### 3.1 The non- linear dynamic model of the UPFC

The dynamic power model of the series SSSC of the UPFC in form of a state- space representation can be defined as

$$ {\displaystyle \begin{array}{l}{\dot{p}}_n=-\omega {p}_n+{x}_{SE}^{-1}{\gamma}_Q\\ {}{\dot{q}}_n=\kern1em \omega {q}_n+{x}_{SE}^{-1}{\gamma}_P\end{array}} $$

(14)

*γ*_{
Q
} and *γ*_{
P
} are the control terms associated with the non-linear dynamic Eq. (17), which are defined as:

$$ \left.\begin{array}{l}{\gamma}_Q=\left[\left({V}_{mD}^2+{V}_{mQ}^2\right)-\left({V}_{mQ}{\vartheta}_{xQ}-{V}_{mD}{\vartheta}_{xD}\right)\right]\\ {}{\gamma}_P=\left[\left({V}_{mD}{\vartheta}_{xQ}-{V}_{mQ}{\vartheta}_{xD}\right)\right]\end{array}\right\} $$

(15)

In the above Eq. (15), the terms *ϑ*_{
xD
} and *ϑ*_{
xQ
} are defined as:

$$ \left.\begin{array}{l}{\vartheta}_{xD}={\vartheta}_{\varsigma D}+{V}_{nD}\\ {}{\vartheta}_{xQ}={\vartheta}_{\varsigma Q}+{V}_{nQ}\end{array}\right\} $$

(16)

Rearranging Eq. (16), we have:

$$ \left.\begin{array}{l}{\vartheta}_{xD}={V}_{mQ}{\gamma}_P+{V}_{mD}{\widehat{\gamma}}_Q/\left[{V}_{mD}^2-{V}_{mQ}^2\right]\\ {}{\vartheta}_{xQ}={V}_{mD}{\gamma}_P+{V}_{mQ}{\widehat{\gamma}}_Q/\left[{V}_{mD}^2-{V}_{mQ}^2\right]\end{array}\right\} $$

(17)

where

$$ {\widehat{\gamma}}_Q=\left[{\gamma}_Q-{V}_{mD}^2-{V}_{mQ}^2\right] $$

(18)

Finally the control objective is attained as:

$$ \left.\begin{array}{l}{\vartheta}_{\varsigma D}={\vartheta}_{xD}-{V}_{nD}\\ {}{\vartheta}_{\varsigma Q}={\vartheta}_{xQ}-{V}_{nQ}\end{array}\right\} $$

(19)

Thus *γ*_{
Q
} and *γ*_{
P
} are the target control terms which are used to control the final dq- axis series injected voltages (Eq. (34)), which subsequently control the proposed AM of the UPFC, respectively.

### 3.2 Proposed adaptive fractional integral terminal sliding mode power control (AFITSMPC) of UPFC

Reformulating the dynamic power model of the series SSSC of the UPFC (Eq. (17)):

$$ \dot{x}= Ax+ Bu $$

(20)

where

$$ A=\left[\begin{array}{cc}-\omega & 0\\ {}0& \omega \end{array}\right],B=\left[\begin{array}{cc}{x}_{SE}^{-1}& 0\\ {}0& {x}_{SE}^{-1}\end{array}\right],x=\left[\begin{array}{l}{X}_1\\ {}{X}_2\end{array}\right]=\left[\begin{array}{l}{p}_n\\ {}{q}_n\end{array}\right],\mathrm{and}\kern0.24em u=\left[\begin{array}{l}{\gamma}_Q\\ {}{\gamma}_P\end{array}\right] $$

(21)

The tracking error functions for the controller, in terms of the active and reactive power of the SSSC of the UPFC are defined as:

$$ \left[\begin{array}{l}{e}_{r1}\\ {}{e}_{r2}\end{array}\right]=\left[\begin{array}{l}{p}_n^{\ast}\\ {}{q}_n^{\ast}\end{array}\right]-\left[\begin{array}{l}{p}_n\\ {}{q}_n\end{array}\right] $$

(22)

Where \( {p}_n^{\ast } \) and \( {q}_n^{\ast } \) are the reference value of the active and reactive powers, which are evaluated in the initial solution.

Taking the derivative of Eq. (22) with respect to time on both sides, we derive:

$$ \left[\begin{array}{l}{\dot{e}}_{r1}\\ {}{\dot{e}}_{r2}\end{array}\right]=\left[\begin{array}{cc}\omega & 0\\ {}0& -\omega \end{array}\right]\left[\begin{array}{l}{p}_n\\ {}{q}_n\end{array}\right]-\left[\begin{array}{cc}0& {x}_{SE}^{-1}\\ {}{x}_{SE}^{-1}& 0\end{array}\right]\left[\begin{array}{l}{\gamma}_P\\ {}{\gamma}_Q\end{array}\right] $$

(23)

The fractional Integral terminal sliding surface (FITSS) are defined as:

$$ \left[\begin{array}{l}{\sigma}_{r1}\\ {}{\sigma}_{r2}\end{array}\right]=\left[\begin{array}{l}{e}_{r1}\\ {}{e}_{r2}\end{array}\right]+\left[\begin{array}{cc}{\alpha}_1& 0\\ {}0& {\alpha}_2\end{array}\right]\left[\begin{array}{l}{e}_{Ir1}\\ {}{e}_{IIr2}\end{array}\right] $$

(24)

where

$$ \left[\begin{array}{l}{\dot{e}}_{Ir1}\\ {}{\dot{e}}_{IIr2}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ {}0& 1\end{array}\right]\left[\begin{array}{l}{e}_{r1}^{\beta_1}\\ {}{e}_{r2}^{\beta_2}\end{array}\right] $$

(25)

With *α*_{
1
}, *α*_{
2
} > 0, β_{1}, β_{2} are fractional numbers satisfying the relation 0 < [β_{1}, β_{2}] < 1.

Integrating the above Eq. (25), and replacing the resultant in Eq. (23), the final expression for the FITSS in terms of the tracking error is defined as,

$$ \left[\begin{array}{l}{\sigma}_{r1}\\ {}{\sigma}_{r2}\end{array}\right]=\left[\begin{array}{l}{e}_{r1}\\ {}{e}_{r2}\end{array}\right]+\left[\begin{array}{cc}{\alpha}_1& 0\\ {}0& {\alpha}_2\end{array}\right]\left[\begin{array}{l}\int {e}_{r1}^{\beta_1}\\ {}\int {e}_{r2}^{\beta_2}\end{array}\right] $$

(26)

where (*β*_{
1
}, *β*_{
2
}) are the fractional powers of the tracking errors (*e*_{
r1
}, *e*_{
r2
}) with initial values (−*e*_{
r1
}(0)/*α*_{
1
}, −*e*_{
r1
}(0)/*α*_{
1
}), respectively.

The derivative of the above Eq. (26) is defined as:

$$ \left[\begin{array}{l}{\dot{\sigma}}_{r1}\\ {}{\dot{\sigma}}_{r2}\end{array}\right]=\left[\begin{array}{l}{\dot{e}}_{r1}\\ {}{\dot{e}}_{r2}\end{array}\right]+\left[\begin{array}{cc}{\alpha}_1& 0\\ {}0& {\alpha}_2\end{array}\right]\left[\begin{array}{l}{e}_{r1}^{\beta_1}\\ {}{e}_{r2}^{\beta_2}\end{array}\right] $$

(27)

Replacing Eq. (23) in Eq. (27), we derive:

$$ \Rightarrow \kern0.5em \left[\begin{array}{l}{\dot{\sigma}}_{r1}\\ {}{\dot{\sigma}}_{r2}\end{array}\right]\kern0.5em =\left[\begin{array}{cc}\omega & 0\\ {}0& -\omega \end{array}\right]\left[\begin{array}{l}{p}_n\\ {}{q}_n\end{array}\right]+\left[\begin{array}{cc}{\alpha}_1& 0\\ {}0& {\alpha}_2\end{array}\right]\left[\begin{array}{l}{e}_{r1}^{\beta_1}\\ {}{e}_{r2}^{\beta_2}\end{array}\right]-\left[\begin{array}{cc}0& {x}_{SE}^{-1}\\ {}{x}_{SE}^{-1}& 0\end{array}\right]\left[\begin{array}{l}{\gamma}_P\\ {}{\gamma}_Q\end{array}\right] $$

(32)

**Theorem 1** *The tracking error functions as defined in* Eq. (25) *will converge to zero in a finite amount of time, and the system will remain robust and stable if the FITSS are chosen as in* Eq. (32), *and the control is designed as follows*:

$$ \left[\begin{array}{l}{\gamma}_P\\ {}{\gamma}_Q\end{array}\right]=\left[\begin{array}{l}{\gamma}_{Pnom}\\ {}{\gamma}_{Qnom}\end{array}\right]+\left[\begin{array}{l}{\gamma}_{\Pr ob}\\ {}{\gamma}_{Qrob}\end{array}\right] $$

(33)

where *γ*_{
Pnom
} and *γ*_{
Qnom
} are the nominal controls, whereas *γ*_{
Prob
} and *γ*_{
Qrob
} are the robust controls, respectively, as introduced by the terminal sliding mode concept and can be derived as follows:

$$ \left[\begin{array}{l}{\gamma}_{Pnom}\\ {}{\gamma}_{Qnom}\end{array}\right]=\left[\begin{array}{cc}0& -\omega {x}_{SE}\\ {}\omega {x}_{SE}& 0\end{array}\right]\left[\begin{array}{l}{p}_n\\ {}{q}_n\end{array}\right]+\left[\begin{array}{cc}0& {\alpha}_2{x}_{SE}\\ {}{\alpha}_1{x}_{SE}& 0\end{array}\right]\left[\begin{array}{l}{e}_{r1}^{\beta_1}\\ {}{e}_{r2}^{\beta_2}\end{array}\right] $$

(34)

$$ \left[\begin{array}{l}{\dot{\gamma}}_{\Pr ob}\\ {}{\dot{\gamma}}_{Qrob}\end{array}\right]=\left[\begin{array}{l}{\gamma}_{\Pr ob V}\\ {}{\gamma}_{Qrob V}\end{array}\right]-\left[\begin{array}{cc}{\beta}_{\Pr ob}& 0\\ {}0& {\beta}_{Qrob}\end{array}\right]\left[\begin{array}{l}{\gamma}_{\Pr ob}\\ {}{\gamma}_{Qrob}\end{array}\right] $$

(35)

where

$$ \left[\begin{array}{l}{\gamma}_{\Pr obV}\\ {}{\gamma}_{QrobV}\end{array}\right]=\left[\begin{array}{cc}{\widehat{\beta}}_{\Pr obV}& 0\\ {}0& {\widehat{\beta}}_{QrobV}\end{array}\right]\left[\begin{array}{l}\mathit{\operatorname{sign}}\left({\sigma}_{r2}\right)\\ {}\mathit{\operatorname{sign}}\left({\sigma}_{r1}\right)\end{array}\right] $$

(36)

$$ \left[\begin{array}{l}{\dot{\widehat{\beta}}}_{\Pr obV}\\ {}{\dot{\widehat{\beta}}}_{QrobV}\end{array}\right]=-\left[\begin{array}{cc}{\beta}_{\Pr obA}& 0\\ {}0& {\beta}_{QrobA}\end{array}\right]\left[\begin{array}{l}\left|{\sigma}_{r2}\right|\\ {}\left|{\sigma}_{r1}\right|\end{array}\right] $$

(37)

In the above Eqs. (34)–(37), {*β*_{
ProbV
}, *β*_{
QrobV
}, *β*_{
ProbA
}, *β*_{
QrobA
}} > 0, are the gains of the controller, whose initial values are mentioned in the Appendix. Eq. (37), which makes the controller adaptive i.e., adjusts the controller gains according to the varying operating conditions, is defined as the fractional integral terminal sliding mode adaptive law for the admittance model of the UPFC.

*Proof of convergence* On the surface \( {\dot{\sigma}}_{r1}=0 \) and \( {\dot{\sigma}}_{r2}=0 \) we have,

$$ \left[\begin{array}{l}{\dot{e}}_{r1}\\ {}{\dot{e}}_{r1}\end{array}\right]=-\left[\begin{array}{l}{\alpha}_1{e}_{r1}^{\beta_1}\\ {}{\alpha}_2{e}_{r2}^{\beta_2}\end{array}\right]\Rightarrow \left[\begin{array}{l}\frac{{\dot{e}}_{r1}}{\alpha_1{e}_{r1}^{\beta_1}}\\ {}\frac{{\dot{e}}_{r2}}{\alpha_2{e}_{r2}^{\beta_2}}\end{array}\right]=-\left[\begin{array}{l}1\\ {}1\end{array}\right] $$

(38)

Integrating the above Eq. (39), and then rearranging the resultant term, the convergence time of the tracking errors are derived as:

$$ \left[\begin{array}{l}{t}_{r1}\\ {}{t}_{r2}\end{array}\right]=\left[\begin{array}{l}\frac{{\left|{e}_{Ir1}\right|}^{1-{\beta}_1}}{\alpha_1^{\beta_1}\left(1-{\beta}_1\right)}\\ {}\frac{{\left|{e}_{Ir2}\right|}^{1-{\beta}_2}}{\alpha_2^{\beta_2}\left(1-{\beta}_2\right)}\end{array}\right]=\left[\begin{array}{l}\frac{{\left|{e}_{r1}\right|}^{1-{\beta}_1}}{\alpha_1\left(1-{\beta}_1\right)}\\ {}\frac{{\left|{e}_{r2}\right|}^{1-{\beta}_2}}{\alpha_2\left(1-{\beta}_2\right)}\end{array}\right] $$

(39)

Equation (39) guarantees a finite time convergence of the tracking error functions [12] as defined in Eq. (25).

*Proof of stability* Let us consider the following Lyapunov function:

$$ {V}_L=\frac{1}{2}{\sigma}_{r1}^2+\frac{1}{2}{\sigma}_{r2}^2+\frac{1}{\beta_{\Pr obA}}{\left({\dot{\widehat{\beta}}}_{\Pr obV}-{\widehat{\beta}}_{\Pr obV}\right)}^2+\frac{1}{\beta_{QrobA}}{\left({\dot{\widehat{\beta}}}_{QrobV}-{\widehat{\beta}}_{QrobV}\right)}^2 $$

(40)

Taking the derivative of the above Eq. (40) w.r.t time on both sides, we derive:

$$ {\dot{V}}_L={\sigma}_{r1}{\dot{\sigma}}_{r1}+{\sigma}_{r2}{\dot{\sigma}}_{r2}+\left({\dot{\widehat{\beta}}}_{\Pr obV}-{\widehat{\beta}}_{\Pr obV}\right)\left|{\sigma}_{r2}\right|+\left({\dot{\widehat{\beta}}}_{QrobV}-{\widehat{\beta}}_{QrobV}\right)\left|{\sigma}_{r1}\right| $$

(41)

Substituting Eqs. (34)–(37) in eq. (32), we simplify as:

$$ \left[\begin{array}{l}{\dot{\sigma}}_{r1}\\ {}{\dot{\sigma}}_{r2}\end{array}\right]\kern0.5em =-{x}_{SE}^{-1}\left[\begin{array}{l}\left\{\left(1-h\right){\beta}_{Qrob}\right\}+\left\{h{\widehat{\beta}}_{Qrob V}\mathit{\operatorname{sign}}\left({\sigma}_{r1}\right)\right\}\\ {}\left\{\left(1-h\right){\beta}_{\Pr ob}\right\}+\left\{h{\widehat{\beta}}_{\Pr ob V}\mathit{\operatorname{sign}}\left({\sigma}_{r2}\right)\right\}\end{array}\right] $$

(42)

where h is the step length equal to 0.01.

Now substituting Eq. (42) in Eq. (41), we derive:

$$ {\displaystyle \begin{array}{l}{\dot{V}}_L=-\left[\left(1+{hx}_{SE}^{-1}\right){\widehat{\beta}}_{Qrob V}-{\dot{\widehat{\beta}}}_{Qrob V}\right]\left|{\sigma}_{r1}\right|\\ {}\kern1.75em -\left[\left(1+{hx}_{SE}^{-1}\right){\widehat{\beta}}_{\Pr ob V}-{\dot{\widehat{\beta}}}_{\Pr ob V}\right]\left|{\sigma}_{r2}\right|\\ {}\kern2em -{x}_{SE}^{-1}\left[\left({\sigma}_{r1}\left(1-h\right){\beta}_{Qrob}\right)+\left({\sigma}_{r2}\left(1-h\right){\beta}_{\Pr ob}\right)\right]\\ {}\Rightarrow \kern1em {\dot{V}}_L\kern1em \le 0\kern1.25em \mathrm{for}\ \mathrm{all}\ {\sigma}_{r1}\ne 0\kern0.75em \mathrm{and}\ {\sigma}_{r2}\ne 0.\end{array}} $$

(43)

From Eq. (43), it is proved that as the value of \( {\dot{V}}_L\le 0 \) for all *σ*_{
r1
} ≠ 0 and *σ*_{
r2
} ≠ 0, which guarantees the system stability. This completes the proof.