**1. Introduction**

Electric power systems are large, interconnected, complex, and highly changeable systems that are always affected by a wide variety of perturbations [1]. Therefore, the control design stage and tuning procedure for multiple controllers is an entangled task [2,3], present interesting approaches on stabilizing procedures in electric power systems that use multiple power system stabilizers with lead and lag compensators. The conventional linear controllers designed around an equilibrium point are useful, but their performance could be degraded if variations are presented in the system. On the other hand, dealing with non-linear controllers is a high demanding and slow task due to the complexity of large-scale power system. In general, for reaching a good performance, these strategies present dependency on the parameters system modeling.

Power system stabilizers (PSS) have been used to generate supplementary signals to control the excitation system to improve the power system dynamic performance by the damping of system oscillations [1]. However, the expected behavior depends entirely on the correct selection of controllers' gains and time constants [2,3]. Moreover, some flexible

**Citation:** Tapia-Olvera, R.; Beltran-Carbajal, F.; Valderrabano-Gonzalez, A.; Aguilar-Mejia, O. A Novel Methodology for Adaptive Coordination of Multiple Controllers in Electrical Grids. *Mathematics* **2021**, *9*, 1474. https://doi.org/10.3390/ math9131474

Academic Editor: Cristina I. Muresan

Received: 19 May 2021 Accepted: 15 June 2021 Published: 23 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

alternating current transmission systems (FACTS devices) are included to solve some specific power systems problems; nevertheless, their operation is also depending upon the positive interaction with other regulation devices. Refs. [4,5] exemplify the problem of simultaneous tuning of multiple controllers in large scale power system including FACTS devices in transmission systems.

There are several methodologies to solve the problem of designing linear controllers to reach good dynamic performance. However, these solutions are complex in implementation; they do not cover a wide range of operating conditions of the power system or they do not have the same behavior with new grid topologies. The main objective of this proposal is to attain an adaptive performance of PSS in large-scale power systems with the possibility of adding new components that change the grid configuration, in this case for exemplifying through a static synchronous compensator (StatCom).

In order to validate the proposed strategy and without loss of generality, this paper presents the control design problem of PSS in power systems including a StatCom, which is one of the most useful FACTS devices in practical power systems. This configuration adds enough complexity to verify the viability of the proposal.

In general, the design control stage has been considered an independent problem, with only one controller. The fact that the system can have other regulation devices, has not been included. Only few works contemplate more than one controller simultaneously in the design stage. However, this is an open research topic due to the electrical grid composition and the continuous topology changing on it [6].

In [2] two objective functions must be solved to obtain coordination between PSS and traditional static VAR compensators (SVC). In order to reduce the high computational load, the genetic algorithm was used for solving the multi-objective optimization problem, adapting it for parallel computing. An analysis based on the power system modeled as a set of hybrid non-linear differential algebraic equations is presented in [3], where the dynamic behavior of the system is studied in various scenarios: no PSS, PSS without dead-band, and PSS with dead-band.

In [7], a single machine infinite bus (SMIB) model is used to tune the PSS, and then new non-specified adjustments are carried out to extend the scheme to the multimachine scenario. Additionally, the tuning stage is very case-dependent. Multi-band PSS are tuned in [8] by using an optimization search method based on modal performance index, but representative linearized system models are required for the optimization procedure.

The PSS tuning based on linear quadratic regulator design is presented in [9]. The state and input matrices of the linearized power system model are required for developing the optimization procedure in a single machine case, and then it is extended to the multimachine case. In [10], a two-level control strategy that blends a local controller with a centralized controller is proposed to diminish low frequency oscillations. In the PSS model, a proportional integral (PI) controller is added. Two extra gains are included in the problem solution. For the tuning procedure, two stages are required, first the design of the local PI controller and then the design of the centralized controller.

A design method using a modified Nyquist diagram with an embedded partial poleplacement capability is presented in [11]. The small signal stability model obtained by the linearization of the power system around an operating point is required. That method evaluates the open loop transfer function along a line of constant damping ratio to design PSS for two test systems.

Additionally, control design based on non-linear theory is used, but in the same sense the procedure is realized separately for each controller. In [12], a scheme called decentralized continuous higher-order sliding mode excitation control is applied. The deviations on the angle of the power are required to obtain the desired system performance, also the estimation of first and second order time derivatives of this angle must be determined. Similarly, in [13] the *H*∞ control with regional pole placement is used to ensure adequate power system dynamic performance, the linearized model around an equilibrium point is also needed. Additionally, deterministic strategies based on artificial intelligence could

be an alternative to the design procedure of multiple controllers in electrical grids [14]. Another important algorithm is the non-linear feed forward control which represents an option of non-linear adaptive control techniques [15]. This kind of strategies has been little explored in applications for electrical power systems. Similarly, other approach that can be extended to large scale power systems is the physics-based control technique [16].

A scheme called networked predictive control (NPC) used to design a damping controller that incorporates a generalized predictive control (GPC) to generate optimal control predictions is presented in [17]. Model identification is required to deal with uncertainties and to provide an adaptive predictive model for GPC. This method describes four steps for designing a NPC for a wide area damping controller: (i) modal analysis of the detailed non-linear model; (ii) determination of the order of the reduced order model of the power system; (iii) obtain the low-order equivalent model via model identification algorithm and use it as the prediction model for the NPC; (iv) selection of parameters like the output prediction horizon, the control horizon, the weighting sequence, and the sampling period.

Finally, artificial intelligence methodologies such as artificial neural networks (ANN), fuzzy logic (FL), or neuro-fuzzy are used for design purposes. In [18] an adaptive fuzzy sliding mode controller with a PI switching surface to damp power system oscillations is proposed. This strategy combines: (a) a sliding surface, (b) a fuzzy controller, (c) a curbing controller, and (d) a wavelet neural network to obtain the best auxiliary signal input to the excitation system. The structure of wavelet neural network is based on three layers, where the inputs are the sliding surface and its derivative.

A so-called hybrid adaptive non-linear controller is proposed in [19]. For the controller design it is necessary to estimate non-linear parts of the system, it is also required to measure data. The controller has a feed forward neural network structure, it is trained offline with extensive test data and it is adjusted online. In [20], the design of a PSS based on a combination of fuzzy logic and sliding mode theory is illustrated. This proposal indicates that a fuzzy-PID controller is composed of fuzzy PI and fuzzy PD controllers, and the response depends on scaling factors, hence selection of these parameters is crucial while designing the controller. The definition of the fuzzy rules is also an important issue for its correct operation.

Other important proposals, including FACTS devices, offer better results working with positive interaction with PSS. In [4], an optimization formulation is used to coordinate one PSS with one unified power flow controller (UPFC), but two objective functions based on eigenvalues of the state are needed for it. The possibility of using different FACTS devices is indicated in [5], the results include a StatCom and a UPFC. The eigenvalues of the power system model are required on the tuning procedure.

In [21], a StatCom and a PSS have been tuned to get a good dynamic power system performance using the seeker optimization algorithm to obtain the controller gains by an objective function. The StatCom model used, includes the components of the current and voltage dynamic in terminals of direct current (DC) capacitor.

Similarly, an objective function in [22] is used to attain a positive interaction between StatCom and PSS with a constraint set. The StatCom model is described with the operating range curve, but no dynamic equations are included. In [23], the dynamic operation of the StatCom is coordinated with a PSS. The tuning procedure depends on an objective function, and the definition of a constrain set.

The changing nature of power systems demands different types of studies due the inclusion of new control devices, renewable energies, and emerging technologies. However, it is difficult to have a unique methodology to solve the problem of the control design in large-scale power system. Although there are different alternatives to solve this problem, these proposals offer a solution limited to the characteristics of the systems under study. In multimachine power systems the control design problem is amplified due to the presence of multiple controllers that must be tuned simultaneously to guarantee a positive interaction for each operating condition.

Therefore, the present contribution considers the non-linear power system nature and it defines an adaptive controllers' behavior. This performance is obtained by the inclusion of some selected dynamic gains that are updated on each sample time to find the best values for every operating condition and system topology. It is possible to update all the controller gains, but to exemplify the relevance of the proposal, only some of them are dynamically calculated. Simultaneous tuning of each controller is obtained.

To validate the proposed scheme based on B-Spline neural networks, PSS are simultaneously coordinated with a StatCom to enhance the power system dynamic response under severe disturbances. An effective control design procedure for power system controllers is demonstrated by the obtained results, improving the overall multimachine system dynamic performance. The proposal avoids the parameters and power system model dependency by using only measurements of some system variables to reach the expected behavior. The main contributions of our methodology are: (i) a new method for tuning multiple controllers in electrical grids is proposed; (ii) a time-domain analysis for damping low frequency oscillations considering different controllers when previous design stage was already performed is included; (iii) different controllers preserving good performance without imposing a particular requirement are considered; (iv) the introduced methodology offers a practical way to obtain adaptive behavior of controllers with simultaneously tuning, and positive interaction; (v) the proposed algorithm is learning online, which means no additional stages for training are required.