC), High Order Sliding Mode (HOSM), and Sliding Mode Extended State Observer (SMESO) offers better effectiveness. However, Type-2 Fuzzy-Neural Network Indirect Adaptive Sliding Mode

Control (T2FNNAS) is the way to achieve the best performance, which is based on the synthesis approach of Lyapunov [52,53].



\*\*\* = Excellent, \*\* = Satisfactory, \* = Acceptable.

#### **7. Numerical Verification of Results for Microgrid Application**

The results obtained so far demonstrate the capabilities of both SMC and LRC to maintain microgrid stability. To ascertain the effectiveness of these methods in real-life conditions, both of them are simulated numerically with data obtained from physical microgrids. These simulations confirm the efficacy of these control systems to sustain stability in real microgrids.

#### *7.1. SMC Technique*

To verify the global stability, we have to calculate the equation below:

$$
\dot{V} = s(d+v),
\tag{92}
$$

where,

$$d = \frac{1}{c} \Delta \mathbf{x}\_1 + \omega \Delta \mathbf{x}\_4 - \frac{1}{c} d\_{P\_{\prime}} \tag{93}$$

$$v = -d \max \ast \operatorname{sat} \left( \frac{s}{\varepsilon} \right); \; \varepsilon > 0,\tag{94}$$

So, .

$$\dot{V} = \left( s \left( \frac{1}{c} \Delta \mathbf{x}\_1 + \omega \Delta \mathbf{x}\_4 - \frac{1}{c} d\_P - d \max \ast \text{sat} \left( \frac{s}{\varepsilon} \right) \right) \right) \tag{95}$$

where,

$$
gamma = \frac{1}{c} \delta\_{\rm x1} + \omega \delta\_{\rm x4} - \frac{1}{c} \delta\_{\rm P} / \left. \delta\_{\rm x3\prime} \right| \tag{96}$$

Putting these all together,

$$\dot{V} = s \left( \frac{1}{c} \Delta \mathbf{x}\_1 + \omega \Delta \mathbf{x}\_4 - \frac{1}{c} d\_p - \left( \frac{1}{c} \delta\_{\mathbf{x}1} + \omega \delta\_{\mathbf{x}4} - \frac{1}{c} \delta\_P / \, \delta\_{\mathbf{x}3} \right) \* \text{sat} \left( \frac{s}{\varepsilon} \right) \right), \tag{97}$$

Now let,

$$
\omega = \text{60 Hz, } \Delta \mathbf{x}\_1 = \text{200 A, } \Delta \mathbf{x}\_4 = \text{50 V, } d\_P = \text{50 A, } \delta\_{\mathbf{x}1} = \text{4000 A, } \delta\_{\mathbf{x}3} = \text{100 A, } \delta\_{\mathbf{x}4} = \text{100 V, } \delta\_{\mathbf{x}5} = \text{20 A, } \delta\_{\mathbf{x}6} = \text{30 A, } \delta\_{\mathbf{x}7} = \text{40 B}
$$

*Energies* **2017**, *10*, 1959

$$\delta\_{\rm P} = 30 \text{ kW}, \varepsilon = 100, \varepsilon = 10 \,\upmu\text{F}$$

Putting these values, we get:

$$\dot{V} = s \begin{pmatrix} \frac{1}{10\mu}(200) + (60)(50) - \frac{1}{10\mu}(50) - \left(\frac{1}{10\mu}(4000) + (60)(100) - \frac{1}{10\mu} \left(\frac{30k}{100}\right)\right) \ast \text{sat}\left(\frac{\varepsilon}{100}\right) \end{pmatrix},\tag{98}$$

$$\dot{V} = s \left[ 15.003 \times 10^6 - \left[ 370.006 \times 10^6 \right] \text{sat} \left( \frac{s}{100} \right) \right] \tag{99}$$

Now, if *<sup>s</sup>* is either positive or negative, we will obtain . *V* ≤ 0, which guarantees global stability.
