The dynamics of multi-area IPS is known for its exceptional complexity and nonlinearity. Due to continuous change in generation-demand balance, IPS states change so frequently within tight tolerance. Moreover, the frequency measured in each control area shows the deviation in tie-line power and the generation-demand mismatch in the area. Thus, the dynamics of IPS while studying LFC needs to consider the tie-line power deviation.

Normally, the LFC scheme in large-scale IPS is carried out by approximating the overall system with several separated subsystem models and executes the control in either distributed or decentralized manner. Distributed control schemes differ from decentralized control as a result of the consideration given to the external state information transmitted from the neighboring subsystems.

The study of LFC is restrained to comparatively low oscillations; the complex nonlinear model of the IPS is derived and linearized [5].

For *i*th CA in a multi-area IPS shown in Fig. 2, the mismatch between generation, *P*^{G} and demand, *P*^{D} on frequency, *f*_{i} is modeled using linearized swing equation as

$$ {\dot{f}}_i=\frac{1}{H_i}\left({P}_i^G-{D}_i{f}_i-{P}_i^{tie}-{P}_i^D\right) $$

(1)

where \( {P}_i^G \) is the *i*th area total generation computed by summing up the power output of all units in the area as illustrated in (2);

$$ {P}_i^G=\sum \limits_{m\in {\mathcal{G}}_i}{P}_{i,\kern0.5em m}^G $$

(2)

where *D*_{i} and *H*_{i} are the *i*th control area damping coefficient and inertia constant respectively. \( {\mathcal{G}}_i \) is the index set of generators in the *i*th control area assumed to form a coherent group. This assumption agrees with assigning a single frequency to each control area.

In conventional LFC design, only phase angles, *δ*_{i}, are changing while the voltage magnitudes, |*V*_{i}|, are maintained constant. This is due to the fact that AVR loop (responsible for voltage control) is much faster than LFC loop. As such, change in tie-line power is normally modeled as in (3) [6].

$$ {\dot{P}}_i^{tie}=\sum \limits_{j\in {\mathcal{A}}_i^{tie}}{\dot{P}}_{ij};\kern0.5em {\dot{P}}_{ij}=2\pi {T}_{ij}^0\left({f}_i-{f}_j\right) $$

(3)

$$ {P}_i^{tie}=\frac{\left|{V}_i\right|\left|{V}_j\right|}{X_{ij}}\mathit{\sin}\left({\delta}_i-{\delta}_j\right) $$

(4)

Where \( {T}_{ij}^0 \) and *X*_{ij} are the synchronizing coefficient and reactance of tie-line between area *i* and *j*. \( {\mathcal{A}}_i^{tie} \) denotes the index set of all the control areas connected to the *i*th area.

However, with the proposed introduction of slow control action, the AVR loop time can be made to be (at least theoretically) as large as that of the LFC loop. As such, a more accurate model accounting for change in frequency (phase angle) as well as change in voltage (*∆*|*V*_{i}|) need to be developed. From (3), the change in *i*th tie-line power flow is obtained as

$$ {\dot{P}}_i^{tie}=\frac{\partial {P}_i^{tie}}{\partial \left|{V}_i\right|}\Delta \left|{V}_i\right|+\frac{\partial {P}_i^{tie}}{\partial \left|{V}_j\right|}\Delta \left|{V}_j\right|+\frac{\partial {P}_i^{tie}}{\partial \left|\left({\delta}_i-{\delta}_j\right)\right|}\Delta \left({\delta}_i-{\delta}_j\right) $$

(5)

Taking the partial derivative of (5), it becomes

$$ {\dot{P}}_i^{tie}={T}_i^0\Delta \left|{V}_i\right|+{T}_j^0\Delta \left|{V}_j\right|+2\pi {T}_{ij}^0\left({f}_i-{f}_j\right) $$

(6)

$$ {T}_i^0=\frac{\left|{V}_j\right|}{X_{ij}}\mathit{\sin}\left({\delta}_i-{\delta}_j\right)\ and\kern0.5em {T}_j^0=\frac{\left|{V}_i\right|}{X_{ij}}\mathit{\sin}\left({\delta}_i-{\delta}_j\right) $$

(7)

The frequency deviation expressed as the function of angular displacement of the rotors referenced to the stator;

$$ {f}_i=\frac{\Delta {\delta}_i}{2\pi } $$

(8)

It can be observed from (7) that voltage perturbations result in added increments in the power. The perturbations are seen to increase the loads in the *i*th CAs by a quantity \( \left(\raisebox{1ex}{$\partial {P}_i^D$}\!\left/ \!\raisebox{-1ex}{$\partial \left|{V}_i\right|$}\right.\right)\Delta \left|{V}_i\right| \), as such this additional term is added as virtual load disturbance. The terms are added at the same point where the actual load disturbances are added. Therefore, the model of the two-area system presented in [6] is modified to incorporate the effect of voltage perturbations as shown in Fig. 2.

The dynamics of the non-reheat turbines (chosen for this study) is governed by the load changes and control input signal computed by the LFC controller. The turbine dynamics of the *m*th unit in the *i*th area is formulated as

$$ {\dot{P}}_{i,\kern0.5em m}^G={sat}_{{\dot{P}}_{i,m}^G}^i\left\{\frac{1}{T_{T_{i,m}}}\left({P}_{i,m}^{Gov}-{P}_{i,m}^G\right)\right\} $$

(9)

where \( {T}_{T_{i,m}} \)and \( {sat}_{{\dot{P}}_{i,m}^G}^i \)are the steam chest time constant and saturation nonlinearity to model GRC of *m*th generator in the *i*th control area.

The change in valve position of *m*th generator as a relation to frequency is approximated to have one time constant, \( {T}_{Gov_{i,m}} \) as

$$ {\dot{P}}_{i,m}^{Gov}=\frac{1}{T_{Gov_{i,m}}}\left({P}_{i,m}^{SC}-{P}_{i,m}^{Gov}-\frac{1}{R_{i,m}}{f}_i\right) $$

(10)

In IPS, deviations in frequency and net tie-line power are the two fundamental variables in the LFC. Their combination referred as area control error (ACE) is utilized as a performance measure and serves as the feedback input signal to the LFC. For *i*th CA, the ACE is computed as [28];

$$ {ACE}_i={\beta}_i{f}_i+\Delta {P}_i^{tie} $$

(11)

$$ \left[\begin{array}{c}{f}_i\\ {}{P}_{i,\kern0.5em m}^G\\ {}{P}_{i,\kern0.5em m}^{Gov}\\ {}{P}_i^{tie}\end{array}\right]=\left[\begin{array}{cccc}-\frac{D_i}{H_i}& \frac{1}{H_i}& 0& -\frac{1}{H_i}\\ {}0& -\frac{1}{T_{T_{i,m}}}& \frac{1}{T_{T_{i,k}}}& 0\\ {}-\frac{1}{R_{i,m}{T}_{Gov_{i,m}}}& 0& -\frac{1}{T_{Gov_{i,m}}}& 0\\ {}\sum \limits_{j\in {\mathcal{A}}_i^{tie}}2\pi {T}_{ij}^0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{f}_i\\ {}{P}_{i,\kern0.5em m}^G\\ {}{P}_{i,\kern0.5em m}^{Gov}\\ {}{P}_i^{tie}\end{array}\right]+\left[\begin{array}{c}-\frac{1}{H_i}\\ {}0\\ {}0\\ {}0\end{array}\right]{P}_i^D+\left[\begin{array}{c}0\\ {}0\\ {}\frac{1}{T_{Gov_{i,\kern0.5em m}}}\\ {}0\end{array}\right]{P}_{i,\kern0.5em m}^{SC}+\left[\begin{array}{c}-\frac{1}{H_i}\\ {}0\\ {}0\\ {}-2\pi \sum \limits_{j\in {\mathcal{A}}_i^{tie}}{T}_{ij}^0\end{array}\right]{f}_j+\left[\begin{array}{c}0\\ {}0\\ {}0\\ {}{T}_i^0\end{array}\right]\Delta \left|{V}_i\right|+\left[\begin{array}{c}0\\ {}0\\ {}0\\ {}{T}_j^0\end{array}\right]\Delta \left|{V}_j\right| $$

(12)

$$ {\mathrm{ACE}}_i=\left({\beta}_i\kern0.5em 0\kern0.5em 0\kern0.5em 1\right){\left({f}_i\kern0.5em {P}_{i,m}^G\kern0.5em {P}_{i,m}^{Gov}\kern0.5em {P}_i^{tie}\right)}^T $$

(13)

The system model described in (1)-(11) for *i*th CA is summarized in the state-space model as in (12) and (13). Where \( {\left[{f}_i\kern0.5em {P}_{i,\kern0.5em m}^G\kern0.5em {P}_{i,\kern0.5em m}^{Gov}\kern0.5em {P}_i^{tie}\right]}^T\in {\mathbb{R}}^4 \) is the state vector (*x*_{i}), \( {P}_i^D \) is the disturbance input while ACE_{i} is the output for area *i*. The vector\( {\left[{P}_{i,\kern0.5em m}^{SC}\kern0.5em \Delta \left|{V}_i\right|\right]}^T\in {\mathbb{R}}^2 \)is the control input (*u*_{i}) comprising of the optimal supplementary control signal, \( {P}_{i,\kern0.5em m}^{SC} \) from the LFC loop and the voltage control input, *∆*|*V*_{i}| for *i*th CA. In the conventional LFC scheme, the control vector is one-dimensional (\( {P}_{i,\kern0.5em m}^{SC} \)) while in the developed coordinated scheme, the size of the control vector doubles for each CA because of the introduction of the voltage control input.