In the current paper, two electrical power system models are engaged to examine the ability of the suggested controller for LFC, which are commonly used in the literature. The test system-I presented in Fig. 1 is 2-area 6-unit system with/without HVDC link [24,25,26,27]. The system comprises a source like thermal hydro-gas. The test system parameters are taken from reference [27]. The readers are advised to refer [27] for the definition and meaning of symbols used in Fig. 1. The generation power rating of every area is 2000 MW, and nominal loading is 1000 MW. The load involvement of thermal, hydro and gas systems is 600 MW, 250 MW and 150 MW, respectively. For more practical power system, time delay (TD) element is incorporated in the test model. In the present paper, the value of TD is taken as 50 ms [39]. In test system-II, a non-reheat-type thermal system [7,8,9,10,11] is considered as presented in Fig. 2. The detailed data of studied systems are available in [7,8,9,10,11].

### Modeling of HVDC linkage

HVDC link (parallel AC–DC) is connected directly with the AC tie-line interconnected power system for improvement of system performance. The structure of two-area arrangement through AC–DC links is presented in Fig. 3 [24, 25, 27]. The modification of output in area-1 of AC tie and HVDC link is as follows:

$$ \Delta P_{{{\text{AC}}}} = \frac{{2\pi T_{12} }}{s}(\Delta F_{1} - \Delta F_{2} ) $$

(1)

$$ \Delta P_{\text{DC}} = \frac{{K_{\text{DC}} }}{{1 + sT_{\text{DC}} }}(\Delta F_{1} - \Delta F_{2} ) $$

(2)

where \(K_{\text{DC}}\) (HVDC link gain) and \(T_{\text{DC}}\) (HVDC link-time constant).

### Optimal controller design

The first step in the development of the design procedure of the controller is the linear representation of the structure. The linear form of the structure is designated in the state space form, as follows:

$$ \mathop {\varvec{x}}\limits^{ \cdot } = {\varvec{A}}x + {\varvec{B}}u +{\varvec{\varGamma}}P_{D} $$

(3)

where **A** (*n***n*) is state matrix, **B** (*n***m*) is a control matrix for *n* number of state variables and *m* number of inputs, and \({\Gamma }\) is a disturbance matrix. Different variables have been defined as:

State variables:

*x*_{1} = ∆*F*_{1}, *x*_{2} = ∆*P*_{Tie}, *x*_{3} = ∆*F*_{2}, *x*_{4} = ∆*P*_{Gt1}, *x*_{5} = ∆*P*_{Rt1}, *x*_{6} = ∆*X*_{t1}, *x*_{7} = ∆*P*_{Gh1}, *x*_{8} = ∆*X*_{h1}, *x*_{9} = ∆*X*_{RH1}, *x*_{10} = ∆*P*_{Gg1}, *x*_{11} = ∆*P*_{FC1}, *x*_{12} = ∆*P*_{VP1}, *x*_{13} = ∆*X*_{g1}, *x*_{14} = ∆*P*_{Gt2}, *x*_{15} = ∆*P*_{Rt2}, *x*_{16} = ∆*X*_{t2}, *x*_{17} = *P*_{Gh2}, *x*_{18} = ∆*X*_{h2}, *x*_{19} = ∆*X*_{RH2}, *x*_{20} = ∆P_{Gg2}, *x*_{21} = ∆*P*_{FC2}, *x*_{22} = ∆*P*_{VP2}, *x*_{23} = ∆*X*_{g2}, *x*_{24} = *∫ACE*_{1}, *x*_{25} = *∫ACE*_{2}, *x*_{26} = *∆P*_{TieDC}.

Control inputs:

*u*_{1} = *∆P*_{C1};* u*_{2} = *∆P*_{C2};* U* = [*u*_{1} *u*_{2}]^{T}

Disturbance inputs:

∆P_{D1;} ∆P_{D2}; **P**_{D} = [∆P_{D1} ∆P_{D2}] ^{T}.

The system state variables (\(\mathop {{\text{x}}_{1} }\limits^{ \cdot } - \mathop {{\text{x}}_{26} }\limits^{ \cdot }\)) equations concerning transfer function block in Fig. 1 can be expressed from which the input matrices **A** are found to be of the order 26 × 26, the matrix **B** is of the order of 26 × 2, and the matrix \({\Gamma }\) is of the order of 26 × 2. The output is given by (4).

$$ \varvec{Y} = \varvec{CX} + \varvec{DU} $$

(4)

For matrix ‘**D**’ is considered as zero.

Therefore, the output is represented as

$$ \varvec{Y} = \varvec{CX} $$

(5)

where ‘**C**’ matrix is the order of (2 × 26) describe the output matrix.

The values of matrices can be calculated with the help of [27]. The 26 states are *x*_{1},* x*_{2}*… x*_{26}.

Hence, finally, the equation for control input can be written as:

$$ {\mathbf{U}} = - ({\mathbf{K}}{\text{X)}} $$

(6)

where ‘**K**’ is a (2 × 26) matrix known as feedback matrix gain and is represented by:

**K** = \(\left[ {\begin{array}{cccccccccccccccccccccccccc} {k_{1,1} } & {k_{1,2} } & {k_{1,3} } & {k_{1,4} } & {k_{1,5} } & {k_{1,6} } & {k_{1,7} } & {k_{1,8} } & {k_{1,9} } & {k_{1,10} } & {k_{1,11} } & {k_{1,12} } & {k_{1,13} } & {k_{1,14} } & {k_{1,15} } & {k_{1,16} } & {k_{1,17} } & {k_{1,18} } & {k_{1,19} } & {k_{1,20} } & {k_{1,21} } & {k_{1,22} } & {k_{1,23} } & {k_{1,24} } & {k_{1,25} } & {k_{1,26} } \\ {k_{2,1} } & {k_{2,2} } & {k_{2,3} } & {k_{2,4} } & {k_{2,5} } & {k_{2,6} } & {k_{2,7} } & {k_{2,8} } & {k_{2,9} } & {k_{2,10} } & {k_{2,11} } & {k_{2,12} } & {k_{2,13} } & {k_{2,14} } & {k_{2,15} } & {k_{2,16} } & {k_{2,17} } & {k_{2,18} } & {k_{2,19} } & {k_{2,20} } & {k_{2,21} } & {k_{2,22} } & {k_{2,23} } & {k_{2,24} } & {k_{2,25} } & {k_{2,26} } \\ \end{array} } \right]\).

The quadratic form of performance index (PI) is as follows

$$ {\text{PI}} = \frac{1}{2}\int\limits_{0}^{\infty } {\left( {x^{T} Qx + u^{T} Ru} \right)} {\text{d}}t $$

(7)

where ‘**Q**’ represents ‘State Weighing Matrix’ and ‘**R**’ represents ‘Control Weighing Matrix.’

The variation of area control errors (ACE) is:

$$ e_{1} (t) = ACE_{1} = B_{1} \Delta F_{1} + \Delta P_{Tie12} = \beta_{1} x_{1} + x_{3} $$

(8)

$$ e_{2} (t) = ACE_{2} = B_{2} \Delta F_{2} + \Delta P_{Tie21} = \beta_{2} x_{2} - x_{3} $$

(9)

The deviations of \(\int {ACE({\text{d}}t} )\) about the steady-state values are minimized. For this case, these deviations are *x*_{24} and *x*_{25}. The deviations of control inputs (*u*_{1} and *u*_{2}) about the steady-state values are minimized. Based on the realistic control specifications requisite of LFC scheme, it is perceived from literature that the best system performance is acquired with minimum values of settling times, peak overshoots, and maximum value of damping ratio in frequency and tie-line power deviations when ITAE is employed as objective function [22]. Therefore, ITAE is elected as a cost function in the current study to determine the parameters of the controller and given by:

$$ J = {\text{ITAE}} = \int\limits_{0}^{{t_{{}} }} {(|\Delta F_{1} | + |\Delta F_{2} | + |\Delta P_{Tie} |} ) \cdot t.{\text{d}}t $$

(10)

For optimal control problem, the objective function is written as:

$$ J = ITAE = \int\limits_{0}^{{t_{sim} }} {(|{\text{x}}_{1} | + |{\text{x}}_{2} | + |{\text{x}}_{3} |} ) \cdot t.dt $$

(11)

### State-space model of Test system-II

The state-space representation of input, control and disturbance vectors for the system under study is:

State vector:

$$ [X]^{T} = \left[ \begin{array}{c} \Delta F_{1} \quad \Delta P_{Tie} \;\Delta F_{2} \quad \Delta P_{Gt1} \quad \Delta P_{Rt1} \quad \Delta X_{t1} \quad \Delta P_{Gh1} \hfill \\ \Delta X_{h1} \quad \Delta X_{RH1} \quad \Delta P_{Gg1} \quad \Delta P_{FC1} \quad \Delta P_{VP1} \quad \Delta X_{g1} \hfill \\ \Delta P_{Gt2} \quad \Delta P_{Rt2} \quad \Delta X_{t2} \quad \Delta P_{Gh2} \quad \Delta X_{h2} \quad \Delta X_{RH2} \hfill \\ \Delta P_{Gg2} \quad \Delta P_{FC2} \quad \Delta P_{VP2} \quad \Delta X_{g2} \;\int {ACE_{1} {\text{d}}t} \;\int {ACE_{2} {\text{d}}t} \hfill \\ \Delta P_{TieDC} \hfill \\ \end{array} \right] $$

(12)

Control vector:

$$ \underline{U} = [\Delta P_{{C1}} \quad \Delta P_{{C2}} ]^{T} $$

(13)

Disturbance vector:

$$ \underline{{P_{d} }} = [\Delta P_{{D1}} \quad \Delta P_{{D2}} ]^{T} $$

(14)

The matrices of state space are found by using the equations of state space. The detailed equations of state space are given in reference [27].

### FOPID controller

Conventional PID controllers may not offered required system performance if it is connected with nonlinearity parameters. Extensive development has been noticed in the growth of intelligent controllers applied to different power systems, but still, it remains a demanding area and a general problem for researchers. The fuzzy logic-based controller needs more fuzzy variables for better accuracy. This will exponentially increase the rues. The PID controller has been effectively used in many applications. The acceptance of the PID is due to the ease of the design processes and acceptable performance. The fractional-order controller design methods are in principle founded on additions of the traditional PID control theory, with an importance on the greater flexibility in the tuning approach ensuing improved control performances as related to classical control. Fractional calculus has become very beneficial in recent times because of its applications in many applied sciences. Persuaded from the positive results of these developments, a FOPID structure is suggested for LFC of power system.

FOPID controller has been suggested in the current paper which includes a fractional-order and PID configuration. Conventional PID controllers are normally not effectual as of their linear arrangement, mainly, if higher-order plants are concerned or if time delay systems and uncertainties are there. On the contrary, the FOPID can handle nonlinearity and uncertainties. The FOPID can be intended to match the necessary performance of the control system. From the literature, it is seen that application of FOPID enhanced the performance of PID/PI. The proposed FOPID controller gets advantages of outstanding ability of a PID in addition to the feedback control mechanism in removing the steady-state error in addition to predicting and controlling future error.

The main advantages of FOPID are that if the parameter of a power system varies, a fractional-order (FO) PID is less responsive over a conventional PID [33, 37]. Additionally, the FO has two additional variables to optimize. Its offers further degrees of freedom to the dynamic properties of FO structure. The FOPID configuration assumed in each generating element is shown in Fig. 4. The inputs to the controllers are the respective ACEs, and outputs of the controllers are the reference power setting of generating units\( \Delta P_{c} \). In Fig. 4, *K*_{P}, *K*_{I,}* K*_{D,} λ and *µ* are to be optimized. The expression of FOPID is given by Eq. (15).

$$G(s)=K_{\text{P}}+\frac{K_{\text{I}}}{S_{\lambda}}+K_{\text{D}}S^{\mu}$$

(15)