Steady state analysis for BESS implementations in power systems are carried out by formulating optimization problems, whereas Transient analysis uses optimization mainly for controller tunning

DER	Distributed Energy Resources	REL	Relaxation of non-convex equations
RES	Renewable Energy Systems	GAMS	General Algebraic Modeling System
PV	Solar Photovoltaic Systems	MINLP	Mixed-Integer Non-Linear Programming
WE	Wind Energy Systems	WOA	Whale Optimization Algorithm
FC	Fuel Cells	SA	Simulated Annealing
HEE	Hydro-Electrical Systems	ABC	Artificial Bee Colony
BESS	Battery Energy Storage Systems	MFABC	Multi-Strategy Fusion ABC
LIB	Lithium-ion Battery	MFABC+	Hybridized MFABC and SA
EV	Electric Vehicles	HHO	Harris Hawks Optimizer
ANN	Artificial Neural Networks	AOA	Arithmetic Optimization Algorithm
SoH	State of Health	hHHO-AOA	Hybridized HHO and AOA
SoC	State of Charge	SOCP	Second Order Cone Programming
DN	Distribution Network	FA	Firefly Algorithm
TN	Transmission Network	HFPSO	Hybridized FA and PSO
UPQC	Unified Power Quality Conditioner	ICSO	Inherited Competitive Swarm Optimization
PID	Proportional-Integral-Derivative Controller	MAG-PSO	Multi-Agent Guiding PSO
FOPID	Fractional Order PID	MFO	Moth Flame Optimization
MPC	Model Predictive Controller	MMFO	Modified MFO
PFR	Primary Frequency Regulation	GOA	Grasshopper Optimization Algorithm
DoD	Depth of Discharge	MOGOA	Multi-Objective GOA
DR	Demand response	MOGWO	Multi-Objective GWO
PSO	Particle Swarm Optimization	TSIO	Two-Stage Interval Optimization
GA	Genetic Algorithm	DHHO	Developed Harris Hawks Optimization
MULTI	Multi-Objective Optimization	ADMM	Alternating Direction Method of Multipliers
MILP	Mixed-Integer Linear Programming	DC-ADMM	Dual-Consensus version of ADMM
STOC	Stochastic Optimization	WOAGA	Hybrid WOA-GA
GWO	Grey Wolf Optimization	MOWOAGA	Multi-Objective WOAGA
BLO	Bi-Layer Optimization	BWOA	Black Widow Optimization Algorithm
RO	Robust Optimization	HSMGWO	Hybridized Halton sequence and Social Motivation Strategy GWO
ML	Machine Learning	ASO	Atom Search Optimization
OPF	Optimal Power Flow	ALA-mQPSO	Hybridized Adaptive Local Attractor-based and Quantum-behaved PSO
MH	Metaheuristics	IPM	Interior-Point Methods
NSGA	Nondominated Sorting Genetic Algorithm	GDM	Gradient Descent Methods
B&B	Branch and Bound method	NM	Newton’s Method
TOPSIS	Technique for Order Preference by Similarity to Ideal Solution
RPNS	Reference-Point-Based Non- Dominated Sorting
GSA	Gravitational Search algorithm
CPSOGSA	Hybridized Chaotic map algorithm with PSO and GSA

Distributed Energy Resources (DER) is a term given to the set of energetic resources that are operated in a decentralized way and are typically, but not necessarily exclusively, driven by uncertain primary resources, like Renewable Energy technologies (RES) such as Solar Photovoltaic (PV) and Wind Energy (WE), or more predictable ones like Hydrogen Energy with Fuel Cells (FC) or Hydro-Electrical Energy (HEE) with micro turbines [1]–[5].The Penetration of DER in power systems has been thrusted recently by a decrease in technological costs, advancements in communication and information technologies, and the social drive to increase efficiencies in energy production, transportation, and consumption with reduced environmental impacts [1], [6] – [8]. This momentum has brought not only technical challenges in its implementation due to the inherent uncertain nature and the mixture of their primary resources [6], [9] – [12], but also changes in the operational frameworks of energy markets due to the decentralized fashion of its implementation and new market agents taking part in energy transactions [13], [14]. During the last decade, these challenges have been faced and extensive research has been published, allowing to find new operational structures, technical advantages, and also new questions to be answered. For example, multiple studies have shown how technically advantageous can be the implementation of DER in distribution networks in terms of power loss reduction, voltage regulation, network loadability, network capacity, system flexibility, frequency regulation, Demand Response, Curtailment, maximization of profit, or minimization of costs [10], [15] – [23]. However, analysis of DER in power systems is usually performed assuming certainty conditions (by means forecasts, study-cases, static behavior, or linearization), thus limiting the scope of obtained results, or by implementing variability compensation systems in the effort to increase the inertial response during electricity supply [24] or the stability [25], [26], for instance, using Battery Energy Storage Systems (BESS), flywheels or hydro-pumped storage [13].

Battery Energy Storage Systems BESS, whose technology is part of DER even though they cannot be considered as proper generation, have the particularity to behave dually: can operate as a load (withdraw energy) or as a support for generation (analogous to a generator). During BESS operation, it storages (charges) or releases (discharges) energy obtained from an external source through electrochemical processes. This behavior, together with the flexibility in controllability and power ramping rate, make their operation especially useful to provide supplementary services in the operation of power systems [27]. The efficiency during operation varies depending on the chemistry and energy density of the unit, i.e., between 72.5 % and 85 % efficiency with energy density ranging between 20 Wh/kg and 30 Wh/kg for Lead-Acid, 85 % - 95 % with 90 Wh/kg - 190 Wh/kg for Lithium-Ion, 72.5 % and 86 % with 150 Wh/kg - 240 Wh/kg Sodium-Sulphur, and 60 % - 72.5 % with 15 Wh/kg - 30 Wh/kg for Redox Flow [28]. Although Lead-acid is now a mature technology and provides availability and good efficiency at lower costs, research has been made in different technologies (chemistries) to overcome some of the downsides (i.e., low cycle life, low energy density, and the highly reduced life cycle under high depths of discharge and temperature [29]). Lithium-ion technology (LIB) shows up as an alternative that not only overcomes some of the mentioned downsides, but also enhances the upsides, by increasing the energy density and the cycle life at least fourfold while improving the efficiency. However, LI life cycle is strongly dependent on temperature and, together with its higher capital costs, might limit its implementation in utility scale applications [30]. Even though LI-BESS is not yet competitive when implemented for ancillary services in power systems, the increasing participation of Electric Vehicles EV (LI main market is now EV) in the electric demand share, the implementation of Vehicle-to-Grid frameworks and the sustained reduction in costs shown since 2013 [28] would make LIB viable for on-grid implementation in few years [31].

As mentioned before, BESS are mainly implemented to provide additional services to power systems either in transmission or distribution [27]. Those services can be classified into technical (where the main concern is to improve the power quality), and economical (increase of profits, reduction of costs) [32], within several timeframes. In Table 1, some services and the timeframe are reviewed.

Table 1 Table 1 Ancillary services provided by BESS adapted from [28]

Service	Category	Timeframe	References
Transient Voltage Stability	Technic (Power Quality)	Very Short (ms)	[33],[34],[35]
Harmonic Mitigation	Technic (Power Quality)	Very Short (ms)	[36],[37],[38]
Peak load and generation mitigation	Technic (Power Quality)	Very Short (ms)	[39],[40]
Primary Frequency Control	Technic (Power Quality)	Very Short – Short (ms-s)	[41],[27],[42]
Virtual Inertia	Technic (Power Quality)	Short (s)	[43],[44],[45]
Black start	Technic (Power Quality)	Short(s)	[46]
RES variability mitigation	Technic (Power Quality)	Medium (min.)	[47],[21],[48]
Voltage Management	Technic (Power Quality)	Medium (min.)	[49],[50],[51]
Secondary Frequency Control	Technic (Power Quality)	Medium (min.)	[52]
Demand Response	Technic /Economic	Long (hrs.)	[53],[22],[20]
Energy arbitrage	Economic	Long (hrs.)	[54],[55],[56]
Off-grid Operation	Technic/Economic (self - consumption)	Long (hrs.)	[57],[58],[59]
Power Loss minimization	Technic (efficiency)	Long (hrs.)	[60],[61],[62]
Congestion Relief	Technic (Power Quality)	Long (hrs.)	[63],[64],[65]
Distribution and Transmission deferral	Economic	Long (hrs.)	[66]

Source: Created by the author.

Modelling BESS for its implementation in power systems has been realized using diverse methods depending on the objective of analysis and its timeframe. For instance, in [27] a Three Time Constant model based in state estimation is proposed in the context of primary frequency and local voltage regulation. In [67], a nonlinear model is proposed for LI batteries using a Hammerstein-Wiener model. Machine Learning techniques (ML) such as Artificial Neural Networks (ANN) are also used to model BESS when data is available [68]. If the chemistry is not considered, BESS can be modelled using efficiency in steady state operation. In [69], an internal resistance model is proposed for efficiency, while in [70] similar structures for particular chemistries are studied including the State of Health (SoH), State of Charge (SoC) and power in longer term contexts.

BESS integration in active distribution networks, or microgrids, is usually analyzed in static BESS frameworks, this means that their mobility is not considered. However, Mobile BESS, MBESS, defines a new structure for operation for BESS, in which different solutions sets for its location, the status (charging, discharging, idle, or transport), and the costs for mobilizing such systems are considered to optimize network operation. Formulating the problem under this operational structure has shown several advantages in comparison with static BESS (Fewer losses, less active and reactive power drawn from substations, and improvements in voltage profiles) [71].

To examine the steady state effects of DER on active distributed networks, or microgrids, an optimal power flow study is typically performed, formulating the set of nonlinear equations resulting from circuit analysis, defining the operational constraints, such as voltage limits, transformer capacities or line current limits, and objective functions, which all depend on the decision variables.

Regardless of DER technology and the corresponding efficiencies based either on construction or operation, either uncertainties, objective function definition, or the modelling of the operation of the DG units might bring non-convexities to optimal power flow formulation, and with it, increased complexity in the steady-state analysis of the system. Then, additional effort is then needed to analyze the system if the objective function(s) and/or any (or every) additional operational constraint has concave properties in a minimization sense of the problem. Therefore, the way the problem is formulated for analysis defines the way it will be solved, and consequently how efficiently it will get to a solution, i.e., in the optimal dispatch of generators if costs or load shaving schemes are defined for even shorter periods, complex topologies and great dimensions in the power system. If this occurs, then metaheuristic techniques (MH) are useful and powerful tools to find approximate (to global) solutions regardless of the formulation [28]. However, some non-linear functions are convex, and some non-convex equations can be relaxed to ensure convexity and, consequently its exactness, if additional constraints are added [72]. MH techniques are general algorithmic frameworks that can be applied to a wide variety of problems, some of them very complex, making few modifications in the implementation [73]. These techniques are often inspired in phenomena observed in the nature and transformed into algorithms that usually start from random initial states and apply the specific search strategy to find solutions that converge the objective(s) function(s) close to a global minimum in complex problems, in a reasonable amount of time [74]. Consequently, due to the heuristic nature of the search strategy, global solutions and exactness are not guaranteed.

In this paper, a review on in optimization methods for operation and implementation of BESS in power systems is presented, and after this introduction, some of the most recent optimization problems regarding BESS operation for ancillary services and their formulations are surveyed in section BESS Optimization Problems. Subsequently, methods used to find the solution are reviewed and categorized with convexity as main criteria and if relaxations were implemented. Finally, results, discussion, and conclusions are presented in their respective sections.

As mentioned in in the previous section, there is an ample variety of applications of BESS implemented to provide services in power systems, in which the optimization of decision variables will provide the technical, economic, or mixed benefits expected from such frameworks. In this section, the formulation objective functions are reviewed in the context of the ancillary services provided with BESS.

As can be observed in the optimization problem section, recent studies implement analysis techniques depending on the formulation of the problem and the timeframe. In this section, a review from the most encountered optimization techniques and frameworks in recent manuscripts is presented. For this, a search in Web of Science is performed with the key *bess AND optimization, filtered for results published from 2019 on. The date of the search is 04/27/2022. From the search results is possible to see that research in optimization of BESS has been increasing and it can be expected to at least be equal as 2021, as can be observed in Figure 1 (results of 2022 correspond to the research published until the date of search and some programmed publications which are not yet published at the date of search).

Figure 1. Figure 1. Publications in BESS optimization from Web of Science search Source: Created by the author.

The list of results is reduced to 200, and a list of optimization techniques and frameworks is obtained from abstracts. This information is filtered and presented ordered by the appearance count in the right side of Table 2., while in the left side, optimization methods (or frameworks If optimization is performed indirectly) are tagged with the base technique if modifications or hybridizations are proposed.

Table 2 Table 2 Labeled technique count left Specific techniques count right

Technique/Framework Label	Count	Proposed Technique	Count
PSO	36	MPC	5
GA	26	SOCP	4
MULTI	20	MAG-PSO	2
MILP	15	DC-ADMM	2
STOC	14	ICSO	2
GWO	13	HHO-AOA	1
BLO	12	HFPSO	1
RO	10	MMFO	1
ML	10	MOGOA	1
REL	7	MOGWO	1
PID	6	MFABC	1
GAMS	6	TSIO	1
MPC	6	DHHO	1
MINLP	6	MOWOAGA	1
WOA	6	MFABC+	1

Source: Created by the author.

From the formulation of optimization problems related to BESS as ancillary services provider could be observed a strong branching in the scope of the analysis to be carried out. When steady state analysis is preferred, then optimization techniques are applied directly over the problem formulation, while, in transient analysis, control strategies are selected, and the optimization is carried out for parameter estimation either online or offline. In this case, Model Predictive Control has been found to be the preferred strategy, since it provides the flexibility of implementing non-linear models and base the action control on predicted behavior of the plant optimizing desired objective functions. This, however, can be a weakness as well since the quality of the predictions depend on the quality of the model.

On the other hand, traditional PID controllers are still being used as control strategy since the model for control is still linear. Although new approaches for its implementation and parameter estimation have been proposed such as FOPID and ANN based control and parameter optimization using MH or ML techniques (Fuzzy logic or ANN) for non-linear models. For steady state analysis, when BESS units are considered behind the meter, the optimization problem is typically constrained by active power balance equations, while in Distribution Networks an AC power flow is used to account power losses. However, the concave nature of AC power flow has also suggested in recent studies to think in linearization (e.g., First order in Taylor Series Expansion, polygon linearization) to simplify the formulation and use convex optimization methods for speeding up the obtention of solution while guaranteeing its exactness. Relaxations on the OPF formulation has been frequently explored in recent studies, specifically by transforming the non-convex quadratic equality constraints present in AC power flow equations (and/or in objectives) into convex second order cone inequality constraints and solving the convex program with SOCP.

During this review, the problem of the optimal allocation (location and sizing) of BESS units was recurrent, and its formulation using AC power flow results in a MINLP problem (MILP if relaxations/linearization eliminate the non-linearity/non convexity in equations). Typically, MILP or MINLP are solved using Branch and Bound Method. (B&B). Such problems include convex transformation of constraints with integer variables and can formulated as an optimization problem using an algebraic approach with GAMS. Due to the flexibility of MH in finding solutions to any kind of problems (convex and non-convex), Multi-objective MINLP programs have been handled with those techniques, achieving good performance while trading exactness off. Techniques in the categories PSO and GA have been found to be the most popular in the last years. As can be observed in Table 2, modifications, or new proposals on PSO or GA techniques can be found in single occurrences, while their use in any other form (original, modified or hybridized) for result comparison are greatly used. Other techniques used in the last studies for BESS implementations are Grey Wolf Optimization (GWO) and Whale Optimization Algorithm (WOA).

Ever since it is desired to achieve better solutions while increasing computational efficiency, hybridization takes relevance, as it is shown in recent studies, since this allows to take the advantageous strategies from several techniques and combine them into a single better technique aiming to achieve greater speed of convergence and diversity in solutions in MULTI frameworks. Optimization problems, as could be observed in the corresponding section, are commonly formulated in mono-objective framework, even when the aim of the problem is to optimize several objectives. This is done so because it simplifies the execution of the program and facilitates any possible linearization or relaxation. However, this reduction in the dimension of the objective space results in the individualization of the solution and the subjectivation of the importance of each objective function.

In multi-objective frameworks, the result of the optimization is a set of solutions that cannot be improved in one objective without degrading the others (non-dominated solutions). This adds complexity to the optimization but delivers flexibility when it is desired to have multiple operation setpoints or if there is no objective information regarding objective weights. As could be observed, the multi-objective framework (MULTI) is recurrent in recent studies, and newly developed metaheuristic techniques are mainly assessed within this framework. It is worth noticing that the pareto dominance criteria is still the most common technique implemented in MO metaheuristic algorithms to select the best solutions. However, the criteria comparing such solutions has also been subject of research, such as TOPSIS, ε-dominance or RPNS.

On the other hand, due to the uncertain nature of the primary resources in RES, Stochastic optimization (STOC) and Robust optimization (RO) have taken relevance in the studies reviewed and are now presented as computational cost-effective alternatives to Monte-Carlo simulations. Within STOC and RO optimization frameworks, LP implementations are possible by introducing relaxations and if probability distributions are represented by convex functions.

Finally, it is worth noticing how multiple optimization stages are now being implemented in BESS research. As observed in Table 2, a Bi-Layer Optimization (BLO) framework has been frequently proposed in recent studies, in which one optimization layer typically optimizes short term operation problems while the other optimizes, partially based on results of the other layer, long term (planning) problems.

In this paper, an overview of the role of BESS in the penetration of RES in power systems and the different advantages of their implementation found in recent literature are presented in the introduction. Then characteristics of BESS chemistries is presented in terms of efficiency and energy density. From this overview, LIB technology is detailed due to trending research and its increasing participation in the operation of power systems, especially in terms of demand patterns for EV and Vehicle to Grid frameworks. Later, a summary of BESS operation and optimization frameworks is presented. Subsequently, a review on the formulation of optimization problems related to BESS as ancillary services provider is presented and objective functions formulated in recent studies are detailed. Next, an overview of optimization frameworks and techniques is presented considering occurrences in literature published in the last three years (since 2019). Finally, it can be concluded that research including BESS optimization has been increasing exponentially in the last decade. The formulation of optimization problems is not only related to ancillary services, but also to support standalone operation or operation support in microgrids and depending on the timeframe of analysis, the optimization may take place within optimal power flow or control frameworks. Given the formulation of the problem and the scope of research, multiple optimization frameworks are being implemented in recent research considering stochasticity, computational efficiency, and dimensionality of objective space. MH techniques dominates complex, multivariate, multi-objective analysis while relaxations, simplifications, linearization, and single objective construction enable the use of traditional, more efficient, and exact techniques. Well known metaheuristic techniques, such as PSO or GA, have been used often as a reference for comparison in the implementation of new methods aiming to find better solutions more efficiently. Hybridization of MH has been studied showing comparable or improved results and presenting possible alternatives to other well-known MH techniques.