Predictive generalised active disturbance rejection for load frequency control with communication delay

Abstract

This paper formulates a generalised active disturbance rejection control technique based on Smith predictor (SP-GADRC) for power systems with communication delay. The SP-GADRC technique employs essential plant information and uses the notion of Smith predictor to alleviate the deterioration of transient response that arises due to the presence of communication delay in a system. The generalised formulae for computation of controller and observer gains for various single and multi-area power systems are derived. Furthermore, the effect of parametric uncertainty, nonlinearities, model-plant mismatch, load disturbances as well as varying time delay is examined to validate the robustness of the SP-GADRC technique. An elaborate comparative analysis is conducted with various existing approaches to exhibit efficacy of the SP-GADRC technique. The simulation results reveal that SP-GADRC scheme demonstrates an improved disturbance rejection ability, better transient performance and is robust to the presence of nonlinearities and disturbances in system.

Keywords

Active disturbance rejection control controller design communication delay load frequency control

Introduction

In the modern large-scale power systems comprising of interconnected distribution, generation and transmission units situated over remotely located geographical areas, the fluctuations in frequency and variations in tie-line power are imminent in the presence of unforeseen and random load demands, dynamically varying environmental conditions, uncertainties in system parameters, imprecise mathematical modelling and structural constraints in speed governor mechanism. Load frequency control (LFC) is an indispensable ancillary service to supply reliable and good quality power output, and maintain synchronism by restoration of the frequency together with the tie-line power to nominal scheduled values, thereby leading to matching of power generation in tandem with the frequently varying load demand (Bevrani, 2009; Hote and Jain, 2018).

The unanticipated load demand perturbations can cause frequency fluctuations and lead to deviations in scheduled tie-line power interchange. The net effect of both these variations is incorporated into a single term known as the area control error, which along with control signals was conventionally transmitted using dedicated communication networks. The delays involved in such networks are insignificant as compared to dynamics of a power system and are usually neglected. But, the emergence of open and distributed communication links has led to the inevitable introduction of substantial communication delay during the transmission of sensor measurements from remote terminal units to the control centre and from the control centre to the generating unit in plant. The presence of communication delay adversely affects the system dynamics, reduces the phase margin, limits the bandwidth of the closed loop control system and hinders the ability of control signal to quickly respond to the errors, thereby leading to deterioration in the performance of controllers, and in worst cases has a potential to cause instability in the system. Therefore, the control of processes having communication delay is a formidable endeavour in the sphere of control system engineering (Sharma et al., 2019).

The increased complexities caused by presence of communication delay have been addressed by several existing works in the literature. Various techniques have been adopted to alleviate the effect of communication delay such as robust control, proportional–integral–derivative (PID) control, optimal control and so on. A delay-dependent two-term $H_{\infty}$ state feedback controller design using the linear matrix inequalities is presented in the study by Dey et al. (2012) for the LFC of two-area interconnected power system with communication delay in the area control error signal. It provides a two-term controller that is superior as compared to the existing one-term controller. The tuning of a PID controller is accomplished via 2 degrees of freedom internal model control technique for LFC of multi-area power systems with communication delay in the study by Tan et al. (2010). The internal model control (IMC) controller is approximated by a PID controller, which involves the approximation of a non-reheated turbine-based power system into a second-order process with time delay. However, as mentioned by the authors, the approximation is not valid for other turbine-based (reheat turbine, hydro turbine) power systems. In the study by Sharma et al. (2019), power systems are modelled via Kharitonov’s theorem and the gains of the PID controller are tuned via the stability boundary locus (SBL) approach (Tan et al., 2006) for LFC of time-delayed power systems. The approach in the study by Sharma et al. (2019) deals with a power system having a communication delay and a fixed parametric uncertainty is considered for design. However, the approach is based on classical feedback technique wherein the control action takes place after the disturbance has led to the deviation in system response away from the set point. An event-triggered technique is formulated by Wen et al. (2016) to update the proportional integral controller parameters and ensure stabilisation of multi-area power systems having communication delay. A decentralised state feedback-based control rule is derived by Ojaghi and Rahmani (2017) by solving semi-definite programming under linear matrix inequalities. A super twisting sliding mode control technique using a disturbance observer is employed to design a prediction-based chattering free controller in the study by Dev et al. (2021) for compensation of time delay effects in interconnected power systems. The effect of phase margin and gain margin on the stability margin of the power system with communication delay is investigated in the study by Sonmez and Ayasun (2019). Finally, the tuning of the PI controller parameters is undertaken by delineating the boundaries of the stability region based on the user-defined phase margin and gain margin for LFC of two-area power systems with communication delays in the study by Sonmez and Ayasun (2018).

Most of the aforementioned existing techniques are either based on ubiquitous PID control strategy or incorporate a complex control mechanism. PID control is based on an error feedback control strategy, wherein the control action acts on error signal subsequent to the occurrence of disturbance. To move beyond the PID control approach that is widely dominant in the industrial environment and offer a simple and better alternative, a new technique, known as the active disturbance rejection control (ADRC) was conceptualised (Han, 2002, 2009). It incorporates the principle of total disturbance rejection and involves the design of a nonlinear extended state observer (NESO) to obtain concurrent estimation of states and the generalised disturbance based on output and the input signals of process. Subsequently, a nonlinear control law is formulated for the compensation of the error and attenuation of generalised disturbance acting on the plant. The ADRC approach enables to undertake the controller design independently of the mathematical model of system and has been established to be a powerful mechanism in dealing with the real-world issues pertaining to disturbances, uncertainties, nonlinearities and so on. (Guo and Zhao, 2016; Sira-Ramírez et al., 2017).

However, the original nonlinear ADRC was ascertained to be complex as it involved nonlinear functions and entailed the tuning of a large number of parameters. To alleviate the complexities associated with the implementation of original ADRC structure, the ESO and the control law were linearised, leading to the evolution of linearised ADRC (LADRC) technique by Gao (2006; also see the study by Huang et al., 2019). The LADRC technique leads to a simplification in the controller design procedure as the complex tuning procedure in ADRC technique is reduced to regulation of only two tuning parameters, namely the controller bandwidth and the observer bandwidth, which are also closely related to performance of the closed-loop plant. Furthermore, the LADRC controller design can be undertaken via the use of only the relative order and high frequency gain of the plant. Using a minimal plant information, the plant model is reduced to a canonical integral form and estimates of the states and the generalised disturbance are obtained in real time and used in the control law to achieve disturbance attenuation (Tan and Fu, 2016). But, the assumption of a simple integral form of the plant model comes at a cost and it was observed that the LADRC technique exhibits performance deterioration for the case of time-delayed systems and non-minimum phase systems (Zhao and Gao, 2010). To improve the system performance further, extra plant information can be incorporated in LADRC approach that leads to development of generalised ADRC (GADRC) scheme (Fu and Tan, 2016; Jain and Hote, 2022; Zhou and Tan, 2015). The GADRC technique is applied for the LFC of power systems without communication delay in the study by Jain and Hote (2020). The application of GADRC approach to time-delayed processes poses fundamental challenges as the presence of time delay leads to the loss of synchronisation between the inputs of the ESO. Therefore, the ADRC techniques have been modified for time-delayed systems, leading to the formulation of several techniques such as approximation of time delay by a transfer function, Smith predictor–based ADRC (SP-ADRC; Zhang et al., 2020; Zheng and Gao, 2014), predictive observer–based ADRC (PO-ADRC; Xue et al., 2016) and delay-designed ADRC (DD-ADRC; Chen et al., 2018; Fu, 2018; Fu and Tan, 2020; Jain and Hote, 2021b; Tan and Fu, 2015; Zhao and Gao, 2014). The DD-ADRC-based techniques involves the incorporation of delay in the path of control input before it feeds into the observer, while the SP-ADRC-based techniques are based on incorporation of an auxiliary structure named as Smith predictor to deal with the communication delay.

In this paper, the Smith predictor–based GADRC (SP-GADRC) technique is formulated for the LFC of power systems comprising of single area as well as multiple interconnected control areas. The SP-GADRC approach is implemented on several power system models such as single-area power systems with non-reheated turbine, hydro turbine as well as multi-area interconnected systems such as benchmark model of a 10 machine, 39 bus New England test system. As compared to the previous works, this work formulates the SP-GADRC technique for the LFC of power systems with communication delay along with the derivation of formulae for computation of controller and observer gains. The major contributions of this work are summarised as follows:

The SP-GADRC technique is formulated for the LFC of single- and multi-area power systems with communication delay.

The generalised formulae for tuning the controller and observer gains of various kinds of power system are derived.

A comprehensive performance and robustness analysis is undertaken in the presence of varying communication delay, parametric uncertainty, nonlinearities, model–plant mismatch and delay mismatch for both single- and multi-area power systems.

To exemplify the efficacy of SP-GADRC scheme, a comparison of frequency deviation response is performed with diverse existing techniques such as PID controller tuned via SBL, fractional order PID controller and various variants of ADRC approach such as GADRC, LADRC, LADRC with Taylor approximation of delay and DD-LADRC.

The remaining paper is divided into several sections as follows. The mathematical models of various kinds of power systems are devised. Subsequently, the the controller tuning procedure via SP-GADRC technique is elucidated for various single- and multi-area power system models. The simulation studies to ascertain and analyse the efficacy and robustness of the SP-GADRC technique in contrast with existing techniques are then conducted. Finally, the conclusions of the paper are given.

Power system modelling

The electric power systems are large-scale systems comprising of innumerable generators and loads, and are highly complicated and inherently nonlinear. But, for the task of LFC, they are exposed to small load perturbations, and can be adequately represented by a linear model about the operating point. The block diagrams of a single-area power system and multi-area power system are shown in Figures 1 and 2, respectively. The ith area of a power system in open loop is mathematically modelled via the following transfer function (Saxena and Hote, 2018)

P_{pi} (s) = β_{i} \frac{G_{gi} (s) G_{ti} (s) G_{pi} (s)}{1 + \frac{G_{gi} (s) G_{ti} (s) G_{pi} (s)}{R_{i}}} e^{- ρ_{i} s}

(1)

where, $G_{pi} (s)$ , $G_{ti} (s)$ , $G_{gi} (s)$ and $κ_{i} (s)$ represent the transfer function of the load and machine, turbine, governor and controller in ith area, respectively. $β_{i}$ denotes frequency bias term, $ρ_{i}$ indicates the total communication delay and $R_{i}$ symbolises the droop characteristics in the ith control area of power system. The transfer function in 1 gives the output as frequency deviations in the power system, which have a reference input of zero. Furthermore, $Δ P_{i}$ signifies tie-line power deviations in ith control area, $Δ P_{di}$ is the load disturbance acting on the ith area and $T_{ij}$ designates the synchronising coefficients between the areas $i$ and $j$ . The specific models of the different power system components along with their respective controller design formulae are enlisted in subsequent sections. The cardinal objective of LFC in a single-area power system is maintenance of frequency at nominal value, while in a system comprising of multiple interconnected control areas, both tie-line power variations and fluctuations in frequency should be mitigated in the presence of multifarious uncertainties such as load disturbance, parametric uncertainty, model–plant mismatch, nonlinearities and varying communication delays.

Figure 1.

Single-area power system with controller.

Figure 2.

Multi-area power system with controller.

SP-GADRC approach for power systems with communication delay

The ADRC technique is an alternative control framework to PID scheme that is characterised by the notion of attenuation of the generalised disturbance using an ESO. Figure 3 illustrates the structure of SP-GADRC control strategy for systems with time delay (Zhang et al., 2020). The SP-GADRC control structure can be categorised into five different parts as follows:

(1) Time-delayed plant $P_{p} (s)$ ;

(2) Mathematical model of the plant $P_{a} (s) e^{- ρ s}$ ;

(3) ESO;

(4) Smith predictor;

(5) State feedback-based control.

Figure 3.

Smith predictor GADRC approach for systems with communication delay.

The ESO undertakes the estimation of system states and the generalised disturbance comprising of external disturbance and the internal uncertainties affecting the system from the input and output signals of a plant. In ADRC, all sorts of disturbances including the external disturbances acting on the system, parametric uncertainties, unmodelled dynamics, nonlinearities, all other ignored and unknown terms and so on are grouped together and represented by a single term, commonly known as generalised disturbance or the total disturbance. Therefore, both endogenous disturbances as well as the exogenous disturbances are treated in a unified framework and the net effect of both of them is considered by a fictitious auxiliary state, which is estimated via an extra dimension in the state observer (Sira-Ramírez et al., 2017). Furthermore, in the absence of Smith predictor, the inputs to the ESO are misaligned due to the presence of time delay in the plant, leading to an inaccurate estimation of the states and loss of synchronisation between the inputs of the observer, which can even cause system instability. The Smith predictor removes the communication delay from the closed loop and provides a delay-less output, thereby ensuring the synchronisation of the inputs that feed into the ESO and also resulting in the improvement of the closed-loop bandwidth (Zheng and Gao, 2013). It also leads to an increase in achievable controller and observer bandwidth. Finally, the estimated state feedback control law utilises the estimate of states as well as the generalised disturbance generated by the ESO and sets up an activation signal that attempts to attenuate the effect of the generalised disturbance and ensure the compensation of the errors between the reference and the output signal. In the following subsections, the SP-GADRC technique is mathematically formulated and the controller and observer gain formulae are derived for various single- and multi-area power systems.

Single-area power system with non-reheated turbine

Consider a single-area power system with non-reheated turbine having the mathematical models of power system components represented by the following transfer functions (Saxena and Hote, 2018)

G_{g} (s) = \frac{1}{δ_{g} s + 1}, G_{t} (s) = \frac{1}{δ_{t} s + 1}, G_{p} (s) = \frac{1}{M_{d} s + D_{c}}

(2)

where, $δ_{g}$ and $δ_{t}$ represent the time constant of the governor and the turbine, respectively. $M_{d}$ denotes the generator inertia constant and $D_{c}$ signifies the damping coefficient. Therefore, the open-loop transfer function of the power system with droop characteristics can be expressed as

P_{p} (s) = \frac{β e^{- ρ s}}{δ_{g} δ_{t} M_{d} s^{3} + (M_{d} δ_{g} + M_{d} δ_{t} + δ_{g} δ_{t} D_{c}) s^{2} + (D_{c} (δ_{g} + δ_{t}) + M_{d}) s + (D_{c} + \frac{1}{R})}

(3)

The state space model of the system in equation (3) can be expressed as

\begin{matrix} \overset{\cdot}{\tilde{x}} (t) = A_{s} \tilde{x} (t) + B_{s} u (t - ρ) + B_{g} g (t) \\ \tilde{y} (t) = C_{s} \tilde{x} (t) \end{matrix}

(4)

where, $A_{s}, B_{s}, B_{g} and C_{s}$ are the corresponding state space matrices and $g (t)$ signifies the total disturbance or the generalised disturbance acting on the system. Furthermore, $\tilde{x} (t)$ denotes the states of the system, $u (t - ρ)$ is the delayed input and $\tilde{y} (t)$ represents the output. The ADRC technique treats all kinds of endogenous and exogenous disturbances, unknown expressions, ignored terms, nonlinearities and unmodeled variables in an aggregate manner and the overall effect of all such terms is represented by $g (t)$ .

Since the ESO should estimate system states and the total disturbance as well, thus, the state space model in equation (4) is extended via inclusion of $g (t)$ as the additional state. The extended state space model of the plant can be written as

\begin{matrix} \overset{\cdot}{\hat{x}} (t) = A_{os} \hat{x} (t) + B_{os} u (t - ρ) + B_{og} h (t) \\ \hat{y} (t) = C_{os} \hat{x} (t) \end{matrix}

(5)

where, $\hat{x} (t) = [\begin{matrix} \tilde{x} & g \end{matrix}]^{T}$ represent the extended states of the system and $h (t) = \overset{\cdot}{g} (t)$ , since $g (t)$ is assumed to be differentiable. The corresponding extended-state space matrices are computed as

\begin{matrix} A_{os} = [\begin{matrix} A_{s} & [\begin{matrix} 0_{(2 \times 1)} \\ 1 \end{matrix}] \\ 0_{(1 \times 3)} & 0 \end{matrix}], B_{os} = [\begin{matrix} B_{s} \\ 0 \end{matrix}] \\ C_{os} = [\begin{matrix} C_{s} & 0 \end{matrix}], B_{og} = {[\begin{matrix} 0_{(3 \times 1)} & 1 \end{matrix}]}^{T} \end{matrix}

(6)

The estimation of the states of extended-state space model is carried out via ESO. The inputs to the ESO are the control input $u (t)$ and the delay-less output of the Smith predictor $y_{p} (t)$ . The output of Smith predictor in Laplace domain is given by $y_{p} (s) = y (s) + P_{a} (s) u (s) - P_{a} (s) u (s) e^{- ρ s}$ . Therefore, in the absence of mismatch between the plant and its mathematical model, the output of the Smith predictor is equal to the output of the nominal plant without delay, leading to the establishment of synchronisation between the inputs of the ESO (Zhang et al., 2020). Hence, the mathematical model of the extended Luenberger observer in state space form is

\overset{\cdot}{\tilde{z}} (t) = A_{os} \tilde{z} (t) + B_{os} u (t) + L_{os} (y_{p} (t) - C_{os} \tilde{z} (t))

(7)

where, $L_{os}$ represents observer gain; ${\tilde{z}}_{1}, {\tilde{z}}_{2}, {\tilde{z}}_{3} and {\tilde{z}}_{4}$ signify states estimated by the observer; and the observer gain is to be tuned such that $(A_{os} - L_{os} C_{os})$ is asymptotically stable. The control law for the plant is propounded as

u (t) = \frac{β \hat{r} - (({\tilde{k}}_{1}^{c} {\tilde{z}}_{1} + {\tilde{k}}_{2}^{c} {\tilde{z}}_{2} + {\tilde{k}}_{3}^{c} {\tilde{z}}_{3} + {\tilde{z}}_{4}) (δ_{g} δ_{t} M_{d}))}{β}

(8)

where, $\hat{r}$ represents the extended reference such that $\hat{r} (t) = F_{re} r (t)$ , such that $r (t)$ is the desired reference signal. Also, the controller gain matrix is given by ${\tilde{K}}_{os} = (δ_{g} δ_{t} M_{d} / β) [\begin{matrix} k_{1}^{c} & k_{2}^{c} & k_{3}^{c} & 1 \end{matrix}]$ and the tuning of $F_{re}$ is undertaken to make the steady-state error as zero.

The bandwidth parameterisation technique is adopted to tune the gains of the observer and controller in equations (7) and (8), respectively, such that all the observer and controller poles are allocated at $- ω_{os}$ and $- ω_{cs}$ , which also represent the corresponding bandwidths, therefore, the poles are chosen as real numbers. Choice of higher value of bandwidth gives rise to quick response times, better tracking of the reference signal, whereas maximum achievable bandwidth is limited by existence of uncertainty and existence of sensor noise (Gao, 2003). Hence, the controller gains are computed as

\begin{matrix} {\tilde{k}}_{1}^{c} = ω_{cs}^{3} - \frac{R D_{c} + 1}{R δ_{g} δ_{t} M_{d}} \\ {\tilde{k}}_{2}^{c} = 3 ω_{cs}^{2} - \frac{(D_{c} (δ_{g} + δ_{t}) + M_{d})}{δ_{g} δ_{t} M_{d}} \\ {\tilde{k}}_{3}^{c} = 3 ω_{cs} - \frac{(M_{d} δ_{g} + M_{d} δ_{t} + δ_{g} δ_{t} D_{c})}{δ_{g} δ_{t} M_{d}} \end{matrix}

(9)

Similarly, the observer gains are computed via Ackermann’s formula as

L_{os} = ψ (A_{os}) {[\begin{matrix} C_{os} \\ C_{os} A_{os} \\ C_{os} A_{os}^{2} \\ C_{os} A_{os}^{3} \end{matrix}]}^{- 1} [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}]

(10)

where, $ψ (s) = (s + ω_{os})^{4}$ . Therefore, the observer gains are ascertained to be

\begin{matrix} {\tilde{β}}_{1}^{o} = 4 ω_{os} - (\frac{M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t}}{M_{d} δ_{g} δ_{t}}) \\ {\tilde{β}}_{2}^{o} = (6 ω_{os}^{2} - (\frac{M_{d} + D_{c} (δ_{g} + δ_{t})}{M_{d} δ_{g} δ_{t}}) + (\frac{{(M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t})}^{2}}{M_{d}^{2} δ_{g}^{2} δ_{t}^{2}}) \\ - (\frac{4 ω_{os} (M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t})}{M_{d} δ_{g} δ_{t}})) \\ {\tilde{β}}_{3}^{o} = (4 ω_{os}^{3} - 4 ω_{os} (\frac{(M_{d} + D_{c} (δ_{g} + δ_{t}))}{(M_{d} δ_{g} δ_{t})} - \frac{{(M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t})}^{2}}{(M_{d}^{2} δ_{g}^{2} δ_{t}^{2})}) \\ - (\frac{6 ω_{os}^{2} (M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t})}{M_{d} δ_{g} δ_{t}}) - (\frac{D_{c} R + 1}{M_{d} R δ_{g} δ_{t}}) \\ + \frac{(\frac{M_{d} + D_{c} (δ_{g} + δ_{t})}{M_{d} δ_{g} δ_{t}} - \frac{{(M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t})}^{2}}{M_{d}^{2} δ_{g}^{2} δ_{t}^{2}}) (M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t})}{M_{d} δ_{g} δ_{t}} \\ + (\frac{(M_{d} + D_{c} (δ_{g} + δ_{t})) (M_{d} δ_{g} + M_{d} δ_{t} + D_{c} δ_{g} δ_{t})}{M_{d}^{2} δ_{g}^{2} δ_{t}^{2}})) \\ {\tilde{β}}_{4}^{o} = ω_{os}^{4} \end{matrix}

(11)

The computation of $F_{re}$ is done to ensure tracking without offset in steady state. Substitution of the control law expression from equation (8), the output of the system to step reference input in the steady state is calculated as

\begin{matrix} lim_{t \to \infty} y (t) = lim_{s \to 0} (C_{s} {(sI - A_{s} + B_{s} {\tilde{K}}_{s} e^{- ρ s})}^{- 1} B_{s}) e^{- ρ s} F_{re} \\ = (C_{s} {(- A_{s} + B_{s} {\tilde{K}}_{s})}^{- 1} B_{s}) F_{re} \end{matrix}

(12)

where, ${\tilde{K}}_{s} = (δ_{g} δ_{t} M_{d} / β) [\begin{matrix} {\tilde{k}}_{1}^{c} & {\tilde{k}}_{2}^{c} & {\tilde{k}}_{3}^{c} \end{matrix}]$ . Therefore, value of $F_{re}$ for perfect tracking is computed as (Jain and Hote, 2021a)

F_{re} = \frac{1}{C_{s} {(- A_{s} + B_{s} {\tilde{K}}_{s})}^{- 1} B_{s}}

(13)

Substitution of parameters gives the value of $F_{re}$ for a single-area power system as

F_{re} = \frac{M_{d} δ_{g} δ_{t} ω_{cs}^{3}}{β}

(14)

Hence, the observer bandwidth and controller bandwidth are two tuning parameters for the design of SP-GADRC controller.

Single-area power system with hydraulic turbine

A single-area power system with hydraulic turbine exhibits non-minimum phase behaviour as well as involves the communication delay, and thereby is a challenging problem from the perspective of control engineering. The transfer functions of different power system components are (Dong et al., 2012)

\begin{matrix} G_{g} (s) = \frac{1}{δ_{g} s + 1} \\ G_{t} (s) = \frac{(- δ_{w} s + 1) (δ_{r} s + 1)}{(0.5 δ_{w} s + 1) (\frac{δ_{r} R_{t}}{R} s + 1)} \\ G_{p} (s) = \frac{1}{M_{d} s + D_{c}} \end{matrix}

(15)

where, $δ_{w}$ , $δ_{r}$ and $R_{t}$ denote the hydro turbine time constant, reset time of hydro unit and the temporary droop coefficient, respectively. The overall open-loop non-minimum phase transfer function of power system consisting of hydro turbine showing is represented as

P_{p} (s) = \frac{e_{2} s^{2} + e_{1} s + e_{0}}{s^{4} + d_{3} s^{3} + d_{2} s^{2} + d_{1} s + d_{0}}

(16)

where

\begin{matrix} d_{3} = (\frac{0.5 D_{c} R_{t} δ_{g} δ_{r} δ_{w} + 0.5 M_{d} R_{t} δ_{r} δ_{w} + 0.5 M_{d} R δ_{g} δ_{w} + M_{d} R_{t} δ_{g} δ_{r}}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}}) \\ d_{2} = (\frac{\begin{matrix} - δ_{r} δ_{w} + 0.5 D_{c} R_{t} δ_{r} δ_{w} + D_{c} R_{t} δ_{g} δ_{r} + 0.5 D_{c} R δ_{g} δ_{w} \\ + M_{d} R_{t} δ_{r} + 0.5 M_{d} R δ_{w} + M_{d} R δ_{g} \end{matrix}}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}}) \\ d_{1} = (\frac{δ_{r} - δ_{w} + D_{c} R_{t} δ_{r} + 0.5 D_{c} R δ_{w} + D_{c} R δ_{g} + M_{d} R}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}}) \\ d_{0} = \frac{D_{c} R + 1}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}} \\ e_{2} = - \frac{β R δ_{w} δ_{r}}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}} \\ e_{1} = \frac{β R (δ_{r} - δ_{w})}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}} \\ e_{0} = \frac{β R}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}} \end{matrix}

(17)

The system in equations (16) and (17) can be represented in state space form as

\begin{matrix} \overset{\cdot}{\tilde{x}} (t) = A_{s} \tilde{x} (t) + B_{s} u (t - ρ) + B_{g} g (t) \\ \tilde{y} (t) = C_{s} \tilde{x} (t) \end{matrix}

(18)

The control law for the plant is given as

u (t) = \hat{r} - \frac{0.5 (({\tilde{k}}_{1}^{c} {\tilde{z}}_{1} + {\tilde{k}}_{2}^{c} {\tilde{z}}_{2} + {\tilde{k}}_{3}^{c} {\tilde{z}}_{3} + {\tilde{k}}_{4}^{c} {\tilde{z}}_{4} + {\tilde{z}}_{5}) (M_{d} R_{t} δ_{g} δ_{r} δ_{t} δ_{w}))}{β R}

(19)

where $\hat{r} (t) = F_{re} r (t)$ and ${\tilde{K}}_{os} = (1 / e_{0}) [\begin{matrix} {\tilde{k}}_{1}^{c} & {\tilde{k}}_{2}^{c} & {\tilde{k}}_{3}^{c} & {\tilde{k}}_{4}^{c} & 1 \end{matrix}]$ . To ensure perfect tracking, the value of $F_{re}$ is computed as

F_{re} = \frac{M_{d} R_{t} δ_{g} δ_{r} δ_{w} ω_{cs}^{4}}{2 R β}

(20)

Using the bandwidth parameterisation technique for placing all the controller poles at $- ω_{cs}$ , controller gains are calculated as

\begin{matrix} {\tilde{k}}_{c}^{1} = ω_{cs}^{4} - (\frac{D_{c} R + 1}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}}) \\ {\tilde{k}}_{c}^{2} = 4 ω_{cs}^{3} - (\frac{δ_{r} - δ_{w} + D_{c} R_{t} δ_{r} + 0.5 D_{c} R δ_{w} + D_{c} R δ_{g} + M_{d} R}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}}) \\ {\tilde{k}}_{c}^{3} = 6 ω_{cs}^{2} - (\frac{\begin{matrix} - δ_{r} δ_{w} + 0.5 D_{c} R_{t} δ_{r} δ_{w} + D_{c} R_{t} δ_{g} δ_{r} + 0.5 D_{c} R δ_{g} δ_{w} \\ + M_{d} R_{t} δ_{r} + 0.5 M_{d} R δ_{w} + M_{d} R δ_{g} \end{matrix}}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}}) \\ {\tilde{k}}_{c}^{4} = 4 ω_{cs} - (\frac{0.5 D_{c} R_{t} δ_{g} δ_{r} δ_{w} + 0.5 M_{d} R_{t} δ_{r} δ_{w} + 0.5 M_{d} R δ_{g} δ_{w} + M_{d} R_{t} δ_{g} δ_{r}}{0.5 M_{d} R_{t} δ_{g} δ_{r} δ_{w}}) \end{matrix}

(21)

Using Ackermann’s formula for placement of all the observer poles at $- ω_{os}$ , the observer gains are computed as

L_{os} = ψ (A_{os}) {[\begin{matrix} C_{os} \\ C_{os} A_{os} \\ C_{os} A_{os}^{2} \\ C_{os} A_{os}^{3} \\ C_{os} A_{os}^{4} \end{matrix}]}^{- 1} [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix}]

(22)

where, $ψ (s) = (s + ω_{os})^{5}$ .

Three-area New England test system

Finally, let us consider a multi-area New England test system (Bevrani, 2009; Canizares et al., 2017). The New England system constitutes 34 transmission lines, 19 loads, 10 generators and 12 transformers. It is a practical power system model, which has topology equivalent to IEEE 39 bus benchmark system (Saxena and Hote, 2017; Sharma et al., 2019). For undertaking the LFC task, the complete system is demarcated into three different control areas as illustrated in Figure 4. The generalised mathematical model of the control areas in New England system, with the incorporation of communication delay is given as

P_{pi} (s) = \frac{β_{i} e^{- ρ_{i} s}}{(2 μ_{i} δ_{ti} δ_{gi}) s^{3} + (D_{ci} δ_{ti} δ_{gi} + 2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti}) s^{2} + (D_{ci} δ_{ti} + D_{ci} δ_{gi} + 2 μ_{i}) s + (μ_{i} + 1 / R_{i})}

(23)

where, $δ_{gi}$ and $δ_{ti}$ represent the time constant of the governor and the turbine, in the ith area, respectively. Furthermore, $μ_{i}$ denotes the inertia constant, $D_{ci}$ signifies the damping coefficient and $R_{i}$ represents the droop characteristics for the ith control area. Using an identical approach as illustrated in the section ‘Single-area power system with non-reheated turbine’, the controller gains for each control area are derived as

\begin{matrix} {\tilde{k}}_{1}^{c} = ω_{cs}^{3} - \frac{R_{i} D_{ci} + 1}{2 R_{i} δ_{gi} δ_{ti} μ_{i}} \\ {\tilde{k}}_{2}^{c} = 3 ω_{cs}^{2} - \frac{(D_{ci} (δ_{gi} + δ_{ti}) + 2 μ_{i})}{δ_{gi} δ_{ti} 2 μ_{i}} \\ {\tilde{k}}_{3}^{c} = 3 ω_{cs} - \frac{(2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + δ_{gi} δ_{ti} D_{ci})}{2 δ_{gi} δ_{ti} μ_{i}} \end{matrix}

(24)

Figure 4.

New England system.

In a similar manner, the gains of the observer are tuned as

\begin{matrix} {\tilde{β}}_{1}^{o} = 4 ω_{os} - (\frac{2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti}}{2 μ_{i} δ_{gi} δ_{ti}}) \\ {\tilde{β}}_{2}^{o} = (6 ω_{os}^{2} - (\frac{2 μ_{i} + D_{ci} (δ_{gi} + δ_{ti})}{2 μ_{i} δ_{gi} δ_{ti}}) + (\frac{{(2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti})}^{2}}{2 μ_{i}^{2} δ_{gi}^{2} δ_{ti}^{2}}) \\ - (\frac{4 ω_{os} (2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti})}{2 μ_{i} δ_{gi} δ_{ti}})) \\ {\tilde{β}}_{3}^{o} = (4 ω_{os}^{3} - 4 ω_{os} (\frac{(2 μ_{i} + D_{ci} (δ_{gi} + δ_{ti}))}{(2 μ_{i} δ_{gi} δ_{ti})} - \frac{{(2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti})}^{2}}{(2 μ_{i}^{2} δ_{gi}^{2} δ_{ti}^{2})}) \\ - (\frac{6 ω_{os}^{2} (2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti})}{2 μ_{i} δ_{gi} δ_{ti}}) - (\frac{D_{ci} R + 1}{2 μ_{i} R δ_{gi} δ_{ti}}) \\ + \frac{(\frac{2 μ_{i} + D_{ci} (δ_{gi} + δ_{ti})}{2 μ_{i} δ_{gi} δ_{ti}} - \frac{{(2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti})}^{2}}{2 μ_{i}^{2} δ_{gi}^{2} δ_{ti}^{2}}) (2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti})}{2 μ_{i} δ_{gi} δ_{ti}} \\ + (\frac{(2 μ_{i} + D_{ci} (δ_{gi} + δ_{ti})) (2 μ_{i} δ_{gi} + 2 μ_{i} δ_{ti} + D_{ci} δ_{gi} δ_{ti})}{2 μ_{i}^{2} δ_{gi}^{2} δ_{t}^{2}})) \\ {\tilde{β}}_{4}^{o} = ω_{os}^{4} \end{matrix}

(25)

Next, the stability analysis of the SP-GADRC technique is undertaken.

Stability analysis

The stability of the SP-GADRC technique can be investigated by transforming it into an equivalent structure of a 2 degrees of freedom internal model control (TDF-IMC) approach (Zhang et al., 2020).

The internal stability of TDF-IMC is guaranteed, if and only if

Λ = \frac{1}{ζ (s)} [\begin{matrix} Q_{imc} (s) & Q_{imc} (s) Q_{d} (s) P_{p} (s) \\ Q_{imc} (s) P_{m} (s) & P_{p} (s) (- P_{m} (s) Q_{imc} (s) Q_{d} (s) + 1) \end{matrix}]

(26)

is stable, where $ζ (s) = (Q_{imc} (s) Q_{d} (s) (P_{p} (s) - P_{m} (s)) + 1)$ . Here, $P_{p} (s)$ , $P_{m} (s)$ , $Q_{imc} (s)$ and $Q_{d} (s)$ denote the transfer function of the plant, model of the plant, set point tracking controller and disturbance rejection controller, respectively. The transformation formulae that related the SP-GADRC and TDF-IMC structure are given as

\begin{matrix} P_{m} (s) = C_{s} {(sI - A_{s})}^{- 1} B_{s} e^{- ρ s} \\ Q_{d} (s) = \frac{({\tilde{K}}_{os} {(sI - A_{os} + L_{os} C_{os})}^{- 1} L_{os})}{F_{re}} \\ Q_{imc} (s) = (1 - K_{os} {(sI - A_{os} + B_{os} K_{os})}^{- 1} B_{os}) F_{re} \end{matrix}

(27)

Using small gain theorem, the sufficient condition for stability of $Λ$ is

| Q_{imc} (j ω) Q_{d} (j ω) | < | {(P_{p} (j ω) - P_{m} (j ω))}^{- 1} |, \forall ω > 0

(28)

Simulation results

In this section, design of SP-GADRC controller is undertaken for LFC of various single-area power systems as well as multiple-area power systems having communication delay, including the realistic New England system. A comprehensive comparative and robustness analysis is carried out to validate the efficacy of SP-GADRC technique.

Example 1: The values of various parameters of the single-area system with nonreheated turbine are considered as (Saxena and Hote, 2018)

δ_{g} = 0.1, δ_{t} = 0.3, M_{d} = 10, D_{c} = 1, R = 0.05, β = 21

(29)

Substitution of the parameter values of the power system constituents yields the state space matrices of the system as follows

\begin{matrix} A_{s} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ - 70 & - 34.667 & - 13.433 \end{matrix}] \\ B_{s} = [\begin{matrix} 0 \\ 0 \\ 70 \end{matrix}] \\ C_{s} = [1 0 0] \end{matrix}

(30)

Subsequently, considering the bandwidth of the controller and observer as $ω_{cs} = 1.8$ and $ω_{os} = 9.5$ , respectively, and using equations (9) and (11), the observer and controller gain matrices are obtained as

\begin{matrix} {\tilde{K}}_{os} = [\begin{matrix} - 0.9167 & - 0.3563 & - 0.1147 & 0.0143 \end{matrix}] \\ L_{os} = {[\begin{matrix} 24.57 & 176.8649 & 132.6082 & 8145.0625 \end{matrix}]}^{T} \end{matrix}

(31)

To appraise the capability of the SP-GADRC controller, a step load variation $Δ P_{d} = 0.01$ p.u. MW is incorporated into system at $2$ seconds. Furthermore, various values of delay, that is, $ρ = 1, 2.28 and 5 seconds$ are taken into consideration to evaluate the efficacy of the SP-GADRC approach. The corresponding disturbance rejection plots are illustrated in Figure 5(a)–(c), respectively. The efficacy of the SP-GADRC technique is analysed by an extensive comparative analysis with various techniques in literature such as GADRC ( $ω_{cs} = 1.8$ and $ω_{os} = 9.5$ ; Fu and Tan, 2016; Zhou and Tan, 2015), LADRC ( $ω_{cs} = 3.5$ and $ω_{os} = 10$ ; Gao, 2006; Tan and Fu, 2016), Fractional order PI (FOPI; Çelik et al., 2017), PI controller (Saxena and Hote, 2018), SBL (Sönmez and Ayasun, 2016), DD-LADRC ( $ω_{cs} = 3$ and $ω_{os} = 100$ ; Tan and Fu, 2015; Zhao and Gao, 2014), DD-GADRC (Jain and Hote, 2021a) and LADRC with Taylor approximation of delay ( $ω_{cs} = 0.05$ and $ω_{os} = 1$ ; Tan and Fu, 2016). It can be ascertained from Figure 5(a) and (b) that the SP-GADRC technique exhibits a minimal oscillations and quick settling time as compared to multifarious existing techniques in literature. Furthermore, when the communication delay is increased to a large value of 5 seconds, various techniques such as GADRC, LADRC, FOPI and SBL exhibit unstable response and the frequency fluctuations increase without limit. However, it can be seen from Figure 5(c) that SP-GADRC, PI, DD-LADRC and LADRC-Taylor exhibit stable response and the SP-GADRC technique depicts quick settling time as compared to other techniques in literature. Furthermore, the performance of the SP-GADRC technique with the existing techniques is also undertaken at the communication delay of $ρ = 2.28$ seconds via the computation of integral time absolute error (ITAE) and the time variance for input and shown in Table 1. It can be observed that SP-GADRC technique shows the minimum value of ITAE and low value of total variance for input, thereby demonstrating the efficacy of SP-GADRC technique.

Figure 5.

Frequency deviation response in Example 1 for various values of delay: (a) $ρ$ = 1 second, (b) $ρ = 2.28 seconds$ , and (c) $ρ = 5 seconds$ .

Table 1.

Performance indices comparison for Example 1.

Method	ITAE	Total variance for input
SP-GADRC	0.008868	4.7974e−04
GADRC	0.2235	0.0046
LADRC	0.01442	0.0038
FOPI	0.1243	0.1018
PI	0.04813	0.0106
SBL	0.1815	0.1292
DD-LADRC	0.02133	0.0920
LADRC-Taylor	0.8568	5.3584e−05
DD-GADRC	0.02145	0.0470

ITAE: integral time absolute error; SP-GADRC: generalised active disturbance rejection control technique based on Smith predictor; GADRC: generalised active disturbance rejection control; LADRC: linearised active disturbance rejection control; FOPI: fractional order; PI; PI: proportional–integral; SBL: stability boundary locus;DD: delay designed.

Next, the robustness of SP-GADRC controller is investigated in the presence of various uncertainties in the system. It is vital that the controller shows acceptable performance even when there exists a perturbation in the nominal system parameters. The occurrence of parametric uncertainty also leads to a mismatch between the actual plant and the mathematical model of the plant. To examine the robustness, the parameters of the plant are perturbed by $\pm 20 %$ in the following manner

\begin{matrix} γ_{g} \in [0.08, 0.12], γ_{t} \in [0.24, 0.36], M_{d} \in [8, 12], \\ D_{c} \in [0.8, 1.2], R \in [0.04, 0.06] \end{matrix}

(32)

Figure 6(a) compares the nominal response versus the response obtained when the parameters of the power system are perturbed by $\pm 20 %$ . It is evident that the perturbed plant is stable and exhibits a swift settling time. Therefore, it can be deduced that the presence of parametric uncertainty does not have a significant effect on the system response and the overall system is robust.

Figure 6.

Robustness analysis for Example 1: (a) $\pm 20 %$ parametric perturbations, (b) $\pm 50 %$ delay mismatch, and (c) effect of GDB and GRC.

Furthermore, the mismatch in communication delay between the delayed plant $P_{p} (s)$ and the mathematical model $P_{a} (s) e^{- ρ s}$ leads to a loss in synchronisation between the inputs of the ESO and can significantly impair the system performance. The capability of SP-GADRC technique to handle the mismatch in communication delay is ascertained by keeping the value of delay in the Smith predictor at $ρ = 2.28 seconds$ , however, the delay time (DT) in the plant $P_{p} (s)$ is perturbed by $\pm 50 %$ , that is, the value of communication delay in the plant is considered as 1.14 and 3.42 seconds. The corresponding response is shown in Figure 6(b). It can be seen that even when there occurs a mismatch in the communication delay, the system is stable and exhibits an acceptable transient behaviour as frequency fluctuations settle to zero in the steady state.

Finally, the robustness of the SP-GADRC approach ( $ω_{cs} = 0.756$ and $ω_{os} = 8.1$ ) is investigated in the presence of nonlinearities such as generation rate constraint (GRC = 10% p.u/minute) as well as GDB = 0.05% (Saxena and Hote, 2017). The combined effect of a large communication delay of 2.28 $s$ as well as the presence of nonlinearities can be highly dangerous for stability of a power system. Figure 6(c) portrays the corresponding frequency excursions. It can be noticed that even in the presence of the combined effect of nonlinearities and communication delay, the SP-GADRC controlled single-area power system still exhibits stable response. Furthermore, the control input required for the implementation of SP-GADRC technique when the system has a communication delay of 2.28 seconds is shown in Figure 7. It can be ascertained that the requirements of the control input are minimal for the SP-GADRC technique. Hence, it can be inferred that the SP-GADRC is an effective approach to control a power system having an associated communication delay.

Figure 7.

Control input in Example 1.

Example 2: Next, let us analyse the efficacy of the SP-GADRC approach on single-area power system with hydraulic turbine. The values of different system parameters are stated as

\begin{matrix} δ_{w} = 1, δ_{r} = 5, δ_{g} = 0.2, M_{d} = 6, D_{c} = 1, R_{t} = 0.38, \\ R = 0.05, β = 21 \end{matrix}

(33)

The system represents fourth order, therefore, fifth-order ESO is considered in the SP-GADRC approach. Using equations (21) and (22), and the controller bandwidth and observer bandwidth as $ω_{cs} = 0.2$ and $ω_{os} = 5$ , respectively, the gain matrices of the controller and observer are computed as

\begin{matrix} {\tilde{K}}_{os} = [\begin{matrix} - 0.99 & - 5.90 & - 7.31 & - 6.9410 & 1.09 \end{matrix}] \\ {\tilde{L}}_{os} = {[\begin{matrix} 42499 & - 8427.8 & 1753.9 & - 305.4 & 3125 \end{matrix}]}^{T} \end{matrix}

(34)

The performance of the SP-GADRC controller is demonstrated via investigation of frequency deviation response, when a step load disturbance $Δ P_{di} = 0.01$ p.u. MW is incorporated in power system at the time $t = 2 seconds$ . The simulations are undertaken for varying values of communication delay, that is, $ρ = 1, 2.28 and 5 seconds$ and corresponding frequency deviation responses are revealed in Figure 8(a)–(c), respectively. The efficacy of SP-GADRC technique is assessed by comparison of closed-loop response with numerous techniques such as GADRC ( $ω_{cs} = 0.2$ , $ω_{os} = 9$ ; Fu and Tan, 2016; Zhou and Tan, 2015), LADRC ( $ω_{cs} = 0.2$ and $ω_{os} = 0.01$ ; Gao, 2006; Tan and Fu, 2016), DD-LADRC ( $ω_{cs} = 0.2$ and $ω_{os} = 0.01$ ; Tan and Fu, 2015; Zhao and Gao, 2014) and SBL ( $k_{p} = 0.3, k_{i} = 0.2$ ; Sönmez et al., 2016). It can be discerned that the response obtained by SP-GADRC technique has minimal oscillations and a quick settling time as compared to the other techniques. When the delay is increased to $2.28 seconds$ , the system controlled via DD-LADRC technique becomes unstable and the other techiques, namely GADRC and SBL exhibit large oscillations. However, the SP-GADRC technique exhibits stable response and the frequency fluctuations settle down to zero quickly, even when the delay is increased to a large value of 5 seconds. Furthermore, the performance of the SP-GADRC technique with the existing techniques is also undertaken at the communication delay of $ρ = 2.28$ seconds via the computation of ITAE and the time variance for input and shown in Table 2. It can be observed that SP-GADRC technique shows the minimum value of ITAE and low value of total variance for input, thereby demonstrating the efficacy of SP-GADRC technique.

Figure 8.

Frequency deviation response in Example 2 for various values of delay: (a) $ρ$ = 1 second, (b) $ρ = 2.28 seconds$ and (c) $ρ = 5 seconds$ .

Table 2.

Performance indices comparison for Example 2.

Method	ITAE	Total variance for input
SP-GADRC	0.5081	0.0042
GADRC	0.9839	0.0322
LADRC	2.495	0.0063
SBL	0.6066	0.0097

Furthermorte, the robustness of the SP-GADRC controller for LFC of power system is examined in the presence of various uncertainties such as parametric uncertainty, model–plant mismatch, delay mismatch and nonlinearities. The $\pm 20 %$ parametric uncertainty is introduced into the plant in the following manner

\begin{matrix} δ_{w} \in [0.8, 1.2], δ_{r} \in [4, 6], δ_{g} \in [0.16, 0.24], M_{d} \in [4.8, 7.2], \\ D_{c} \in [0.8, 1.2], R_{t} \in [0.304, 0.456], R \in [0.04, 0.06] \end{matrix}

(35)

Figure 9(a) depicts the system response, when parametric perturbations of $\pm 20 %$ are introduced in the plant $P_{p} (s)$ , while keeping the mathematical model identical to the nominal plant, since the variation occurs only in the plant, which is exposed to environment, but not in the corresponding mathematical model of the plant $P_{a} (s)$ . It can be observed from Figure 9(a) that SP-GADRC method can handle the parametric uncertainty as well as the model–plant mismatch efficiently and the adjustment in controller parameters is not required, when there exists parametric perturbations in the parameters of system.

Figure 9.

Robustness analysis for Example 2: (a) $\pm 20 %$ parametric perturbations, (b) $\pm 50 %$ delay mismatch, and (c) effect of GDB and GRC.

Furthermore, the effect of communication delay mismatch between the delayed plant $P_{p} (s)$ and the mathematical model of the plant $P_{a} (s) e^{- ρ s}$ is studied by introducing $\pm 50 %$ variations in the delay time of the plant $P_{p} (s)$ . Figure 9(b) illustrates the corresponding frequency fluctuations response, when the delay time in the plant is considered as $1.14$ and $3.42$ seconds, while the delay in mathematical model has a value of $2.28 seconds$ . It can be deduced that the overall system is robust to the mismatch of the commmunication delay occurring in the system as the response obtained by varying delay time is stable and does not show significant variation from the nominal response.

Finally, the robustness of the SP-GADRC controller is examined in the presence of nonlinearities, namely GDB = 0.05% and GRC = 2.7 and –3.6 p.u/minute (Sahu et al., 2016) along with the existence of communication delay of 2. 28 seconds. It can be noted from Figure 9(c) that the existence of nonlinearities and a communication delay in tandem does not lead to a deterioration in the frequency deviation response. Furthermore, the control input requirements of an SP-GADRC-controlled nominal system are portrayed in Figure 10 and demonstrate that a minimal control input is involved in its implementation. Hence, SP-GADRC technique is a viable alternative to mitigate frequency excursions in a single-area power system with hydro turbine.

Figure 10.

Control input in Example 2.

Example 3: Finally, let us consider a multi-area New England test system (Bevrani, 2009; Canizares et al., 2017).

The values of system parameters are stated as follows (Saxena and Hote, 2017)

\begin{matrix} δ_{g 1} = 0.15, δ_{g 2} = 0.1, δ_{g 3} = 0.1, δ_{t 1} = 0.2, δ_{t 2} = 0.2, \\ δ_{t 3} = 0.2, μ_{1} = 35.8, μ_{2} = 26.4, μ_{3} = 34.5, D_{c 1} = 10, \\ D_{c 2} = 8, D_{c 3} = 14, R = 0.05, β_{1} = 30, β_{2} = 28, β_{3} = 35 \end{matrix}

(36)

Using equations (24) and (25), and the controller bandwidth and observer bandwidth as $ω_{cs} = 8$ and $ω_{os} = 1.5$ , respectively, the controller and observer gain matrices for the control areas are obtained as

\begin{matrix} {\tilde{K}}_{os 1} = [\begin{matrix} 53.988 & 18.084 & 1.488 & 0.107 \end{matrix}] \\ {\tilde{L}}_{os 1} = {[\begin{matrix} - 4.139 & 31.856 & - 221.050 & 5.063 \end{matrix}]}^{T} \end{matrix}

(37)

\begin{matrix} {\tilde{K}}_{os 2} = [\begin{matrix} 27.965 & 8.862 & 0.595 & 0.057 \end{matrix}] \\ {\tilde{L}}_{os 2} = {[\begin{matrix} - 7.485 & 79.078 & - 805.908 & 5.063 \end{matrix}]}^{T} \end{matrix}

(38)

\begin{matrix} {\tilde{K}}_{os 3} = [\begin{matrix} 30.172 & 9.495 & 0.637 & 0.061 \end{matrix}] \\ {\tilde{L}}_{os 3} = {[\begin{matrix} - 7.536 & 79.473 & - 807.092 & 5.063 \end{matrix}]}^{T} \end{matrix}

(39)

The performance and robustness of the SP-GADRC technique is demonstrated by taking identical values of $ω_{cs}$ and $ω_{os}$ , but considering widely different values of communication delays in the three control areas. The values of communication delay in the three areas are $ρ_{1} = 7 seconds$ , $ρ_{2} = 5 seconds$ and $ρ_{3} = 1 second$ . Identical values of $ω_{cs}$ and $ω_{os}$ for the three control areas, each having different plant parameters serves as a tool to investigate the performance of the New England system to uncertainties in system parameters. Furthermore, dead bandwidth of 0.036 Hz is also considered to ascertain the efficacy of SP-GADRC scheme. However, the generation rate constraint is not considered in the given example. Also, different load disturbances are assumed in each control area such as $Δ P_{d 1} = 0.038$ p.u. MW in bus 8 at $t = 1 second$ , $Δ P_{d 2} = 0.064$ p.u. MW at $t = 100 seconds$ in bus 16 and $Δ P_{d 3} = 0.038$ p.u. MW in bus 3 at $t = 200 seconds$ . Thus, New England test system, having a similar topology to IEEE 39 bus system, is an ideal example to demonstrate the disturbance rejection ability and the robustness of controller in the presence of parametric uncertainty, varying and large values of communication delay and nonlinearities concurrently.

Figures 11 and 12 illustrate the frequency fluctuations and tie-line power deviations from the nominal values in the three control areas. It can be observed that the deviations in frequency as well as tie-line power converge to zero, even in the presence of widely varying communication delays in the three control areas, parametric uncertainty as well as load disturbances.

Figure 11.

Frequency deviation response for the control areas in Example 3.

Figure 12.

Tie-line power variations for three control areas in Example 3.

Hence, the SP-GADRC technique can be a feasible and practical alternative to existing control algorithms for the LFC of various single- and multiple-area interconected power systems.

Conclusion

In this paper, an SP-GADRC technique, that uses the available information about the system to be controlled is comprehensively investigated and the observer and controller gains are derived for the LFC of single- as well as multi-area interconnected power systems with communication delay. The SP-GADRC technique has only two tuning parameters and is validated on widely different type of single- and multi-area interconnected systems. In each case, it is observed that the SP-GADRC technique is robust to the presence of large and varying communication delay, perturbations in system parameters, load disturbance, mismatch between the model and the plant and nonlinearities in power system. A comprehensive comparison is undertaken with multifarious techniques in literature also demonstrating the effectiveness of SP-GADRC technique. This study considered the fixed parameter controller for various values of communication delay. In future, this study can be extended to formulate time varying controller and observer gains such that they adapt online to changes in communication delay and external disturbances acting on the system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Science and Engineering Research Board (SERB), India (grant no. MTR/2019/001580).

ORCID iD

Shivam Jain

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