Economic impact assessment of hydraulic operating constraints in brazilian hydropower plants

Context

The Brazilian power matrix is predominantly composed of hydroelectric plants, which are responsible for most of the country’s electricity generation. The National Electric System Operator (ONS) coordinates and controls the operation of the National Interconnected System (SIN).

To ensure safe and economic operation, it is essential to consider the operational limitations of these plants, both in real-time operation and in the representation of constraints within electrical-energy models for planning and operation scheduling.

Among the examples of hydroelectric plant variables subject to Hydraulic Operating Constraints (COPHIs, from the Portuguese acronym) are water levels (upstream and downstream) and flow rates (outflow, turbined, and spilled), for which minimum and/or maximum value limits may be established, as well as rates of increase or decrease. COPHIs are necessary to ensure:

• Multiple uses of water, compliance with federal and state regulations, and adherence to demands related to socio-environmental issues.
• Adherence to the operating guidelines of each reservoir, which are declared to promote better management of SIN’s hydro-energy resources, including definitions from the National Water and Sanitation Agency (ANA) resolutions.
• Safety in the execution of activities and services that require the control of hydraulic variables at the sites.
• The performance of interventions on plant structures (spillways, powerhouses, etc.) that result in some type of restriction on hydraulic variables.

The occurrences of the main upstream and downstream hydraulic operating constraints registered and in effect were observed in the ONS “Hydraulic Restriction Update Request Form” (FSARH acronym in Portuguese) extract conducted on November 28, 2024.

	Maximum Water Elevation	Minimum Water Elevation	Water Elevation decrease rate	Water Elevation increase rate
Upstream Constraints	50%	40%	6%	3%

	Max. Water Outflow	Min. Water Outflow	Outflow increase rate	Outflow decrease rate	Ecological minimum Flow	Other
Downstream Constraints	31%	33%	17%	8%	6%	5%

COPHIs are established in Submodule 4.7 of the ONS Grid Procedures (“Inclusion and updating of hydraulic operating constraints for hydroelectric projects”). They are highly relevant to the planning, scheduling, and real-time operation processes, given their implications for hydroelectric plant operation.

Recent years have seen a significant increase in COPHIs registered with the ONS to ensure the multiple uses of water, compliance with federal and state regulations, and adherence to socio-environmental and energy-related demands. In aggregate, these COPHIs reduce the operational flexibility of hydroelectric plants, which has economic implications for system operation. For example, it may be necessary to dispatch a gas-fired thermal plant to compensate for short-term variations in renewable sources or to keep it synchronized during periods when hydroelectric plants are constrained.

In 2025, the ONS contracted PSR, seeking improvements in the analysis, evaluation, acceptance, and management process of the COPHI constraints that comprise the SIN. One of the activities developed by PSR in this project was a methodology to assess the economic impact of each COPHI without judging its necessity. However, by applying the methodology, it is possible to establish a ranking of which COPHIs are the costliest. The expectation is that this exercise will guide more objective discussions on whether it is possible to act to reduce or make these COPHIs more flexible.

A little over a decade ago, during a prolonged drought that particularly affected the São Francisco River, it became clear that it was possible to reduce the downstream flow of some plants, such as the Três Marias HPP. This flow was being used to maintain a specific level required for urban water supply intake. The installation of floating pumps—a simple project with a relatively small budget—allowed for a reduction in these flows, preserving water in the plant’s reservoir to prevent it from emptying. This measure provided a benefit far exceeding the cost of the adaptation.

The ranking of the highest-impact COPHIs can help guide where to look first. The Três Marias case illustrates how targeted interventions outside the traditional power-sector scope may yield benefits far greater than their cost, even if there is no guarantee that such success will always be replicated.

This article discusses the project component where PSR develops a methodology for evaluating COPHIs by considering their consequences and repercussions under various operating conditions of the National Interconnected System (SIN). The methodology is grounded in the concept of marginal costs.

Marginal costs

Marginal costs represent the rate of change of the objective function with respect to infinitesimal variations in the independent terms of the constraints (i.e., the Right-Hand Side or RHS of the constraints). In practical terms, they indicate how much the optimal value of the objective function (e.g., total cost or profit) changes when the availability of a resource (such as generation capacity, flow constraints, etc.) is slightly increased or decreased.

In the context of power system optimization problems, marginal costs are widely used to evaluate the impact of constraints. In a hydroelectric plant dispatch problem, for example, the marginal cost of a minimum flow constraint indicates how much the total system cost would increase if the minimum flow were raised. Constraints with high marginal costs are identified as critical to the system and relaxing them can bring significant benefits. Thus, marginal costs are useful for identifying bottlenecks and can guide decisions on where to expand plant capacity or transmission lines.

The marginal cost of the k^th COPHI, at a stage  and a hydrological scenario , expressed by , measures the variation in the SIN operating cost (in R$) resulting from a change in the COPHI. Some illustrative examples are presented next.

Impact Assessment of Permanent COPHIs

There are currently about 600 active COPHIs, the majority of which are permanent constraints. One possible approach is to evaluate the impact of each COPHI separately by deactivating them one by one. To do this, the difference  is calculated between the operating cost of the SIN simulation considering all active COPHIs (base case) and the cost of the simulation when the ^th COPHI is removed.

If the operating cost of the SIN with all COPHIs “deactivated” is , and if  is the base case cost includes all COPHIs, then the economic impact of the ^th COPHI is calculated as:

$$\text{Impacto}k = (Z – Z_0) \frac{\Delta Z_k}{\sum{j=1}^{K} \Delta Z_j}$$

This cost allocation method, known as “last-addition”, or the “first-addition” alternative (which measures the difference between the SIN [National Interconnected System] cost when only the ^th COPHI is activated and the cost when no COPHI is active), has the disadvantage of requiring  simulations, which is computationally expensive. Furthermore, it covers only one of the many possible combinations of COPHIs that may be activated or deactivated. Thus, strictly speaking, it would be necessary to evaluate the contribution of each COPHI for all possible combinations of which ones are active.

The Shapley Allocation Method, in turn, is used to distribute costs or benefits among participants fairly, considering each one’s contribution to the total result. This method is applied in various fields, such as economics, data science, and resource management. Its basic concepts involve:

• Cooperative Game: when agents cooperate to obtain a collective benefit.
• Characteristic Function (v): represents the value for any subset of players. For a set of players, v(S) is the value.
• Marginal Contribution: the difference a player i (COPHI) adds when entering a group S, expressed by: v(S ∪ {i}) − v(S)

For a player i, the Shapley value φᵢ is given by:

$$\phi_i = \sum_{S \subseteq N \setminus {i}} \frac{|S|!(n – |S| – 1)!}{n!} [v(S \cup {i}) – v(S)]$$

Where:

• N corresponds to the set of all players.
• S is a subset of participants excluding i.
• v(S) is the value (or cost) of coalition S.
• v(S ∪ {i}) is the value of the coalition formed by adding player i to S.
• |S| corresponds to the size of subset S.
• The fraction |S|!(n − |S| − 1)! / n! is the weight given to all formation orders of the groups.

For example, take three power plants (A, B, C) generating energy. The Shapley method calculates how much each plant should be compensated based on its contribution to all possible combinations (A alone, A+B, A+C, A+B+C, etc.).

The practical limitation of applying this method is the number of combinations that would need to be tested for a set of hundreds of COPHIs. To bypass this issue, an extension of this method is suggested: the Aumann-Shapley (A-S) method, usually used in situations where factors are not discrete, but continuous. It is especially useful for cost allocation problems in systems such as power grids or supply chains.

While the Shapley method deals with individual elements, the Aumann-Shapley method deals with continuous fractions of contribution. Instead of allocating costs to individual plants, it allocates costs to shared resources. Its operation is related to:

1. Integration of Marginal Contributions: The method calculates the marginal contribution of each continuous fraction (e.g., each MWh generated) and integrates these contributions along a continuous path.
2. Integration Path: The allocation is made along a path from “no contribution” to “total contribution.”

For a continuous resource , the allocated cost φ(x) is given by:

$$\Phi(x) = \int_0^1 \frac{\partial C(t \cdot x)}{\partial x} dt$$

Where:

• C(t · x) is the total cost when resource x is scaled by a factor t.
• ∂C(t · x) / ∂x is the marginal cost of resource x at point t · x.

It is suggested to apply the A-S method to evaluate the impact of COPHIs:

• The system operating cost is the function to be allocated.
• The objective is to quantify the marginal impact of each COPHI on the total operating cost of the SIN, considering their interdependence.

O método A-S permite uma alocação justa e eficiente. Em sua versão discreta por fração, todas as COPHIs são escaladas conjuntamente de seu valor atual para um valor mínimo (para restrições de limite inferior — “maior ou igual a”) ou um valor máximo (para restrições de limite superior — “menor ou igual a”). Passos discretos são usados para medir o incremento de custo do SIN. O impacto marginal de cada restrição é obtido por meio dos custos marginais. O algoritmo A-S possui quatro etapas principais:

Step 1: Define discrete steps

• Choose N (e.g., N = 10).
• Define aₙ = n/N, where n = 0, …, N, to scale all types of COPHIs, for all plants, stages, and hydrological scenarios. The increment is 1/N.
  • When aₙ = 1, a COPHI k receives its original value.
  • When aₙ = 0, the COPHI k is relaxed.

Example of operating flow limits for COPHI k:

The premises for the minimum and maximum constraint values must be carefully defined. The relaxed minimum flow could be, for example, the historical minimum daily flow of a plant, while the relaxed maximum flow can be the value used in calculating the flood control volume. This definition is crucial for a more realistic assessment of the economic impact of COPHIs.

Step 2: Simulate the system with tiered constraints

For each step n = 0, …, N:

• Scale all COPHIs for all plants by the factor αₙ.
• Solve the SIN (National Interconnected System) optimization problem using SDDP for medium-term operation planning for these COPHIs that includes different generation technologies (hydropower plants, thermal power, solar power, wind power), the transmission network and storage technologies, such as pumped hydro and battery.
• • Calculate the system operating cost zₙ and the incremental cost Δzₙ = zₙ − zₙ₋₁.

Step 3: Calculate the individual contribution of each COPHI  to the cost increment.

Seja Δk,n = ΣₜΣₛ (πk,t,sⁿ · COPHIkⁿ − πk,t,sⁿ⁻¹ · COPHIkⁿ⁻¹) e ΔIk,n = Δzₙ · Δk,n / ΣₖΔk,n

O impacto econômico final de cada COPHI k é Ik = Σₙ ΔIk,n

Etapa 4: Gerar o Ranking das Restrições

Ordenar as COPHIs por seus impactos em valores absolutos. O ranking deve fornecer uma desagregação por tipo de COPHI (e.g., vazão mínima, vazão máxima, etc.) e por UHE (Usina Hidrelétrica).

Exercício Numérico

O procedimento Aumann-Shapley (A-S) para avaliação do impacto econômico das COPHIs foi inicialmente testado com o SDDP, devido à facilidade de implementação pela PSR e à disponibilidade dos componentes necessários (modelagem de restrições e preparação dos custos marginais associados). As seguintes premissas foram utilizadas:

• Deck do PMO de maio de 2025 (Programa Mensal de Operação), com horizonte de 5 anos e 200 cenários sintéticos de afluências gerados pelo modelo PAR(p) do SDDP, sem uso de reamostragem de cenários na política.
• O A-S foi implementado em código Julia para alocar impactos econômicos usando 10 pontos (0%, 10%, 20%, …, 100%), ou seja, 11 rodadas do SDDP com cálculo de política e simulação para cada caso utilizando os 200 cenários de hidrologia.
• Foram adotados valores para cada COPHI, conforme apresentado na Tabela 1.

COPHI	Premissa para o caso 100% relaxado
Vazão mínima	Vazão total mínima diária histórica
Vazão máxima	Vazão utilizada no cálculo do volume de controle de cheias, se aplicável. Caso contrário, o mínimo entre 2x a vazão máxima e a vazão de projeto do vertedouro.
Volume mín./máx.	Volume operativo mínimo/máximo
Rampas mín./máx. de nível	Zero
Rampas mín./máx. de vazão	Zero

Tabela 1 – Premissas para os valores das COPHIs no caso relaxado

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Results

The proposed procedure was executed, consisting of 11 SDDP runs (for points ranging from 0% to 100% in 10% steps), the May 2025 PMO deck with a 5-year horizon, 200 hydrological scenarios, and the inclusion (for now) of the previously presented COPHIs:

1. Minimum outflows
2. Minimum water level per plant
3. Minimum turbined flow
4. Minimum spilled flow

The following figure presents the allocation of the increase in operating costs in the 5-year horizon in proportion to the product of the marginal cost and the COPHI increment for the 15 plants with the highest values obtained using this methodology.

Among the results presented, it is interesting to highlight the costs of the following plants:

• Pimental: presents the highest value (not shown to avoid distorting the others), as it represents the effect of the outflow required to meet the hydrograph; this water is no longer turbined at Belo Monte (a plant with ~90 m of head) and is instead turbined at Pimental (~20 m of head) or spilled. The average impact is R$ 850 million. The following figure shows the economic impact value for each simulated scenario. This helps understand the dispersion of the economic impact results. The Pimental site, for instance, has a minimum environmental flow rate (monthly values) that must be kept in the Xingu River, thus does not generate electricity. Depending on the SIN supply condition, this energy loss can have more limited or more severe effects.
• Porto Primavera and Xingó: consider their structural minimum outflow values, which are 4,600 m³/s and 800 m³/s, respectively.
• Jupiá: reflects minimum flow constraints.
• Barra Bonita, Três Irmãos, and Ilha Solteira: reflect minimum level constraints for the maintenance of the Tietê-Paraná waterway operations.
• Sobradinho and Itaparica: reflect minimum flow constraints and the impact of operating guidelines.
• Funil (Paraíba do Sul): reflects the minimum storage level of 30% defined in Joint Resolution ANA/INEA/DAE/IGAM No. 1382/2015.
• Machadinho: reflects the minimum turbined flow conditioned on the reservoir level, as stated in FSAR-H 2858/2022, which defines a minimum flow of 295 m³/s when the level is above 469.9 m and 250 m³/s when below that elevation.

Evaluation Criteria and Cost-Benefit Analysis

The ranking of COPHIs by their economic impact, derived from the Aumann-Shapley method, serves as a critical strategic guide. It does more than identify constraints: it helps indicate where the largest potential savings may lie.

While the ranking identifies the costliest constraints, it is important to recognize that “relaxing” a COPHI is not always feasible or appropriate. In practice, the scope for flexibility tends to fall into two categories:

• Inflexible Constraints: Some COPHIs are essential for non-negotiable socio-environmental protections. For example, a minimum flow requirement may be vital to ensure the survival of an endangered and endemic species in a specific river reach. In such cases, the “cost” to the ecosystem of relaxing the constraint would be infinite, making flexibility impossible regardless of the economic impact on the power sector.
• Flexible Constraints via Investment: Many constraints exist to protect specific human activities that can be adapted with targeted investment. A primary example is the Três Marias HPP, where downstream flow was maintained at a high level solely for urban water supply intake. By investing in floating pumps – a simple, low-budget project – the system allowed for reduced flows and preserved reservoir levels, yielding a benefit that far exceeded the adaptation cost.

To move from a theoretical ranking to practical action, a marginal cost-benefit analysis is required. This involves comparing two distinct curves:

• The Benefit Curve: Represents the cumulative reduction in SIN operating costs as a COPHI is incrementally relaxed. This is calculated by simulating the system (e.g., using the SDDP model) under varying constraint limits.
• The Cost Curve: Represents the investment required (structural or non-structural) to enable that flexibility.

ADICIONAR FIGURA

Figure 2 – Cost-benefit curve graph

The optimal degree of flexibility is found at the intersection of these two curves. By applying this logic, COPHIs can be prioritized to guide decision-making considering low-to-moderate cost solutions with rapid or medium-term implementation that provide significant gains. On the other hand, complex structural interventions with high costs and long timeframes may only be justified if the benefit to the SIN is substantial.

This analytical approach is directly relevant to electricity consumers, since any reduction in SIN operating costs ultimately translates into lower tariffs. A significant regulatory hurdle remains, however: the power sector often lacks the legal mandate to fund or execute adaptations outside its immediate domain, even when those interventions would clearly reduce system costs. The Três Marias case illustrates this point well: the installation of floating pumps for urban water intake made it possible to reduce downstream flow requirements, preserve reservoir storage, and generate benefits far greater than the cost of the adaptation. A proactive inter-institutional framework would help enable similar cross-sector investments whenever the benefit to electricity consumers exceeds the cost of the required external intervention.