## Abstract

Owing to integrating the dense range of distinct electric power sources, high volume of power generation units, abrupt and continuous changes in load demand, and rising utilization of power electronics, the electric power system (EPS) is striving for high-performance control schemes to counterwork the concerns depicted above. Additionally, it is highly creditable to have the controller structure as simple as possible from a viewpoint of practical implementation. Thus, this paper describes a virgin application of fractional order proportional integral–fractional order proportional derivative (FOPI–FOPD) cascade controller for load frequency control (LFC) of electric power generating systems. The proposed controller includes fractional order PI and fractional order PD controllers connected in cascade wherein orders of integrator (\(\lambda\)) and differentiator (\(\mu\)) may be fractional. The gains and fractional order parameters of the controller are concurrently tuned using recently proposed dragonfly search algorithm (DSA) by minimizing the integral time absolute error (ITAE) of frequency and tie-line power deviations. DSA is the mathematical model and computer simulation of static and dynamic swarming behaviors of dragonflies in nature, and its implementation in LFC studies is very rare, unveiling additional research gap to be bridged. Performance of the advocated approach is first explored on popular two-area thermal PS with/without governor dead band (GDB) nonlinearity and then on three-area hydrothermal PS with suitable generation rate constraints. To highlight the prominence and universality of our proposal, the work is extended to single-/multi-area multi-source EPSs. Several comparisons with DSA optimized FOPID controller and the relevant recent works for each test system indicate the contribution of proposed DSA optimized FOPI–FOPD cascade controller in alleviating settling time/undershoot/overshoot of frequency and tie-line power oscillations.

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## Appendix A

### Appendix A

### A.1 Nominal parameters of test system-1 are (Ali and Abd-Elazim 2011; Panda et al. 2013; Sahu et al. 2015, 2016)

\(f = 60\) Hz, \(B = 0.425\) p.u MW/Hz, \(R = 2.4\) Hz/pu, \(T_{\text{g}} = 0.03\) s, \(T_{\text{t}} = 0.3\) s, \(K_{\text{ps}} = 120\) Hz/pu, \(T_{\text{ps}} = 20\) s, \(T_{12} = 0.545\) p.u MW/rad.

### A.2 Nominal parameters of test system-2 are (Panda et al. 2013; Sahu et al. 2016; Çelik 2020; Gozde and Taplamacioglu 2011)

\(f = 60\) Hz, \(B = 0.425\) p.u MW/Hz, \(R = 2.4\) Hz/pu, \(T_{\text{g}} = 0.2\) s, \(T_{\text{t}} = 0.3\) s, \(K_{\text{ps}} = 120\) Hz/pu, \(T_{\text{ps}} = 20\) s, \(T_{12} = 0.444\) p.u MW/rad.

### A.3 Nominal parameters of test system-3 are (Panda et al. 2013; Çelik 2020; Nanda et al. 2006)

\(f = 60\) Hz, \(B = 0.425\) p.u MW/Hz, \(R = 2.4\) Hz/pu, \(T_{\text{g}} = 0.08\) s, \(K_{\text{r}} = 0.5\), \(T_{\text{r}} = 10\) s, \(T_{\text{t}} = 0.3\) s, \(K_{p} = 1.0\), \(K_{d} = 4.0\), \(K_{i} = 5.0\), \(T_{\text{w}} = 1\) s, \(K_{\text{ps}} = 120\) Hz/pu, \(T_{\text{ps}} = 20\) s, \(T_{12} = T_{23} = T_{13} = 0.086\) p.u MW/rad.

### A.4 Nominal parameters of test system-4 are (Mohanty et al. 2014; Padhy et al. 2017; Parmar et al. 2012; Barisal 2015)

\(f = 60\) Hz, \(R = 2.4\) Hz/pu, \(T_{\text{sg}} = 0.08\) s, \(K_{\text{r}} = 0.3\), \(T_{\text{r}} = 10\) s, \(T_{\text{t}} = 0.3\) s, \(T_{\text{gh}} = 0.2\) s, \(T_{\text{rs}} = 5\) s, \(T_{\text{rh}} = 28.75\) s \(T_{\text{w}} = 1\) s, \(b_{\text{g}} = 0.05\) s, \(c_{\text{g}} = 1\), \(X_{\text{c}} = 0.6\) s, \(Y_{\text{c}} = 1\) s, \(T_{\text{cr}} = 0.01\) s, \(T_{\text{f}} = 0.23\) s, \(T_{\text{cd}} = 0.2\) s, \(K_{\text{T}} = 0.543478\) pu, \(K_{\text{H}} = 0.326084\) pu, \(K_{\text{G}} = 0.130438\) pu, \(K_{\text{ps}} = 68.9566\) Hz/pu MW, \(T_{\text{ps}} = 11.49\) s

### A.5 Nominal parameters of test system-5 are (Mohanty et al. 2014; Padhy et al. 2017; Padhy and Panda 2017)

\(f = 60\) Hz, \(B = 0.4312\) pu, \(R = 2.4\) Hz/pu, \(T_{\text{sg}} = 0.08\) s, \(K_{\text{r}} = 0.3\), \(T_{\text{r}} = 10\) s, \(T_{\text{t}} = 0.3\) s, \(T_{\text{gh}} = 0.2\) s, \(T_{\text{rs}} = 5\) s, \(T_{\text{rh}} = 28.75\) s \(T_{\text{w}} = 1\) s, \(b_{\text{g}} = 0.05\) s, \(c_{\text{g}} = 1\), \(X_{\text{c}} = 0.6\) s, \(Y_{\text{c}} = 1\) s, \(T_{\text{cr}} = 0.01\) s, \(T_{\text{f}} = 0.23\) s, \(T_{\text{cd}} = 0.2\) s, \(K_{\text{T}} = 0.543478\) pu, \(K_{\text{H}} = 0.326084\) pu, \(K_{\text{G}} = 0.130438\) pu, \(T_{12} = 0.0433\), \(K_{\text{ps}} = 68.9566\) Hz/pu MW, \(T_{\text{ps}} = 11.49\) s

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Çelik, E. Design of new fractional order PI–fractional order PD cascade controller through dragonfly search algorithm for advanced load frequency control of power systems.
*Soft Comput* **25, **1193–1217 (2021). https://doi.org/10.1007/s00500-020-05215-w

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### Keywords

- Load frequency control
- Governor dead band
- Generation rate constraint
- Fractional order PI–fractional order PD (FOPI–FOPD) cascade controller
- Dragonfly search algorithm
- Optimization