Intraday Dispatch Strategies of Battery Energy Storage Systems to Smooth the Duck Curve: A Real-Life Brazilian Case

Abstract

The growing penetration of centralized and distributed solar photovoltaic (PV) generation has introduced significant operational challenges in power systems, most notably the “duck curve” phenomenon, which is characterized by steep net load ramp-up at dusk. This issue has become increasingly evident worldwide, including in Brazil, where solar PV capacity is expanding rapidly. In this context, this paper proposes optimization-based strategies for dispatching Battery Energy Storage Systems (BESS) to smooth the duck curve, using real data from the state of Minas Gerais, Brazil. Three distinct optimization objectives are investigated: load derivative compensation, variable generation compensation, and average load tracking. Energetic and electric analyses were performed using the real data. For the evaluated Minas Gerais case study, both analyses indicate that the BESS operation based on load derivative compensation provides the most effective reduction in load ramps among the three strategies investigated.

1. Introduction

The increasing penetration of Renewable Energy Sources (RES), particularly Photovoltaic (PV) generation, is transforming net load patterns in electrical grids worldwide. One of the most emblematic effects is the so-called “duck curve” phenomenon, characterized by a midday net load dip caused by high PV generation and a steep ramp-up during evening hours when solar production drops while demand increases [1]. This effect was first observed and popularized in 2013 by the California Independent System Operator (CAISO) in the United States as a way to illustrate electrical grid load behavior under high solar generation [2].

This pattern introduces several technical and operational challenges for system operators, such as: (1) the system must deploy fast-ramping generation to meet the evening demand surge, placing stress on thermal generation fleets that are often not designed for such flexibility [3], (2) midday PV overproduction can lead to negative pricing and forced renewable curtailment, especially in systems without sufficient flexible resources [4], and (3) the variability also increases the need for reserve provisioning, frequency regulation, and voltage support [5]. In countries like the United States, the duck curve has already led to curtailment of solar generation and increased reliance on fast-ramping resources.

Battery Energy Storage Systems (BESS) have emerged as promising solutions to mitigate these impacts [6]. By absorbing excess PV generation and injecting power during the transition between increasing demand and decreasing PV generation, BESS can smooth verified electrical grid load profiles and support grid stability. Thus, an appropriate BESS operation strategy could promote benefits in a power system with high PV penetration.

The state of Minas Gerais, Brazil, has been experiencing significant growth in PV generation, driven by the region’s social, economic, and geographic characteristics. The state is already beginning to face the impacts of high load variability and lack of grid support, posing operational challenges for the national system operator.

This paper proposes an optimal dispatch strategy for BESS in the state of Minas Gerais, Brazil, aiming to smooth the duck curve caused by high PV penetration. Energetic and electrical evaluations are conducted to assess the impact of different optimization objectives by using an alternating procedure of both energetic and optimal power flow solutions.

The remainder of the paper is organized as follows: Section 2 provides an overview of the duck curve phenomenon and BESS applications. Section 3 describes the method proposed in this research and the considerations made. Section 4 details the proposed BESS dispatch methodology. Section 5 presents the analysis and simulation results. Section 6 presents the conclusions and future research directions.

2. Literature Review

The duck curve phenomenon is not unique to the United States. Several countries beyond the United States are experiencing similar behavior. In Australia, the Australian Energy Market Operator (AEMO) has reported duck curve characteristics in the southern region of the country due to high residential PV penetration [7]. Likewise, Germany and Italy, with their extensive distributed solar resources, have also faced over-generation and ramping issues during specific seasons. Emerging economies, like Chile and Brazil, are observing initial signs of these trends as solar expansion continues [8].

To address the operational challenges posed by the duck curve, the CAISO proposed two complementary strategies: “fattening” and “flattening” the curve [9]. Figure 1 illustrates the impact of these two mitigation approaches. The fattening strategy involves shifting flexible loads such as electric vehicle (EV) charging to midday hours, thereby increasing demand when PV generation is abundant and reducing the magnitude of the evening ramp. In contrast, the flattening strategy employs BESS to store excess solar energy during midday and discharge it during peak demand in the evening. As shown in the green shaded area, the BESS effectively smooths net load variability, enhancing grid stability and operational flexibility.

Batteries have gained attention as a strategic resource due to their fast response time. Their key contributions to mitigate the duck curve include: (1) energy shifting, that is, BESS can store excess PV energy during low-demand hours and discharge during peak demand, reducing ramping stress on traditional generation; (2) firming of renewable output, that is, by smoothing short-term PV generation, BESS improve predictability and dispatchability of RES; (3) curtailment reduction, that is, properly dispatched storage systems reduce the need to curtail PV output, enhancing renewable utilization and economic efficiency; and (4) provision of ancillary service, where BESS can participate in frequency regulation, spinning reserves, and voltage control, adding resilience during dynamic system conditions [10].

Several studies in the literature have addressed the optimal dispatch of BESS. For instance, the authors in [11] propose a Mixed-Integer Linear Programming (MILP) approach for the optimal siting and sizing of BESS, primarily to alleviate transmission network congestion. Although its primary focus is not on the duck curve phenomenon, the methodology offers insights into how storage placement can affect system performance. The work of [12] presents a comprehensive study that simulates the year-round operation of a combined heat and power-dominated energy system using a high-resolution chronological approach. That work evaluates the optimal combination of Pumped Hydro Storage (PHS) and electric boilers to enable a higher penetration of RES in China, with the objective of minimizing both operational costs and CO₂ emissions. In [13], the authors propose a novel microgrid configuration that integrates PV generation with mini-hydro pumped storage systems, leveraging the gravitational potential available in high-rise buildings. A cost-based multiobjective optimization strategy for the optimal integration of BESS into distribution networks was proposed in [14]. In that work, the optimization process simultaneously considers both system operation costs and battery degradation (cycling) costs, and the methodology is implemented using the interior-point algorithm with an analytical Jacobian to improve convergence. The authors of [15] address the optimal sizing of energy storage systems in distribution networks, particularly under high uncertainty in demand and distributed generation. Furthermore, they propose a scenario-based two-stage stochastic optimization framework. In the first stage, the model determines the optimal storage capacity required to prevent operational issues such as overvoltages and undervoltages. The second stage computes optimal control strategies for the storage systems based on specific realizations of demand and generation. In [16], the authors propose an energy management strategy aimed at mitigating power fluctuations associated with PV generation. Rather than relying on conventional BESS, the study explores the use of PHS due to its advantages in terms of cost-effectiveness, environmental friendliness, extended life cycle, and low maintenance requirements.

References [17,18] investigate strategies to address the challenges posed by the duck curve phenomenon, which emerges from the growing penetration of PV generation and its impact on net load profiles. Both studies emphasize that the significant midday generation from PV leads to steep net load ramps, increasing the start-up and fuel costs of thermal generators due to frequent on/off cycling. To mitigate these effects, the authors propose the integration of alternative energy storage systems, specifically, concentrated PV and pumped storage hydroelectricity as more cost-effective solutions compared to conventional BESS. Additionally, ref. [18] incorporates real-time price-based demand response and the use of controllable fuel cell loads in smart homes to enhance grid flexibility. The authors in [9] propose an Energy Management System (EMS) based on a Model Predictive Control (MPC) framework for coordinating batteries, load shedding, PV curtailment, and generation unit scheduling. The optimization is formulated as a MILP problem.

In addition to analytical and mathematical optimization approaches, demand-side flexibility plays a crucial role in mitigating the duck curve phenomenon. Distributed batteries, when installed behind the meter, offer a promising solution for reshaping the local load curve [19]. A significant portion of installed PV capacity in several countries originates from the demand side, for instance, approximately 30% in the United States, highlighting the importance of behind-the-meter PV systems, such as rooftop solar [20]. These systems contribute directly to the formation of the duck curve. Although system operators on the supply side typically lack visibility into behind-the-meter PV generation, end-users can locally monitor their generation and operate batteries accordingly to support grid stability.

One of the most effective mechanisms to enable such demand-side flexibility is the implementation of Demand Response (DR) programs. DR schemes, often categorized into time-of-use, critical peak pricing, and Real-Time Pricing (RTP), encourage consumers to shift or reduce consumption in response to price signals [21]. Among these, RTP has shown significant potential to influence user behavior dynamically. The coordination of these distributed resources is typically managed by Resource Aggregators (RA), which act as intermediaries between the grid and prosumers. RA are responsible for aggregating storage, flexible loads, and DR-capable consumers to participate in energy and ancillary service markets, thereby enhancing the system’s operational flexibility and contributing to duck curve mitigation [22].

Many researchers have addressed the mitigation of the duck curve through model-based strategies, particularly focusing on energy management systems and Demand Response (DR) coordinated by aggregators. For instance, the study in [18] proposed an optimal thermal unit commitment strategy using MILP under an RTP scheme, aiming to reduce the disparity between peak and off-peak demand. The potential of demand-side flexibility, particularly from large fleets of electric vehicles, to reduce steep ramp-up requirements was explored in [23]. In [24], the authors demonstrated the effectiveness of pre-cooling strategies in residential households as a means to shift cooling loads and alleviate the duck curve. Additionally, reference [25] presented a dynamic pricing-based DR model for heating, ventilation, and air conditioning systems, formulated as a single-level optimization problem to minimize peak electricity demand.

On the other hand, statistical and rule-driven methods continue to play a central role in the mitigation of the duck curve. These methods are characterized by transparent formulations with operational constraints. Recent studies have extended classical explicit approaches by incorporating stochastic programming to explicitly capture uncertainty in renewable generation and demand variability [26]. In [27], the authors present the concepts of probabilistic ramp curve and probabilistic duck curve to model the set of uncertainties of the system, thus enabling the development of more beneficial and secure strategies for system operation planning. In [28], the authors applied Monte Carlo simulation for a probabilistic assessment of the Spanish electrical system for different scenarios in 2030, and the results revealed many “duck curve” scenarios that could be detrimental to the power system. Again, prior knowledge of these future scenarios through Monte Carlo simulation can favor the development of possible mitigation strategies. In [29], the authors present some strategies for dealing with the duck curve by taking advantage of smart grid technologies. In [30], the authors proposed a convex optimization model to smooth the duck curve through the potential use of electric vehicles in the system. In [31], the authors propose a Predictive Control Model to control the system’s power flow, resulting in a range of benefits, one of which is the prevention of the duck curve. Compared to implicit learning-based strategies, explicit approaches typically require less historical data, offer guaranteed feasibility with respect to grid constraints, and provide clearer insights into cause–effect relationships between control actions and system behavior [32].

Different from the aforementioned references, the present paper proposes a strategy for intraday BESS dispatch (charge and discharge) based on the minimization of the rate of change of load (the load derivative). Aiming to prove its effectiveness, the proposed strategy is compared to two other strategies for BESS dispatch definition. One is based on the compensation of variable renewable resources (such as wind and PV power plants) and the other is based on the idea of making the liquid load as flat as possible. The results presented in this paper, obtained for the Minas Gerais case study, show that the minimization of the load derivative provides the most effective ramp-mitigation performance among the three investigated strategies.

3. Proposed Approach

While these studies provide valuable insights through mathematical optimization frameworks and metaheuristic techniques for optimal allocation of energy storage, this paper evaluates the impact of BESS in mitigating load curve variations caused by the increasing penetration of PV generation in the state of Minas Gerais, Brazil. This state is the national leader in PV solar generation, ranking first in both centralized plants and distributed rooftops. In this paper, the following three important definitions are used:

• PV—centralized PV generating plants connected to the extra-high-voltage transmission network that is operated by the Brazilian Transmission System Operator (ONS);
• DG—small-scale distributed hydro and thermal generation connected to the distribution network that are not operated by ONS;
• MMDG—micro and mini distributed generation basically composed by rooftop PV connected to the low- and median-voltage radial distribution feeders, respectively.

This remarkable growth is driven by multiple factors, such as state-level tax incentives for this energy source, high solar irradiance across the territory, and the availability of large areas of low-cost land, resulting from rural poverty and unfavorable climatic conditions for agribusiness. Additionally, the state also has significant volumes of Distributed Generation (DG), which includes non-solar sources and therefore exhibits a more stable generation profile throughout the day.

The combination of these factors contributes to the intensification of the duck curve phenomenon, which is already strongly observed in the state. These fluctuations create steep hourly ramps in the operation of the Brazilian Interconnected Power System (BIPS), driven by the limited flexibility of many generation units. The operational challenge arises from the time lag required by these sources to adjust active power output, whether due to system inertia or technical response constraints. These limitations can degrade power quality, impair voltage and frequency stability, overload equipment, and elevate the risk of systemic failures.

To assess the effectiveness of BESS in mitigating these impacts, this study assumes that each centralized PV plant in Minas Gerais is equipped with an associated BESS unit. The optimization problem is formulated as a Non-Linear Programming (NLP) model based on the hourly operation of multiple BESS.

A flexibility analysis is carried out to evaluate the system’s performance under three distinct operational strategies:

• Load derivative compensation: this strategy aims to minimize hourly load variations by smoothing the derivative of the verified load profile (i.e., after accounting for DG, MMDG, and centralized PV generation). The goal is to reduce operational stress from steep ramps, thereby improving system reliability;
• Variable generation compensation: this strategy seeks to minimize the difference between the verified load including all sources (DG, MMDG, and centralized PV) and the electrical grid load considering only DG. By doing so, it compensates for high variability from MMDG and centralized PV generation, smoothing fluctuations and enhancing operational stability;
• Average load tracking: this strategy minimizes the deviation between the verified load (with all sources’ contributions) and the average load (considering only DG). The objective is to flatten the load curve, reducing operational interventions and improving system stability through a more predictable dispatch pattern.

4. Mathematical Formulation

This section presents the proposed approaches for the optimal dispatch of BESS based on three different strategies, each designed to mitigate the impacts of PV generation. The models aim to enhance reliability by smoothing load ramps, compensating variability, and flattening the load profile [33,34].

The following notation is used in the formulations of three distinct optimization models.

t	Hour index, $t\in \{0,1,\dots ,23\}$.
$P_t^{\mathrm{bess}}$	BESS power dispatch at hour t (MW);
	positive: discharge, negative: charge.
$E_t^{\mathrm{bess}}$	Energy stored in the BESS at
	the end of hour t (MWh).
$V{L}_t$	Verified load at hour t, including DG,
	MMDG and PV.
$G{L}_t$	Global load for compensation
	at hour t.
$G{L}_{\mathrm{average}}$	Average global load, without
	renewable compensation.
$E_{\min }^{\mathrm{bess}}$, $E_{\max }^{\mathrm{bess}}$	Minimum and maximum energy storage
	of the BESS.
$P_{\min }^{\mathrm{bess}}$, $P_{\max }^{\mathrm{bess}}$	Maximum and minimum charging/discharging
	power of the BESS.
$E_0^{\mathrm{bess}}$, $E_{23}^{\mathrm{bess}}$	Initial and final energy of the BESS.
$P_0^{\mathrm{bess}}$ to $P_{23}^{\mathrm{bess}}$	The variables to be optimized (BESS dispatch).

GL_t is the total system load at hour t used as reference for compensation, excluding PV, DG and MMDG contributions; VL_t is the measured load at hour t including contributions from PV, DG and MMDG; and GL_average is the average global load over 24 h.

Figure 2 illustrates the proposed BESS optimization framework, including data input, formulation of the objective functions and constraints, solver configuration (initialization, bounds, and tolerances), and post-processing steps to obtain BESS dispatch, state of charge, and net load profiles. Based on the system’s operational priorities, one of three objective functions can be selected.

It is worth noting that the optimal BESS dispatch obtained from the energy-based optimization could be evaluated by using an AC Optimal Power Flow to ensure compliance with network operational constraints, such as voltage limits and line loading. So, in cases where violations are detected, an AC-OPF is solved with the objective of minimizing generator redispatch deviations, while keeping the BESS dispatch fixed, ensuring that all operational constraints are satisfied. A systematic evaluation of potential violations can be implemented using the procedure described in Algorithm 1. This algorithm ensures consistency between the energy-layer optimization and the electrical-layer validation, preventing unintended modifications of the BESS dispatch.

Algorithm 1 Coupling between energy optimization and electrical validation

1: Input: Load and PV data ONS, network model, generator limits   2: Output: Feasible operating point and validated BESS dispatch   3: Solve energy-layer optimization problem   4: Obtain optimal BESS dispatch $P_t^{bess}$   5: Initialize generation dispatch from base case   6: Run AC power flow with $P_t^{bess}$   7: if no violations are detected then   8:       Accept solution and store results   9: else 10:       Solve OPF problem with:     Objective: minimize redispatch deviation     Control variables: generator active power     Constraints: voltage limits, line loading, generator limits     Fixed variable: $P_t^{bess}$ 11:       Obtain adjusted generation dispatch 12:       Run AC power flow with updated dispatch 13:       Store validated results 14: end if 15: Return: Validated system operation

4.1. Objective Functions

4.1.1. Derivative Compensation (DER)

The objective is to minimize the discrete-time derivative (ramp) of the net load after BESS compensation. Thus, the compensated net load is defined as

$$\begin{array}{cc}\hfill & N_t=V{L}_t-P_t^{\mathrm{BESS}},\hfill \end{array}$$

Here, the DER objective minimizes the squared discrete-time ramp of the compensated net load:

$$\begin{array}{cc}\hfill & J_{\mathrm{DER}}={\displaystyle \sum _{t=1}^{T-1}}{\left[{\displaystyle \frac{N_t-N_{t-1}}{\Delta t}}\right]}^2.\hfill \end{array}$$

(1)

In this case, $T=24$ and $\Delta t=1$ h. The summation starts at $t=1$ because the ramp at each time step is calculated relative to the preceding hourly value.

4.1.2. Variable Generation Compensation (GEN)

This strategy aims to smooth the impact of high-variability sources (MMDG and centralized PV) by aligning the verified load curve with the global load considering only DG:

$$\min {\mathcal{J}}_{\mathrm{GEN}}={\displaystyle \sum _{t=0}^{23}}{\left[P_t^{\mathrm{bess}}-V{L}_t+G{L}_t\right]}^2$$

(2)

4.1.3. Average Load Tracking (FLAT)

This formulation seeks to flatten the verified load around the average value of the global load:

$$\min {\mathcal{J}}_{\mathrm{FLAT}}={\displaystyle \sum _{t=0}^{23}}{\left[P_t^{\mathrm{bess}}-V{L}_t+G{L}_{\mathrm{average}}\right]}^2$$

(3)

4.2. Constraints

4.2.1. Energy Balance

Constraint (4) defines the energy balance over the day, i.e., ensures energy balance over the 24-h period. This means that the total energy charged/discharged by the battery over the day must result in the final energy level $E_{23}^{bess}$ equaling the initial level $E_0^{bess}$, adjusted by the electrical grid energy exchange (sum of power dispatches). This constraint prevents the battery from ending the day overcharged or undercharged, which ensures consistency in daily operations. One should notice that it is being considered an ideal efficiency of the charging and discharging of the BESS. If desired, it is necessary to include a weight multiplying the active power to represent the BESS efficiency.

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{t=0}^{23}}{P}_t^{bess}\Delta t+E_{23}^{bess}-E_0^{bess}=0\hfill \end{array}$$

(4)

4.2.2. Energy and Power Dispatch Limits

The energy and power dispatch limits are defined by constraints (5) and (6). These constraints bound the battery’s energy $E_i^{bess}$ and power dispatch $P_i^{bess}$ at every hour t; that is, it ensures that the battery operates within its physical and technical limits.

$$\begin{array}{cc}\hfill & E_{\min }^{bess}\le E_t^{bess}\le E_{\max }^{bess}\forall t=1,\dots ,23\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & P_{\min }^{bess}\le P_t^{bess}\le P_{\max }^{bess}\forall t=1,\dots ,23\hfill \end{array}$$

(6)

4.2.3. Final Dispatch Power and Energy Capacity Bound

Constraint (7) enforces the final dispatch power bound; that is, this limits the absolute value of power at the final hour to the maximum battery power rating. This is often used to ensure that the final dispatch is respecting a desired value ($P_{desired}^{bess}$), still within the system’s controllability, avoiding extreme values that may impact system stability at the end of the time horizon, and to represent the final energy capacity bound, constraint (8) ensures that the energy stored in the BESS at the end of the day does not exceed its desired final energy ($E_{desired}^{bess}$).

$$\begin{array}{cc}\hfill & |P_{23}^{bess}|\le P_{desired}^{bess}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & E_{23}^{bess}\le E_{desired}^{bess}\hfill \end{array}$$

(8)

4.2.4. State of Charge (SoC)

Constraint (9) models the state of energy of the battery at each time step t as the initial energy minus the cumulative dispatch up to that time. This dynamic constraint tracks how the energy in the battery evolves over time due to charging (negative $P_t^{bess}$) or discharging (positive $P_t^{bess}$). This ensures that the BESS energy is computed for all hours and remains within the storage capacity limits.

$$\begin{array}{c}\hfill E_t^{bess}=E_{t-1}^{bess}-P_t^{bess}\Delta t,\forall t\in 1,\dots ,23\end{array}$$

(9)

4.3. BESS Operational Constraints

To improve the BESS model, charging and discharging efficiencies are incorporated into the energy balance equation. The BESS operation is modeled using separate charging and discharging power variables, where $P_t^{\mathrm{ch}}$, $P_t^{\mathrm{dis}}$ are the charging and discharging power at hour t (MW) with $P_t^{\mathrm{ch}}\ge 0$ and $P_t^{\mathrm{dis}}\ge 0$, ${\eta }^{\mathrm{ch}}$ and ${\eta }^{\mathrm{dis}}$ represent, respectively, charging and discharging efficiency of BESS. If $P_t^{\mathrm{ch}}\ge 0$, then $P_t^{\mathrm{dis}}=0$ and vice versa.

The BESS power injection is defined as:

$$P_t^{\mathrm{bess}}=P_t^{\mathrm{dis}}-P_t^{\mathrm{ch}}$$

(10)

and the energy evolution of the BESS is given by:

$$E_t^{\mathrm{bess}}=E_{t-1}^{\mathrm{bess}}+{\eta }^{\mathrm{ch}}{P}_t^{\mathrm{ch}}\Delta t-{\displaystyle \frac{1}{{\eta }^{\mathrm{dis}}}}{P}_t^{\mathrm{dis}}\Delta t$$

(11)

and the operational constraints of the BESS are defined as:

$$\begin{array}{c}\hfill P_{\min }^{\mathrm{bess}}\le P_t^{\mathrm{bess}}\le P_{\max }^{\mathrm{bess}}\end{array}$$

(12)

$$\begin{array}{c}\hfill E_{\min }^{\mathrm{bess}}\le E_t^{\mathrm{bess}}\le E_{\max }^{\mathrm{bess}}\end{array}$$

(13)

Additionally, the initial and final energy conditions are imposed as:

$$E_0^{\mathrm{bess}}=E_{23}^{\mathrm{bess}}$$

(14)

In this work, ${\eta }^{\mathrm{ch}}={\eta }^{\mathrm{dis}}=1$ are assumed in order to isolate the impact of the objective functions on ramp mitigation.This assumption was adopted to isolate the effect of the proposed dispatch objective functions under identical BESS operating conditions. However, practical BESS units operate with non-ideal efficiencies, and their round-trip efficiency is typically lower than 100%.

Furthermore, it is focused on the technical comparison of three BESS dispatch objectives over a single 24-h operating horizon. Accordingly, battery degradation, calendar aging, cycle-dependent aging, state-of- health evolution, and lifetime constraints are not included in the reported optimization model. This assumption allows the effect of each objective function on net-load ramp mitigation to be evaluated under the same BESS power and energy limits. However, the energy throughput and the number of equivalent full cycles may differ among the DER, GEN, and FLAT strategies. Therefore, the strategy that achieves the greatest ramp reduction may not necessarily provide the lowest long-term degradation or operating cost. The results presented here should consequently be interpreted as a technical one-day comparison of ramp-mitigation performance, rather than as a long-term techno-economic sizing or lifetime dispatch assessment.

4.4. Optimal Solution Approach

The optimization is implemented in Python using the SLSQP method from the scipy.optimize.minimize function with explicitly defined bounds and convergence tolerances ($f_{tol}=1e^{-6},ma{x}_{iter}=1000$). The formulation is fully vectorized to ensure reproducibility. Depending on the scenario analyzed, one of the previous objective functions is selected, and the solution provides the optimal hourly dispatch of the BESS while satisfying all operational constraints.

The optimization was initialized with a zero-vector guess for the decision variables, i.e., $P_{\mathrm{t}}^{bess}=0$, $\forall t=0,\dots ,23,$ at the start of the SLSQP iterations. This choice represents a neutral starting point in which no BESS charging or discharging is assumed a priori.

4.5. Energetic and Electrical Integrated Solution

The proposed optimal BESS dispatch formulation has a previous generator unit commitment solution as a base scenario, where it is added the BESS dispatches obtained through the proposed optimization problem and the system is minimally redispatched aiming to meet the BESS active power injection or consumption.

At this point, the system redispatch can enable more renewable resources to reduce curtailment and can disable thermal power plant to supply peak demand, since it will be done through BESS.

After the energetic solution, the alternating procedure is started, where multiple power flow executions are performed, aiming to evaluate the substation voltages and equipment loading. If voltage or loading levels are violated in any scenario, an optimal power flow can be performed to meet such limits, using a minimal redispatch objective function.

Thus, by performing all the aforementioned steps, the optimal operation of the electrical system can be obtained.

4.6. BESS Cycling Metrics

This part defines two auxiliary metrics used to characterize the cycling behavior of the BESS under each dispatch strategy: the energy throughput and the equivalent number of full cycles (EFC).

The energy throughput quantifies the total energy processed by the BESS through charging and discharging over the 24-h horizon, and is defined as

$$E_{\mathrm{throughput}}={\displaystyle \sum _{t=0}^N}\left|P_t^{\mathrm{bess}}\right|\Delta t,$$

(15)

where N is the last analyzed hour (which is 23 in this work), $P_t^{\mathrm{bess}}$ is the BESS power dispatch at hour t and $\Delta t$ is the duration of the dispatch interval.

Based on the energy throughput, the equivalent number of full cycles is computed as

$$N_{\mathrm{EFC}}={\displaystyle \frac{E_{\mathrm{throughput}}}{2E_{\mathrm{bess}}^{\max }}},$$

(16)

where the factor of two accounts for the charging and discharging energy associated with one complete cycle.

These two metrics, together with the maximum evening ramp, the ramp reduction relative to the No-BESS case, and the computational performance indicators, are reported in Table 5.

5. Analysis and Results

The data used in this study are based on real operational measurements from the BIPS, obtained from the ONS [35]. The dataset consists of a 24-h load profile with hourly resolution.

The analysis was structured into two main components: energy evaluation and electrical assessment of BESS deployment in the state of Minas Gerais, Brazil. This system configuration is shown in Figure 3, which includes PV generation, BESS, and transmission lines of different voltage levels.

The BESS operational parameters adopted in this study are explicitly defined to ensure full reproducibility. The maximum charging and discharging power limits are given by $P_{\max }^{\mathrm{bess}}=2400$ MW. The energy storage limits are defined as $E_{\min }^{\mathrm{bess}}=0$ and $E_{\max }^{\mathrm{bess}}=4800$ MWh. The initial and terminal energy conditions are imposed as $E_0^{\mathrm{bess}}=0$ and $E_T^{\mathrm{bess}}=0$ ensuring a daily cyclic operation of the storage system. The optimization problem is solved over a 24-h horizon ($t=0,\dots ,23$) using the SLSQP solver. The decision variables correspond to the hourly BESS dispatch profile.

It is worth noting that the BESS power and energy capacities used in this study were defined as case-study assumptions. The maximum BESS power was set to 50% of the installed PV capacity considered in the Minas Gerais system. This ratio was selected to provide a common storage resource for comparing the DER, GEN, and FLAT dispatch objectives. It is not intended to represent a universal BESS-to-PV sizing rule or the outcome of a techno-economic sizing optimization.

Two objective functions were implemented: (i) derivative compensation and (ii) variable generation compensation. A binary flag is used to select the objective function during the optimization process.

5.1. Energy Evaluation

This part presents the energy evaluation of BESS aimed at mitigating the duck curve introduced by the increasing penetration of PV generation.

5.1.1. Load, DG and MMDG

To conduct this part, real-life load and generation data for Minas Gerais were retrieved from the Open Data Portal of the ONS. The month of June of 2024 was selected as the reference period to reflect the most recent available scenario at the time of the study here presented.

From this dataset, only weekdays were considered, and average hourly curves were computed for the global load, DG, and MMDG. These averages were used to derive two net load curves: (i) the net load considering only DG (defined as the difference between the global load and DG), and (ii) the net load considering both DG and MMDG (defined as the difference between the global load and the sum of DG and MMDG). These curves are illustrated in Figure 4.

One should notice that the DG curve exhibits a relatively uniform profile throughout the day during the evaluated period. This behavior results from the fact that DG includes all generation units connected to the distribution network that are not operated by the ONS, such as thermal power plants, especially biomass in the region, small hydroelectric plants, PV systems, and others. In contrast, the MMDG is predominantly composed of PV installations, which explains the pronounced diurnal variation characteristic of solar generation observed in its profile.

For clarity, all curves presented in Figure 4, Figure 6, Figure 8, Figure 10 and Figure 12 represent load quantities, expressed in MW. “Total system load” denotes the aggregate electrical load before accounting for generation resources. The generation sources indicated in parentheses identify the resources subtracted from the total load to obtain the corresponding net-load profile. Thus, the labels do not represent generation curves; they indicate the composition of the net load.

The net-load profiles are defined as follows:

$$\begin{array}{cc}\hfill & L_t^{DG}=L_t-P_t^{DG},\hfill \\ \hfill & L_t^{DG+MMDG}=L_t-P_t^{DG}-P_t^{MMDG},\hfill \\ \hfill & L_t^{DG+MMDG+PV}=L_t-P_t^{DG}-P_t^{MMDG}-P_t^{PV},\hfill \\ \hfill & L_t^{DG+MMDG+PV+BESS}=L_t-P_t^{DG}-P_t^{MMDG}-P_t^{PV}-P_t^{BESS},\hfill \end{array}$$

where, $L_t$ denotes the total system load, while $P_t^{DG}$, $P_t^{MMDG}$, and $P_t^{PV}$ denote the power generated by DG, MMDG, and centralized PV, respectively, and $P_t^{BESS}$ denotes the BESS power injection, with positive values representing discharge and negative values representing charging.

5.1.2. Centralized PV Generation

To evaluate the impact of centralized PV plants on the load curve, an estimate of approximately 4800 MW of installed centralized PV capacity in Minas Gerais for 2024 was obtained from the ONS Open Data Portal [35].

To determine the representative hourly capacity factor curve for these plants, verified historical data from centralized PV installations in Minas Gerais provided by ONS and covering the period from 2018 to 2022 were analyzed. The hourly capacity factor curve was derived using the 50th percentile (P50) of the observed data, as shown in Figure 5 [35].

The P50 represents the median value of the dataset, meaning that half of the historical observations fall below this value and half above. In the context of PV generation, it provides a statistically balanced estimate of the expected generation profile, avoiding overestimation due to outliers or extreme conditions. This curve is applied to the estimated 4800 MW of installed capacity to obtain the corresponding hourly generation profile of the centralized PV plants.

The PV generation profile used in this study represents a deterministic operating condition. Although this approach provides a representative chronological profile for comparing the dispatch strategies, it does not explicitly represent cloud-induced fluctuations, forecast errors, or rapid changes in solar irradiance. However, these details are included in the historical data utilized for the simulations.

5.1.3. Impact of PV Generation on the Load Curve and Application of BESS

The impact of centralized PV generation on the load curve was evaluated by subtracting this generation from the net curve that already includes distributed generation and micro-mini distributed generation, resulting in the curve labeled “Verified Load (DG + MMDG + PV)” as shown in Figure 6.

The figure shows the considerable influence of MMDG and PV on the intraday load curve, particularly during daylight hours. This variation presents several operational challenges for the system operator, such as the need to curtail generation during periods of low demand coinciding with peak solar output, and the difficulty in managing steep ramping requirements during transitions from solar to conventional generation sources.

A promising strategy to address these challenges is the integration of BESS. These systems can be co-located with centralized PV plants to mitigate the variability introduced by solar generation. By storing excess energy during periods of low demand and releasing it during peak hours or rapid ramping events, BESS enhances reliability and flexibility.

5.1.4. Application of BESS for Smoothing Load Variations Caused by the Duck Curve

In this case, it was assumed that each centralized PV plant in Minas Gerais would be coupled with a dedicated BESS. The sizing of each storage system was based on the premise that the usable power of the BESS would correspond to 50% of the installed capacity of the associated PV plant. Based on this criterion, the total usable power of all BESS installations was estimated at approximately 2400 MW. Additionally, the maximum energy capacity of each BESS was set to twice its usable power (4800 MWh), following established technical guidelines. It is important to note that, irrespective of the location and size of the individual systems, this evaluation assumes uniform operational behavior across all BESS units, with energy dispatches scaled proportionally to their respective usable power.

To evaluate the role of BESS in mitigating the duck curve phenomenon, the three dispatch strategies presented in Section 2 and formulated in Section 3 were analyzed: (1) load derivative compensation, (2) variable generation compensation and (3) average load tracking.

The results of the derivative compensation algorithm, including the hourly dispatch and storage energy (or State-of-Charge, SoC) of the BESS fleet, are presented in Figure 7, while its impact on the daily load profile is shown in Figure 8. The outcomes for the variable generation compensation algorithm are depicted in Figure 9, with the corresponding influence on the load curve illustrated in Figure 10. Likewise, the results for the average load tracking algorithm are provided in Figure 11, and its effect on the system load is presented in Figure 12.

From the results of the three proposed strategies, it is evident that each offers energetically advantageous behavior for the real-time operation of the Brazilian electrical system, with specific operational characteristics.

The variable generation compensation and average load tracking strategies produced similar system-level impacts. The generation compensation algorithm aligned the BESS dispatch closely with the variability of the net distributed generation load. Meanwhile, the average load tracking strategy aimed to flatten the net load by equalizing energy use and injection throughout the day. Both approaches effectively increased consumption during periods of high PV generation and reduced it during peak demand, enhancing energy efficiency and system balance.

Regarding the system load ramping from 4 p.m. to 8 p.m. (period of four hours), originally, the system was presenting a ramp of 1125 MW/h. In this same period of time, the variable generation compensation strategy presented a ramp of 930 MW/h, the average load tracking strategy presented a ramp of 883 MW/h, and the load derivative compensation proposal reached a ramp of 725 MW/h.

The most effective algorithm for addressing sharp electrical grid load variations caused by the significant integration of PV generation in Minas Gerais was the load derivative compensation strategy. It significantly reduced load ramping, for instance, around 35% reduction in the 4 p.m. to 8 p.m. ramp compared to the original scenario. Additionally, while less impactful on peak shaving than the other strategies, it still managed to reduce peak demand by approximately 600 MW. It also modestly increased demand during PV oversupply periods, contributing to spillover reduction and helping to mitigate energy imbalances between generation and demand.

One performs a sensibility analysis regarding the scale of BESS capacity and its impact on the load ramp between 4 p.m. and 8 p.m.

Table 1. Sensitivity analysis regarding BESS scale.

Dispatch Methodology	BESS	BESS	BESS	BESS	BESS
	0%	25%	50%	75%	100%
Original System (MW/h)	1125	-	-	-	-
Generation Compensation (MW/h)	-	1030	930	830	661
Derivative Compensation (MW/h)	-	888	725	609	499
Average Load Tracking (MW/h)	-	988	883	783	668

presents the load ramps obtained through the methodologies, considering a BESS capacity of 25%, 50%, 75% e 100%, regarding the PV installed capacity of Minas Gerais State.

5.2. Electrical Evaluation

To assess the electrical impacts of using BESS for mitigating load curve variations induced by PV generation, a quasi-static analysis approach was adopted. This methodology is suitable given the nature of the phenomena under study, which evolve over hourly time intervals rather than on a sub-second timescale. Consequently, power flow analyses were conducted for each hour of the day, resulting in 24 distinct evaluations. This approach mirrors the granularity of the energy evaluation and enables a comprehensive understanding of system behavior under varying operational conditions throughout the day.

The electrical analysis presented in this section is grounded in the outcomes of the energy evaluation, with a particular focus on the derivative-based algorithm, which demonstrated superior performance in reducing ramping effects and balancing the load curve.

5.2.1. Power Flow Database

To ensure consistency between the electrical evaluation and the previously conducted energy analysis, the weekday scenarios were adopted from the ONS studies for the June 2024 Short-Term Planning monthly horizon. These scenarios were obtained from the SINTEGRE platform [35].

The weekday cases are subdivided into three load levels: light, median, and heavy, and each level has specific details regarding the expected load per bus, as shown in

Table 2. Bus level load forecasts on weekday.

Load Level	Bus-Level Load Forecast for Weekday
Light	Minimum monthly global load forecast by the agent
	for the period between 00:00 and 07:00.
Median	Maximum monthly global load forecast by the agent
	for the periods between 07:00–18:00 and 21:00–24:00.
Heavy	Maximum monthly global load forecast by
	for the period between 18:00 and 21:00.

.

5.2.2. Load, DG and MMDG

To incorporate the values defined in the energy evaluation of weekday load curves (as illustrated in Figure 4) into the power flow cases, the hourly data for system load, DG, and MMDG were segmented according to the load level classifications presented in Table 2. The results of this segmentation are summarized in

Table 3. Hourly curve of total load and the sum of DG and MMDG with the corresponding load level.

Load Level	Hour	Total Load	DG + MMDG
		(MW)	(MW)
Light	1	8089.51	737.73
Light	2	7701.05	734.46
Light	3	7487.39	732.99
Light	4	7360.31	731.52
Light	5	7325.05	730.76
Light	6	7442.98	728.92
Light	7	7748.50	767.04
Median	8	7916.89	1112.20
Median	9	8197.49	1696.20
Median	10	8348.71	2230.09
Median	11	8502.89	2686.69
Median	12	8698.48	2920.27
Median	13	8682.40	2983.70
Median	14	8646.43	2846.12
Median	15	8821.51	2584.74
Median	16	8861.41	2205.34
Median	17	8956.37	1719.02
Median	18	8718.27	1188.15
Heavy	19	8905.22	822.87
Heavy	20	9335.31	738.87
Heavy	21	9381.15	740.14
Median	22	9512.34	740.64
Median	23	9157.28	740.08
Median	24	8639.79	738.58

.

To accurately represent the hourly values of total system load and the combined output of DG and MMDG, power flow cases were used to identify the buses associated with distribution networks, consumers connected to the transmission network, and self-generators located in the state of Minas Gerais. These buses correspond to the energy data used in the analysis, with a total number of 591 components.

For each of the three load levels, light, median, and heavy; the proportions of both total load and the sum of DG and MMDG at these buses were calculated relative to the values from the corresponding energy evaluation cases. These proportions were used to derive the new hourly values of total load and generation (DG + MMDG) for each bus by multiplying the respective proportion by the total hourly values from the energy analysis. Importantly, the adjusted hourly values adhered to the load level classifications to maintain consistency.

5.2.3. Centralized PV Generation with BESS

An evaluation of the ONS Short-Term Planning case studies enabled the identification of the approximate installed capacity of centralized PV plants at each connection point, totaling around 4800 MW. This value aligns with the energy evaluation previously conducted. Additionally, the maximum power capacity of the BESS was defined as 50% of the installed PV capacity at each connection point. The installed BESS are shown in Figure 3, and the corresponding data are listed in

Table 4. Installed capacity of PV plants and BESS at each connection point.

Connection Point	Voltage	PV Capacity	BESS Power	BESS Energy
	(kV)	(MW)	(MW)	(MWh)
Araxá 4	138	90.00	45.00	90.00
Arinos 2	500	299.20	149.60	299.20
Jaíba	230	1181.31	590.65	1181.31
Jaíba	138	128.50	64.25	128.50
Janaúba 3	500	1022.70	511.35	1022.70
Paracatu 4	138	694.80	347.40	694.80
Pirapora 2	345	495.04	247.52	495.04
Pirapora 2	138	320.99	160.49	320.99
Três Marias	138	71.40	35.70	71.40
Várzea da Palma 4	345	500.00	250.00	500.00

.

The dispatch levels used in the electrical simulations for each centralized PV plant and each hourly case were derived from the hourly capacity factors obtained in the energy assessment, as shown in Figure 5. These factors resulted in proportionally allocated dispatches per connection point, which are illustrated in Figure 13.

In the implementation of BESS in the power flow simulations, these systems were modeled as an active power injection (such as a generator, when discharging) or absorption (such as a load, when charging), with no reactive power supplied. Each BESS unit was connected to its respective connection point via a low-impedance transmission line. The BESS dispatches were based on the values presented in Figure 9, which represent the results of the load derivative algorithm applied in the energy evaluation. These dispatches and consequently the SoC were scaled proportionally for each connection point, according to the effective power of the respective BESS unit, as shown in Figure 14 and Figure 15.

5.2.4. Impact of Using BESS on Electrical Evaluation

To assess the impact of BESS on the electrical network, 24 case studies were conducted, each representing different hourly load scenarios. Voltage control resources within the modeled network were employed to ensure that all buses of the basic grid remained within acceptable operational limits.

Figure 16 presents the loadings of main extra-high-voltage transmission lines under normal operating conditions across all hourly scenarios. These transmission lines represent critical corridors for energy transfer within Minas Gerais state. Figure 17 shows the corresponding hourly voltage profiles at relevant substations across the state. It can be observed that the daily voltage profile remained reasonably flat.

The inclusion of BESS in the simulations enabled the elimination of approximately 600 MW of thermal generation dispatch during the nighttime peak, which had previously been required in the heavy load scenarios of the original ONS Short-Term Planning studies. This modification not only reduced operational costs but also contributed to a cleaner energy mix. Furthermore, the analysis revealed a notable improvement in the utilization of renewable energy sources from the Northeast region. During periods of high PV output, an increase of 500 MW in the exported active power from the Northeast was observed in the median load scenario compared to the original ONS case. This was made possible by the additional demand from BESS charging during solar peak hours, which facilitated greater renewable integration.

To provide a quantitative comparison focused on the critical evening transition, the maximum absolute hourly ramp was calculated between 4 p.m. and 8 p.m. interval (This interval targets the peak hourly ramp and is therefore distinct from the four-hour average ramp (4 p.m.–8 p.m) reported in Section 5.1). The reduction obtained with each BESS strategy was computed relative to the corresponding original system (No-BESS condition). In addition to the ramping performance, the computational time, number of SLSQP iterations, number of objective-function evaluations, and convergence status were recorded for each optimization run.

As shown in

Table 5. Evening-ramp mitigation, computational performance, and BESS cycling metrics for the evaluated dispatch strategies.

Strategy	Max. Evening Ramp (MW/h)	Reduction vs. No BESS (%)	Execution Time (s)	SLSQP Iterations	Function Evaluations	Convergence	Energy Throughput (MWh)	EFC
No BESS	106.00	0.00	—	—	—	—	—	—
DER	33.95	67.97	0.0176	8	216	Successful	1489.65	1.0640
GEN	72.36	31.74	0.0175	12	315	Successful	1401.50	1.0011
FLAT	100.28	5.40	0.0246	13	358	Successful	1400.00	1.0000

, the DER strategy achieved the largest reduction in the maximum evening ramp, decreasing it from 106.00 MW/h in the No-BESS case to 33.95 MW/h, corresponding to a reduction of 67.97%. The GEN and FLAT strategies achieved reductions of 31.74% and 5.40%, respectively. Therefore, for the evaluated Minas Gerais case and the selected evening interval, derivative compensation was the most effective strategy for mitigating the critical net-load ramp.

On the other hand, battery degradation was not explicitly represented as an objective-function term or as an operational constraint in the present study. In practice, degradation is influenced by cumulative energy throughput, equivalent full cycles, depth of discharge, and calendar aging. A degradation-related cost could be incorporated into the optimization through a term associated with energy throughput or cycle aging. Alternatively, the model could include constraints on cumulative throughput, the number of equivalent full cycles, or the minimum allowable state of health. These extensions would allow the optimization to represent the trade-off between ramp mitigation, energy losses, operating cost, and battery lifetime.

6. Conclusions

This study evaluated the integration of BESS into the BIPS, focusing on the state of Minas Gerais, which is the region with the highest penetration of solar PV generation and, consequently, the largest electrical grid load reduction in the country, resulting in a pronounced duck curve. The analysis combined both energy and electrical assessments, and the results confirm that BESS effectively mitigate the impacts of high PV penetration, particularly in terms of smoothing the verified load curve and enhancing the operational flexibility of the system.

Three dispatch strategies were proposed; while all strategies contributed to reducing PV curtailments and supporting peak load, they presented distinct characteristics. The average load and generation compensation strategies provided a more uniform energy balance throughout the day. However, the load derivative compensation strategy, focused on minimizing hourly load ramps, achieved the most favorable results in smoothing the duck curve. This strategy significantly reduced ramping events, eliminated the need for thermal generation dispatch during the night peak (saving around 600 MW), and enabled an increase of approximately 500 MW in renewable sources utilization from the Northeast region during solar peak hours.

Additionally, the results of the electrical assessment reinforced the benefits observed in the energy analysis. The operation adjusted by the derivative compensation strategy proved to be technically feasible, with no overloads or voltage violations observed in the network equipment under normal operating conditions.

As future work, the integration of energy and electrical analyses into a unified co-optimization framework could be investigated, improving the consistency between dispatch strategies and network constraints. The main idea of this future research is to deal with the complex integration between the energetic solution and the electrical power flow solution, trying to obtain an automated computational tool. Another relevant direction is the development of an algorithm for the optimal allocation of BESS across the network. Additionally, the present analysis relies on static, historical weekday profiles rather than real-time data. Extending the proposed framework to a real-time or rolling-horizon setting, incorporating short-term load and PV forecasts, represents an important direction for future work, and would allow a more direct assessment of the practical value of the three dispatch strategies under operational uncertainty.