TOPICS IN POWER SYSTEM GENERATION AND EXPANSION PLANNING
OBJECTIVES
The purpose of this summary is to provide a comprehensive and structured review material for the course EELT7030 – Planning of Power System Operation and Expansion.
This document organizes 60 essential study topics, distributed across six main sections that cover the fundamental pillars of the discipline:
Overview and Structure of the Electric Power Sector
Thermal and Hydraulic Generation
Hydrothermal Planning and Optimization
Reliability and Emerging Energy Sources
Distribution Networks, Microgrids, and Uncertainties
Power System Expansion Planning
In addition, the compendium includes five detailed appendices, which delve into specific methods, tools, and models:
Appendix A: Technical Summary of DESSEM
Appendix B: Strategies of DECOMP and NEWAVE
Appendix C: EPE’s MDI Model
Appendix D: Optimization Methods (LP, MILP, QP, MIQP, NLP, MINLP)
Appendix E: Uncertainties and D-OPF Formulations
Thus, this summary offers a unified and complementary view of the theoretical and practical aspects necessary for a solid understanding of the subject and for good performance in essay-based evaluations.
Happy studying!
OVERVIEW AND STRUCTURE OF THE ELECTRIC POWER SECTOR
FULL COMPETITION
Full competition in the electric power sector refers to a market model in which all consumers—regardless of size—are free to choose their electricity suppliers. In this scenario, the exclusivity of service by the local distribution company is eliminated, allowing direct negotiations between consumers and generators or energy traders. This arrangement aims to increase the sector’s economic efficiency, foster competition, and reduce costs for the end user. However, such competition also requires strong regulatory oversight to ensure fair access to transmission and distribution networks, preventing discriminatory practices. In practical terms, full competition is an important step toward modernizing the market and aligning the sector with international best practices.
SPOT MARKET AND SETTLEMENT PRICE (PLD)
The short-term market, or spot market, is the environment where the settlement of differences between contracted and effectively consumed or generated energy takes place. This market is priced by the Settlement Price for Differences (PLD), calculated weekly based on optimization models that reflect the system’s operating conditions. The PLD seeks to indicate the marginal cost of system operation, considering both hydrological conditions and thermal plant availability. As such, it provides economic signals that influence consumption, contracting, and investment decisions. The PLD is therefore a central mechanism in the operation of the Brazilian electricity market, affecting both free and captive market participants.
FUNCTIONS OF THE NATIONAL SYSTEM OPERATOR (ONS)
The National System Operator (ONS) performs three essential functions: technical coordination, security, and economic optimization.
Technical coordination involves real-time control of generation and transmission, ensuring balance between supply and demand.
System security seeks to reduce the risk of rationing or blackouts through reliability criteria and preventive planning.
Economic optimization aims to dispatch energy resources in a way that minimizes total costs—both present and future—while respecting physical and environmental constraints.
Thus, ONS is a key institution ensuring both the reliability and the economic efficiency of the interconnected Brazilian power system.
RENEWABLE ENERGY VS. CLEAN ENERGY
Although often used interchangeably, renewable energy and clean energy are not identical concepts.
Renewable energy refers to sources that naturally regenerate on human timescales, such as solar, wind, biomass, and hydropower.
Clean energy, on the other hand, refers to the environmental impact of the source, being characterized by very low or zero emissions of pollutants and greenhouse gases.
Consequently, a source may be renewable without being entirely clean—for example, biomass, which, despite being renewable, can emit pollutants. Conversely, nuclear energy is not renewable, but it can be considered clean from the perspective of direct emissions.
ENERGY TRANSITION
Energy transition is the process of transforming the energy matrix by reducing the share of fossil fuels in favor of renewable and low-impact sources. This transformation arises from the need to mitigate climate change, lowering greenhouse gas emissions and promoting long-term sustainability. Moreover, energy transition encompasses technological innovation, new consumption models, and profound socio-economic shifts. It is directly linked to the concept of sustainable development, as it aims to meet current energy needs without compromising the welfare of future generations. In the electric power sector, this transition is expressed through the growing integration of wind, solar, biomass, and green hydrogen resources.
THERMAL AND HYDRAULIC GENERATION
ENERGY TRANSITION
The energy transition is the process of transforming the energy matrix by reducing the participation of fossil fuels in favor of renewable and low-impact sources. This transformation arises from the need to mitigate the effects of climate change, lowering greenhouse gas emissions and promoting greater sustainability. In addition, it involves technological innovation, new consumption models, and deep socioeconomic changes. The concept is directly linked to sustainable development, aiming to meet current energy demands without compromising the welfare of future generations. In the electric sector, the transition is characterized by the growing integration of wind, solar, biomass, and green hydrogen technologies.
ADVANTAGES OF THERMAL POWER PLANTS
Thermal power plants offer several operational advantages. One of the main benefits is the possibility of being located near load centers, reducing transmission losses and improving system reliability. They also have relatively short construction times, allowing a faster response to demand growth. Another key aspect is their independence from hydrological conditions, making them crucial during drought periods. Furthermore, thermal plants provide operational flexibility, complementing renewable generation when its availability is low. These factors justify their strategic importance, despite their higher environmental and economic costs.
DISADVANTAGES OF THERMAL POWER PLANTS
Despite their strategic role, thermal power plants have notable drawbacks. Environmentally, they contribute to air pollution, greenhouse gas emissions, and, in the case of nuclear plants, radioactive waste that is difficult to manage. From a cost perspective, they tend to be more expensive to operate compared to low marginal-cost renewable sources. Other disadvantages include the thermal pollution of nearby water bodies due to cooling systems and social and environmental impacts related to biomass land use. Therefore, while indispensable in power systems, their use must be balanced with cleaner and more sustainable alternatives.
ECONOMIC DISPATCH
The economic dispatch problem determines the optimal generation allocation among available units to meet demand at the lowest possible cost. It is a continuous optimization problem considering the variable production costs of each plant and their operational limits. Unlike Unit Commitment, it does not decide whether a unit should be turned on or off — only how much each online unit should generate. This method is widely used in daily system operations because it minimizes costs while respecting power limits. Although relatively simple, it forms the basis for more complex models such as Unit Commitment (UC).
UNIT COMMITMENT
The Unit Commitment (UC) problem extends the economic dispatch by deciding which generating units should be turned on or off at each time period, in addition to determining their production levels. It introduces binary (integer) variables, representing the discrete nature of on/off decisions. The model includes additional constraints such as minimum up/down times, ramping limits, and hot/cold startup costs. Due to its complexity, UC is typically solved using advanced mathematical programming techniques. It is an essential tool for short-term operation planning, ensuring cost-effective and reliable generation scheduling.
LAGRANGE MULTIPLIERS
In the context of power system operation, Lagrange multipliers have an important economic interpretation. They represent the shadow price of a constraint — that is, the rate at which the total cost would change if the constraint were relaxed by one unit. For example, if the generation limit of a thermal unit increased by 1 MW, the multiplier indicates how much the minimum cost would decrease. When associated with demand balance constraints, they represent the marginal operating cost of the system. Therefore, Lagrange multipliers are crucial for understanding price formation and guiding planning and investment decisions.
RESERVOIRS
Hydropower plants can be classified as run-of-river, which have negligible storage capacity, or reservoir plants, which store significant amounts of water. The latter provide flexibility by allowing water storage during wet periods for use during dry seasons, acting as natural energy batteries that ensure supply reliability. In computational models, reservoir representation varies with planning horizon — more aggregated in the medium term and more detailed in the short term. Efficient reservoir management is thus a central element of hydrothermal operation.
WATER BALANCE EQUATION
The water balance equation is the fundamental relationship governing the operation of a hydropower plant with a reservoir. It relates the final storage volume to the initial volume, plus inflows, minus the outflows due to turbining and spillage. Mathematically, it expresses the conservation of mass in the hydrological system. This balance is critical in both short- and long-term studies, as it predicts future generation availability and ensures that operation complies with physical reservoir limits. It also links present water use with its future value, forming the basis of hydrothermal optimization models.
CASCADING HYDRO PLANTS
Hydropower plants arranged in cascades are hydraulically coupled since the discharge from an upstream plant becomes the inflow of the downstream one. This interdependence adds significant complexity to system operation, requiring models that capture these interactions. Coordinated management of cascaded reservoirs allows maximizing global energy generation while avoiding water spillage. It is also necessary to account for water travel times between plants, introducing temporal delays in the hydraulic balance. Consequently, the planning of cascaded hydropower systems must be approached integratively and systematically.
HYDROPOWER PRODUCTION FUNCTION
The hydropower production function establishes the relationship between turbined flow and generated power. It depends on the net head height and the turbine-generator efficiency. In simplified form, power can be approximated as the product of flow and plant productivity. However, in reality, this function is nonlinear, since the net head varies with reservoir level and hydraulic losses. Understanding this function is essential for optimizing water use, maximizing generation efficiency, and ensuring economic dispatch within tariff constraints.
HYDRAULIC LOSSES
Hydraulic losses occur during water flow through penstocks and other hydraulic structures of a plant. They are typically modeled as quadratic or cubic functions of the turbined flow. As flow increases, losses grow, reducing the available net head for generation. These losses must be considered in operation models since they directly affect the effective power output. Ignoring them leads to overestimation of generation potential. Accurate modeling of hydraulic losses thus contributes to more realistic simulations of hydropower performance.
TURBINE-GENERATOR EFFICIENCY
The turbine-generator efficiency is a key determinant of hydropower plant performance, representing the proportion of hydraulic energy converted into useful electrical energy. In practice, efficiency is non-constant, varying nonlinearly with both turbined flow and available head. Efficiency curves typically feature a peak region of optimal operation and zones of reduced performance. Operating close to the optimal range enhances generation efficiency and reduces costs. Planning and optimization models frequently include these efficiency curves for a more accurate physical representation of plant behavior.
PROHIBITED OPERATING ZONES
Some hydropower plants have prohibited operating zones due to mechanical, hydraulic, or environmental constraints. These correspond to power or flow ranges that should be avoided to prevent vibrations, cavitation, or negative ecological impacts. Operational dispatch must respect these restrictions to ensure equipment safety and longevity. In mathematical models, such zones are represented as non-feasible intervals within the generation domain. Compliance with these constraints is therefore both a technical and environmental requirement in hydropower operation.
SPILLAGE
Spillage occurs when a portion of the reservoir inflow cannot be used for generation and must be released through spillways. This typically happens during high inflow periods when turbine capacity limits are reached or hydraulic restrictions must be met. While necessary for plant safety, spillage represents a loss of potential energy. Planning models seek to minimize spillage by maximizing the utilization of turbined water. In some cases, however, controlled spillage is mandated for environmental reasons, such as maintaining minimum downstream flow.
TEMPORAL COUPLING
Temporal coupling is a key characteristic of reservoirs: the decision to release water in one period directly affects future storage availability. This means operational decisions must be intertemporal, not isolated by stage. In optimization models, this coupling is captured through the reservoir volume as a state variable. It justifies the use of techniques such as dynamic programming, which handle sequential dependencies. Correctly modeling temporal coupling ensures more efficient and sustainable water use throughout the planning horizon.
SPATIAL COUPLING
Spatial coupling arises in systems with multiple hydraulically connected reservoirs, such as cascaded plants. In such systems, the operation of one reservoir affects not only its own storage but also downstream conditions. This interdependence increases the complexity of the optimization problem since interactions between multiple reservoirs must be considered simultaneously. In practical terms, upstream releases can increase or decrease downstream generation capacity, depending on cascade topology. Therefore, integrated hydro system planning is essential to maximize economic and energetic benefits.
HYDROTHERMAL PLANNING AND OPTIMIZATION
MAIN OBJECTIVE OF HYDROTHERMAL PLANNING
The main goal of hydrothermal system operation planning is to minimize the total operating cost of the power system while ensuring reliable demand supply. This process considers the physical constraints of generating units—such as generation limits and hydraulic balance—as well as electrical network constraints. It must also comply with environmental and regulatory requirements, including minimum flow levels and emission limits. The objective function incorporates both thermal generation costs and energy deficit penalties. Thus, hydrothermal planning seeks to balance economic efficiency, energy security, and environmental sustainability.
MEDIUM-TERM PLANNING (NEWAVE)
The NEWAVE model is used for medium-term planning, representing the hydroelectric park through Equivalent Energy Reservoirs (REEs). This aggregation simplifies the problem by grouping plants within hydrologically equivalent regions, reducing computational complexity. The adopted solution method is Stochastic Dual Dynamic Programming (SDDP), which handles inflow uncertainty probabilistically. NEWAVE produces Marginal Operating Cost (CMO) series, which act as price signals for market agents. Despite its simplifications, NEWAVE provides a consistent foundation for defining long-term operational strategies across multiple years.
SHORT-TERM PLANNING (DECOMP)
The DECOMP model is employed for short-term planning, with a horizon of weeks to months. Unlike NEWAVE, it represents individual hydro plants, allowing for a more detailed description of operational constraints. Its solution method is Deterministic Dual Dynamic Programming (DDDP), in which inflows are represented through fixed scenarios rather than stochastic processes. DECOMP enables the evaluation of detailed operation strategies, accounting for hydraulic losses, efficiencies, and network constraints. It serves as the main operational planning tool used by the National System Operator (ONS) to guide mid-term dispatch decisions.
VERY SHORT-TERM PLANNING (DESSEM)
The DESSEM model operates on a very short-term horizon, ranging from hours to days, with hourly time resolution. It explicitly represents the electrical network using DC power flow or, in more advanced versions, full AC formulations. DESSEM integrates multiple generation types—hydroelectric, thermal, wind, solar—and storage systems, in a unified framework. Its formulation is typically a Mixed-Integer Linear Programming (MILP) problem due to the presence of discrete variables (e.g., unit commitment decisions). DESSEM is used for daily energy dispatch, providing high-precision scheduling that supports real-time operation of the national grid.
MARGINAL OPERATING COST (CMO)
The Marginal Operating Cost (CMO) represents the incremental cost of meeting an additional 1 MW of demand. It corresponds to the Lagrange multiplier associated with the power balance constraint in the optimization problem. The CMO reflects the economic value of stored water, since current reservoir use directly impacts future thermal generation costs. Moreover, it functions as a price signal in the short-term electricity market, influencing both generation and consumption decisions. In summary, the CMO is a central indicator of system efficiency and economic equilibrium in the electric sector.
FUTURE COST (CF)
The future cost quantifies the economic value of water storage for future periods, embodying the intertemporal nature of hydro operation. It is estimated using methods such as Dynamic Programming or Benders Decomposition, which approximate the multi-stage cost function. The future cost acts as a proxy for the benefit of conserving water, discouraging excessive generation in the present. Consequently, it guides the optimal turbining policy at each stage. Proper representation of future cost ensures balance between short-term and long-term operation, promoting sustainable system management.
IMMEDIATE COST (CI)
The immediate cost refers to the actual operating cost incurred in a specific stage, considering thermal dispatch, spillage, deficits, and imports. It is computed based on the generation decisions made within that period, without accounting for future effects. This concept complements the future cost, as together they form the expected total cost. Minimizing only the immediate cost may lead to myopic decisions, such as inefficient reservoir depletion. Therefore, optimal planning requires balancing both immediate and future costs to achieve efficient intertemporal operation.
BELLMAN’S PRINCIPLE OF OPTIMALITY
Bellman’s Principle of Optimality states that an optimal policy possesses the property that, regardless of the initial state or decision, subsequent decisions must form an optimal policy for the resulting state. This principle underpins Dynamic Programming, enabling the decomposition of a global optimization problem into smaller, sequential subproblems. In energy planning, it governs the decision-making process for turbining and water storage in each stage, ensuring intertemporal consistency. Bellman’s principle is therefore a fundamental theoretical foundation for hydrothermal system modeling.
CURSE OF DIMENSIONALITY
The curse of dimensionality arises when the number of state and decision variables in an optimization problem grows exponentially with the number of reservoirs, time steps, or discretization granularity. This phenomenon is a major limitation of Dynamic Programming, where detailed reservoir representation leads to intractable problem sizes. To mitigate this issue, aggregation, approximation, and decomposition techniques are employed. A common example is the use of Equivalent Energy Reservoirs (REEs) in the NEWAVE model. Thus, the curse of dimensionality remains one of the greatest computational challenges in hydrothermal planning.
FUTURE COST FUNCTION USING BENDERS CUTS
Benders cuts are a decomposition technique used to approximate the future cost function in multi-stage problems. The method solves the problem iteratively, adding linear cuts at each iteration to refine the approximation of the cost-to-go function. This approach significantly reduces computational complexity, making large-scale optimization feasible. In the context of hydrothermal planning, Benders cuts approximate the future value of stored water, ensuring intertemporal consistency between stages. As a result, they enable robust and efficient operation planning under hydrological uncertainty.
RELIABILITY AND EMERGING ENERGY SOURCES
ADEQUACY
Adequacy is one of the key pillars of power system reliability. It refers to the system’s structural capacity to meet demand under normal operating conditions. Adequacy depends on having sufficient installed generation, transmission, and distribution capacity to supply projected loads. In planning, it involves ensuring reserve margins that guarantee service even under load growth scenarios. This assessment relies on historical consumption data, future projections, and resource availability. Therefore, adequacy measures the long-term structural robustness of the power system.
SECURITY
Security is the second pillar of reliability and relates to the system’s ability to withstand unexpected disturbances such as transmission line outages, generator failures, or extreme weather events. It is evaluated using criteria such as the N-1 rule, which requires the system to remain stable after the loss of any single component. Security encompasses transient and voltage stability as well as protection coordination. In summary, while adequacy ensures sufficient resources, security ensures resilience and operational continuity in the face of disruptions.
TRADITIONAL RELIABILITY INDICATORS
Traditional reliability indicators quantify the expected performance of the power system under uncertainty. The most commonly used include:
LOLP (Loss of Load Probability): probability that demand will not be met;
LOLE (Loss of Load Expectation): expected number of hours of load curtailment within a period;
EENS (Expected Energy Not Supplied): expected value of unserved energy.
These indices provide a quantitative foundation for planning and operational decisions and are widely used in generation expansion studies and the definition of long-term energy security criteria.
NEW RELIABILITY INDICATORS WITH RENEWABLES
With the increasing share of variable renewable energy sources, new reliability metrics have been developed to capture their specific characteristics. Examples include:
ESEC (Expected Solar Energy Curtailment): measures solar energy that could not be utilized;
EWES (Expected Wind Energy Spillage): analogous metric for wind generation losses.
These indices allow a more realistic assessment of renewable integration challenges by accounting for both technical curtailments and oversupply scenarios. Thus, they complement classical indicators and provide more appropriate metrics for renewable-dominated systems.
DUNKELFLAUTE
The German term Dunkelflaute, meaning “dark lull,” describes extended periods with little to no wind or sunlight. This phenomenon poses a critical challenge for systems with high shares of intermittent renewables, as it may threaten supply reliability. During such periods, backup resources—such as thermal plants, hydropower, or energy storage systems—are essential. Dunkelflaute highlights the importance of energy diversification and technological integration. Planning for this phenomenon is crucial for countries pursuing deep decarbonization of their power sectors.
BIOENERGY
Bioenergy is produced from biological resources such as ethanol, biodiesel, biogas, and biomethane. It is considered renewable, as its feedstocks can be replenished over time. Beyond electricity generation, bioenergy plays an important role in transportation and industrial processes. However, its classification as “clean energy” is relative, since combustion may release pollutants. One of its main advantages is dispatchability—the ability to be stored and used on demand, unlike intermittent sources such as wind or solar. Thus, bioenergy remains a key component in energy diversification strategies.
LOW-CARBON HYDROGEN
Low-carbon hydrogen—often referred to as green hydrogen when produced using renewable electricity—is one of the most promising energy carriers for future power systems. It can be generated through water electrolysis powered by renewable sources, avoiding carbon emissions in the process. Hydrogen offers significant potential as an energy vector, suitable for transport, industry, and power generation. Its flexibility also makes it ideal for long-term energy storage, helping to address renewable intermittency. Despite its promise, hydrogen still faces challenges related to costs and infrastructure for production and transport.
ENERGY STORAGE SYSTEMS (ESS)
Energy Storage Systems (ESS) are essential components of modern power systems, enabling energy shifting from periods of surplus to periods of scarcity. Technologies include batteries, pumped hydro storage, and hydrogen-based systems. ESS provide services such as frequency regulation, renewable smoothing, and enhanced reliability. From a planning perspective, they reduce operating costs by limiting the dispatch of high-cost thermal units. Additionally, storage systems play a key role in smart grid development and the integration of electric vehicles into the power network.
ELECTRIC VEHICLES (EVs)
Electric Vehicles (EVs) represent both a new type of load and an opportunity for grid flexibility. When widely adopted, they can significantly increase power demand, especially during peak hours. However, with smart and bidirectional charging technologies, EVs can function as distributed storage systems. The Vehicle-to-Grid (V2G) concept allows EVs to return energy to the grid when needed, contributing to system stability, flexibility, and renewable integration. Proper modeling of EV penetration is therefore crucial for future operation and planning studies.
CARBON CAPTURE, UTILIZATION, AND STORAGE (CCUS)
Carbon Capture, Utilization, and Storage (CCUS) technologies aim to reduce emissions from fossil-fueled power plants, mitigating their environmental impacts. The process involves capturing CO₂ emissions, transporting them, and storing them in geological formations or using them in industrial applications. This approach allows continued operation of thermal plants while supporting decarbonization goals. Although costly and infrastructure-intensive, CCUS is considered a strategic transitional technology, bridging the gap between the current fossil-based energy system and a renewable, low-carbon future.
DISTRIBUTION NETWORKS, MICROGRIDS, AND UNCERTAINTIES
DISTRIBUTION NETWORK ASSETS
The distribution network comprises various assets that ensure the reliable delivery of electricity to end users. Among the most important are feeders, which carry power from substations to load centers, and power transformers, which adjust voltage levels for transmission and distribution. Other critical components include voltage regulators, capacitor banks, and protection devices such as circuit breakers and reclosers. Proper coordination of these assets is essential to maintain power quality and supply continuity. Accurate modeling of these elements is fundamental for distribution system planning and operation studies.
ACTIVE DISTRIBUTION NETWORKS (ADNs)
Active Distribution Networks (ADNs) represent the evolution of traditional distribution grids into active and intelligent networks. In these systems, distributed generation, energy storage systems, and demand response resources interact in a coordinated manner. Active management allows the network not only to consume power but also to supply it back to the grid when needed. This enhances flexibility, improves reliability, and reduces losses. ADNs are therefore a cornerstone of the energy transition, enabling greater integration of renewables and advanced services. Their planning requires sophisticated optimization and control techniques.
MICROGRIDS (MGs)
Microgrids are small-scale power systems that integrate distributed generation, energy storage, and controllable loads under centralized or local coordination. They can operate either connected to the main grid or in islanded mode, providing autonomy during outages. This concept increases energy resilience, allowing critical loads to remain supplied during network disturbances. Additionally, microgrids promote the use of local energy resources such as solar and biomass. From a planning perspective, they introduce new coordination challenges but also offer innovative solutions for efficient distributed resource integration.
DEMAND RESPONSE (DR)
Demand Response (DR) enables consumers to reduce or shift their electricity usage in response to price signals or financial incentives. It helps balance the system during peak demand periods or generation shortages. Beyond improving reliability, DR lowers operational costs by avoiding the dispatch of expensive peaking plants. There are two main types of DR programs: price-based and incentive-based. In operational planning, DR should be modeled as a dispatchable resource, capable of providing flexibility comparable to generation units.
SMART TRANSFORMERS (ST)
Smart Transformers (STs) are advanced electronic devices that function as energy routers within hybrid AC/DC microgrids. In addition to traditional voltage transformation, they incorporate power converters capable of independent control of active and reactive power. This enables bidirectional power flow management and enhances voltage quality. STs also facilitate the integration of renewables and storage systems, reducing the need for auxiliary equipment. In operational planning, they represent a technological breakthrough, expanding control capabilities in complex and interactive grids.
CHALLENGES OF DISTRIBUTED ENERGY RESOURCES (DERs)
The integration of Distributed Energy Resources (DERs)—such as solar PV and wind—poses significant challenges for distribution networks. One is reverse power flow, which can stress equipment designed for unidirectional operation. Another is voltage regulation, affected by renewable intermittency. Higher DER penetration may also increase losses and dispatch complexity. These issues demand new control, planning, and investment strategies. Hence, DERs bring both benefits and challenges to grid modernization efforts.
CURTAILMENT
Curtailment refers to the intentional reduction of renewable generation when production exceeds demand or when technical constraints prevent full utilization. Although it results in the loss of clean energy, curtailment may be necessary to maintain system stability and adhere to safety limits. With growing renewable penetration, curtailment is becoming increasingly frequent. In planning models, it can be represented as a decision variable with zero generation cost but constrained by the grid’s absorption capacity. Thus, curtailment reflects a trade-off between efficiency and operational security.
HIERARCHICAL OPTIMIZATION IN ADNs
The operation of Active Distribution Networks can be structured under a hierarchical optimization framework. In this configuration, a master level coordinates global decisions such as power flows and resource dispatch, while local levels (slaves) optimize the performance of individual microgrids or subsystems. This approach reduces computational complexity by distributing decision-making across scales. It also enhances data privacy and local autonomy. In planning applications, hierarchical optimization is viewed as a promising solution for managing systems with numerous distributed resources.
ROBUSTNESS AGAINST UNCERTAINTIES
Operational planning must contend with uncertainties in renewable availability, demand variability, and equipment failures. Purely deterministic methods are often inadequate for such conditions. Consequently, stochastic, robust, and data-driven optimization techniques are employed. Robust optimization seeks secure solutions under worst-case scenarios, whereas data-driven methods leverage historical information to build realistic uncertainty sets. These approaches minimize the risk of inefficient or insecure operations. Therefore, robustness to uncertainty is a key requirement in modern energy planning models.
THE THREE Ds OF ENERGY
The so-called Three Ds of Energy—Decarbonization, Digitalization, and Decentralization—represent the major trends shaping the 21st-century power sector.
Decarbonization aims to cut greenhouse gas emissions through renewable expansion and energy efficiency.
Digitalization integrates communication and control technologies, enabling smart grids and greater consumer participation.
Decentralization emphasizes distributed generation, microgrids, and local autonomy.
In some contexts, a fourth D—Democratization—is added, referring to consumer empowerment in energy markets. Together, these principles define the future trajectory of global power systems.
GENERATION AND TRANSMISSION EXPANSION
CONCEPT AND PURPOSE
The Investment Decision Model (MDI) is a Mixed-Integer Linear Programming (MILP) formulation developed by the Energy Research Office (EPE) to determine the optimal generation and transmission expansion plan for the Brazilian National Interconnected System (SIN). Its main goal is to minimize the expected total cost of expansion—comprising investment and operating costs—over a ten-year planning horizon, while accounting for uncertainties related to hydrological and wind conditions. The MDI ensures an efficient and economically sustainable expansion of the generation fleet, maintaining supply security and adequacy to meet projected demand.
STRUCTURE OF THE MDI
The MDI is implemented in the Pyomo modeling environment and solved using the CPLEX solver, employing a graph-based representation of interconnected subsystems. Each subsystem is defined by its average energy demand and peak load, while generation plants—existing, contracted, or candidate—are explicitly modeled. The framework also represents interconnections between subsystems and their transmission exchange capacities. The formulation incorporates stochastic uncertainty treatment, enabling the optimal expansion plan to be derived by minimizing the expected cost, weighted by the probability of hydrological and wind scenarios.
DEMAND REPRESENTATION
System demand is modeled using four load segments—light, medium, heavy, and peak—each associated with a specific duration and intensity. This segmentation preserves monthly average energy while capturing the temporal variability of consumption. Modeling multiple load levels allows for performance assessment of different generation technologies across daily and seasonal variations, while capacity constraints ensure that available power satisfies instantaneous peak demand plus reserve requirements. This guarantees both energy and capacity adequacy throughout the planning horizon.
MODELED GENERATION SOURCES
The MDI represents all major generation technologies in detail.
Hydropower plants are modeled using energy and power series derived from the SUISHI and NEWAVE hydrothermal models.
Thermal plants are characterized by their Variable Operation Cost (CVU), inflexibility levels, and load-level dispatch.
Non-dispatchable renewable sources (wind, solar, and small hydro) are represented through average contribution factors and seasonal generation profiles.
Energy storage technologies, including batteries and pumped-storage plants, can be incorporated to transfer energy between load levels, accounting for losses and associated costs.
OBJECTIVE FUNCTION
The objective of the MDI is to minimize the expected total expansion cost, encompassing investment, operation, and penalty costs, while optionally including a perpetuity term. This perpetuity component ensures that investment decisions made near the end of the planning horizon are properly valued, avoiding the so-called “end-of-horizon problem.” The function sums discounted costs using a defined rate, thereby representing the present value of all investment and operation decisions. The resulting plan defines the economically optimal expansion strategy, balancing cost minimization with system reliability.
MAIN CONSTRAINTS
The MDI includes several constraint categories to ensure technical and operational feasibility:
Energy balance constraints ensure that the sum of generation, exchanges, and deficits—minus stored energy—is sufficient to meet demand in each load level and subsystem.
Capacity constraints guarantee that peak demand is covered with an adequate operational reserve margin.
Resource availability constraints prevent decommissioning of existing units.
Transmission constraints limit energy exchanges between subsystems based on interconnection capacities.
Investment and regulatory constraints govern the pace, timing, and proportion of expansion in line with national energy policy.
Together, these constraints ensure that the expansion plan is technically viable and consistent with real-world regulatory and operational conditions.
UNCERTAINTY TREATMENT
The MDI addresses hydrological and wind uncertainties through the generation of multiple scenarios, each associated with a probability of occurrence. These scenarios are organized in a decision tree structure, where investment decisions must remain robust across future conditions. The model applies the deterministic equivalent approach, converting the stochastic problem into a single optimization problem by weighting expected costs according to scenario probabilities. This ensures that the proposed expansion is resilient and cost-effective under diverse natural resource conditions.
DECISION VARIABLES
The MDI’s decision variables describe the system state and decisions at each time step, including:
Investment decisions (\(x_{i,t}\))
Generation per load level and period (\(G_{i,p,t}\))
Installed capacity (\(C_{i,t}\))
Inter-subsystem energy and power exchanges (\(I_{ij,p,t}\))
Energy and capacity deficits (\(D_{p,t}\))
Stored energy (\(B_{a,p,t}\))
Together, these variables allow the model to jointly optimize investment, dispatch, and exchange decisions, subject to technical and economic constraints. Their mixed (continuous and binary) nature makes the problem computationally challenging but ensures realism and accuracy in system representation.
MODEL OUTPUTS
The outputs of the MDI provide a comprehensive view of the optimal expansion plan for the power system. Key results include:
Commissioning schedule for new plants;
Marginal expansion and operation costs (CME and CMO);
Inter-regional power flows;
System reliability indicators.
Additionally, the model identifies transmission bottlenecks and quantifies the contribution of each generation source to load supply. These results serve as technical input for the Ten-Year Energy Expansion Plan (PDE), guiding public policy and investment decisions based on consistent techno-economic criteria.
RELATIONSHIP WITH THE MARGINAL EXPANSION COST (CME)
The Marginal Expansion Cost (CME) is directly derived from the MDI through the dual multipliers associated with energy and capacity constraints. It represents the incremental cost of meeting an additional unit of demand, simultaneously reflecting the need for energy expansion and peak capacity. The CME is calculated by introducing a 1 MW increase in demand for each load level and observing the resulting impact on total expansion cost.
In the PDE 2030 horizon, the average CME was estimated at R$ 187.46/MWh, with energy and capacity components of R$ 105.65/MWh and R$ 688.18/kW/year, respectively. This metric provides a unified reference for assessing economic efficiency and investment adequacy in Brazil’s long-term power sector planning.
TECHNICAL APPENDIX: SUMMARY OF THE DESSEM MANUAL
PURPOSE OF DESSEM
DESSEM is the computational model used for very short-term operation planning, with a horizon of up to seven days and hourly discretization. Its main objective is to determine the optimal dispatch of hydro, thermal, and complementary resources while minimizing the total system cost. The model explicitly represents the electrical network, ensuring operational security and feasibility. As such, DESSEM provides the initial conditions for real-time operation and supports the centralized dispatch performed by the National System Operator (ONS).
MATHEMATICAL FORMULATION
The mathematical formulation of DESSEM is based on Mixed-Integer Linear Programming (MILP), incorporating both continuous and binary variables.
Continuous variables represent generation levels and network power flows.
Binary variables represent Unit Commitment decisions, such as switching thermal units on or off.
This formulation allows a realistic representation of operational constraints but also increases computational complexity. Consequently, DESSEM employs decomposition techniques and high-performance algorithms to ensure convergence within practical computational times.
HYDRAULIC REPRESENTATION
The hydraulic subsystem in DESSEM models the hourly operation of hydroelectric plants, accounting for reservoir volumes, inflows, turbine discharges, spillage, and minimum downstream releases. Water balance constraints ensure mass conservation at each stage. The model also incorporates plant-specific productivity coefficients, hydraulic losses, and turbine flow limits. This detailed representation ensures that hydraulic energy is optimally utilized in compliance with environmental and physical restrictions.
THERMAL REPRESENTATION
Thermal generation units are modeled considering their variable operating costs, minimum and maximum generation limits, startup and shutdown costs, and ramping constraints. The inclusion of Unit Commitment logic ensures temporal consistency by enforcing minimum up and down times. This level of detail captures the actual operational characteristics of thermal plants, allowing the model to determine both the generation output and the on/off status of each unit at every hour.
ELECTRICAL NETWORK REPRESENTATION
DESSEM uses a DC power flow (linearized) formulation to represent transmission constraints and approximate network losses. Each bus in the network is subject to active power balance equations, and transmission lines and transformers are modeled with capacity limits. Contingencies and inter-regional exchange limits can also be included. This network representation ensures that dispatch results are not only economically optimal but also electrically feasible, integrating both the energy and network perspectives of system operation.
MODULAR STRUCTURE OF THE MODEL
DESSEM is organized into modular components, each responsible for a specific aspect of the problem:
Hydraulics – reservoir operation and water balance;
Thermal – generation and commitment;
Network – transmission constraints;
Operational reserves – system security margins.
These modules are combined within a main optimization framework, enabling parallel computation and efficient scalability. This modular design improves maintainability, flexibility, and computational performance, making DESSEM suitable for large-scale systems such as the Brazilian SIN.
INTEGRATION WITH OTHER MODELS
DESSEM operates in conjunction with other models in the planning chain.
It receives initial conditions—such as reservoir volumes and operating policies—from the medium-term models DECOMP and NEWAVE.
DESSEM refines these results with hourly precision, bridging the gap between medium-term planning and real-time operation.
Its outputs then serve as inputs for the ONS’s real-time dispatch.
This integration ensures coherence across planning horizons, avoiding inconsistencies between short-, medium-, and long-term decision processes.
PRACTICAL APPLICATIONS
In practice, DESSEM is used to:
Determine the hourly Marginal Operating Cost (CMO);
Identify the optimal dispatch of each generation resource;
Evaluate inter-regional energy exchanges;
Set short-term market prices;
Conduct electrical security analyses, such as transmission limits and load shedding assessments.
The model is particularly critical in systems with high shares of variable renewable generation, as it can integrate hourly wind and solar forecasts. In doing so, DESSEM provides a robust decision-support framework for operation planning in increasingly complex and dynamic energy environments.
MATHEMATICAL MODELING OF DESSEM
HYDROPOWER PLANTS
SETS AND INDICES
\(\mathcal{T} = \{1,\dots,T\}\) – hourly periods
\(\mathcal{H} = \{\text{UHE}_1, \dots, \text{UHE}_{n_h}\}\) – hydropower plants
\(\mathcal{U}(h) \subseteq \mathcal{H}\) – set of upstream hydropower plants of \(h\)
PARAMETERS (DATA)
\(a_{h,t}\) – natural inflow to plant \(h\) in period \(t\) (m\(^3\)/s)
\(d_t\) – demand in period \(t\) (MWh/h)
\(\zeta_{\text{vol}}\) – volume–flow conversion factor (hm\(^3\)/h) \(\Rightarrow \frac{3600}{10^6}\)
\(\zeta\) – hydraulic constant \(\Rightarrow \frac{9.81}{1000}\) (MW per m\(^3\)/sm of head)
\(V_{{\min}_h},\,V_{{\max}_h}\) – volume limits (hm\(^3\))
\(Q_{{\min}_h},\,Q_{{\max}_h}\) – turbined flow limits (m\(^3\)/s)
\(V_{{\text{ini}}_h}\) – initial reservoir volume (hm\(^3\))
\(V_{{\text{meta}}_h}\) – target terminal volume (hm\(^3\))
\(C_{{\text{def}}}\) – unitary cost of deficit ($/MWh)
\(\alpha_{h,k},\,\beta_{h,k},\,\gamma_{h,k}\) – polynomial coefficients for \(h_{\text{up}}\), \(h_{\text{down}}\), \(h_{\text{loss}}\)
\(\rho_{h,k}\) – polynomial coefficients of the specific productivity \(\rho_h\)
DECISION VARIABLES
\(Q_{h,t} \ge 0\) – turbined flow (m\(^3\)/s)
\(S_{h,t} \ge 0\) – spillage (m\(^3\)/s)
\(V_{h,t} \ge 0\) – stored volume (hm\(^3\))
\(G_{h,t} \ge 0\) – hydropower generation (MWh/h)
\(D_{b,t} \ge 0\) – energy deficit (MWh/h)
HYDRAULIC POLYNOMIAL FUNCTIONS FOR PLANT \(h\)
\(\rho(Q, H_{\text{net}}) = \zeta \,\Big( \rho_0 + \rho_1 Q + \rho_2 H_{\text{net}} + \rho_3 Q H_{\text{net}} + \rho_4 Q^2 + \rho_5 H_{\text{net}}^2 \Big), \text{ (CEPEL, 2023)}\)
\(h_{\text{up},h}(V) = \sum_{k=0}^{K_a} \alpha_{h,k}\, V^k, \)
\(h_{\text{down},h}(q) = \sum_{k=0}^{K_b} \beta_{h,k}\, q^k,\)
\(h_{\text{loss},h}(Q) = \sum_{k=0}^{K_\gamma} \gamma_{h,k}\, Q^k.\)
HYDROPOWER PRODUCTION FUNCTION (HPF)
\(H_{{\text{net}}_{h,t}} = h_{\text{up},h}(V_{h,t})- h_{\text{down},h}(Q_{h,t}+S_{h,t}) - h_{\text{loss},h}(Q_{h,t}),\)
\(G_{h,t} = \zeta \, Q_{h,t}\, \rho_h(Q_{h,t}, H_{{\text{net}}_{h,t}})\, H_{{\text{net}}_{h,t}} \quad \textbf{(Exact HPF)},\)
\(G_{h,t} = \zeta \mathrm{PE} H_{{\mathrm{net}}_{h,t}} Q_{h,t} \quad \textbf{(HPF with constant PE)},\)
\(G_{h,t} = \mathrm{P} Q_{h,t} \quad \textbf{(Linearized HPF)}.\)
TOTAL INSTANTANEOUS INFLOW (CASCADE WITHOUT DELAY)
CONSTRAINTS
GENERATION
RESERVOIR MASS BALANCE
\(V_{h,1} = V_{{\text{ini}}_h} + \zeta_{\text{vol}} ( I_{h,1} - Q_{h,1} - S_{h,1}),\)
\(V_{h,t} = V_{h,t-1} + \zeta_{\text{vol}} ( I_{h,t} - Q_{h,t} - S_{h,t}), \quad \forall t=2,\dots,T.\)
TARGETS AND OPERATIONAL LIMITS
\(V_{h,T} \ge V_{{\text{meta}}_h},\)
\(V_{{\min}_h} \le V_{h,t} \le V_{{\max}_h},\)
\(Q_{{\min}_h} \le Q_{h,t} \le Q_{{\max}_h},\)
\(S_{h,t},\, G_{h,t},\, D_t \ge 0.\)
THERMAL POWER PLANTS
SETS AND INDICES
\(\mathcal{T} = \{1,\dots,T\}, \quad\)
\(\mathcal{G} = \{\text{UTE}_1,\dots,\text{UTE}_{n_g}\}.\)
PARAMETERS
\(d_t\) – system demand (MW)
\(P_{{\min}_g},\, P_{{\max}_g}\) – generation limits of plant \(g\) (MW) (CEPEL, 2023)
\(a_g,\, b_g,\, c_g\) – thermal cost coefficients of plant \(g\)
\(SC_g\) – startup cost of plant \(g\)
\(R U_g,\, R D_g\) – ramp-up/ramp-down rates (MW per interval)
\(t_{{\uparrow}_g},\, t_{{\downarrow}_g}\) – minimum up/down times (h)
\(C_{\text{def}}\) – deficit penalty cost (R$/MWh)
\(u_{g,0},\, p_{g,0}\) – initial on/off status and generation (from initial status data)
DECISION VARIABLES
\(p_{g,t}\ge 0,\quad\)
\(u_{g,t}, y_{g,t}, w_{g,t}\in\{0,1\},\quad\)
\(D_t\ge 0.\)
(where \(u\) = on, \(y\) = startup, \(w\) = shutdown.)
CONSTRAINTS
POWER BALANCE
CONDITIONAL CAPACITY
STARTUP/SHUTDOWN LOGIC
RAMPING LIMITS
\(p_{g,t} - p_{g,t-1} \le RU_g + P_{{\max}_g} y_{g,t}\)
\(p_{g,t-1} - p_{g,t} \le RD_g + P_{{\max}_g} w_{g,t}\)
MINIMUM UP/DOWN TIME
\(\sum_{\tau=t-t^{\uparrow}_g+1}^{t} y_{g,\tau} \le u_{g,t},\)
\(\sum_{\tau=t-t^{\downarrow}_g+1}^{t} w_{g,\tau} \le 1-u_{g,t}.\)
CONSISTENT INITIAL CONDITIONS
RENEWABLE ENERGIES AND STORAGE
SETS AND INDICES
\(\mathcal{T} = \{1,\dots,T\},\quad\)
\(\mathcal{R} = \{1,\dots,R_{n_r}\},\quad\)
\(\mathcal{S} = \{1,\dots,S_{n_s}\}.\)
PARAMETERS
\(\overline{g}_{r,t}\) – exogenous renewable availability profile (MW avg)
\(\Delta t\) – time step (typically 1 h)
\(E_{{\min}_s},\,E_{{\max}_s}\) – energy storage limits (MWh)
\(E_{{\mathrm{ini}}_s}\) – initial energy (MWh)
\(\overline{P}_{{\mathrm{ch}}_{s}},\,\overline{P}_{{\mathrm{dis}}_{s}}\) – max charge/discharge power (MW)
\(\eta_{{\mathrm{c}}_{s}},\,\eta_{{\mathrm{d}}_{s}}\) – charge and discharge efficiencies
DECISION VARIABLES
\(g_{{\mathrm{ren}}_{r,t}} \ge 0\) – dispatched renewable generation (MW)
\(E_{s,t} \ge 0\) – stored energy (MWh)
\(P_{{\mathrm{ch}}_{s,t}},\,P_{{\mathrm{dis}}_{s,t}} \ge 0\) – charging/discharging power (MW)
CONSTRAINTS
RENEWABLE GENERATION — AVAILABILITY LIMIT
STORAGE — ENERGY BALANCE (SoC)
STORAGE — STATE OF CHARGE (SoC) LIMITS
STORAGE — POWER LIMITS
TRANSMISSION LINES AND CONNECTION BARS
SETS AND INDICES
\(\mathcal{T} = \{1,\dots,T\}\) – time periods (typically hourly)
\(\mathcal{L} = \{\text{LINE}_{1}, \dots, \text{LINE}_{n_\ell}\}\) – transmission lines
\(\mathcal{CB} = \{\text{BAR}_{1}, \dots, \text{BAR}_{n_b}\}\) – connection bars (buses)
\((i,j) \in \mathcal{E} \subseteq \mathcal{CB}\times\mathcal{CB}\) – ordered pair of bars defining endpoints of line \(\ell\)
\(\mathcal{L}(b)\) – set of lines incident to bar \(b\)
\(\mathcal{H}(b)\), \(\mathcal{G}(b)\), \(\mathcal{R}(b)\), \(\mathcal{S}(b)\) - generator attached to bar \(b\)
PARAMETERS
\(b_{\ell}\) – susceptance of line \(\ell\) (p.u. or 1/x)
\(\overline{F}_{\ell}\) – transmission capacity limit (MW)
\(\theta_{b,0}\) – reference (slack) bus angle (rad)
\(p_{\text{base}}\) – system base power (MW)
\(b \in \mathcal{CB}\) – bar with associated demand \(d_{b,t}\) and deficit \(D_{b, t}\)
DECISION VARIABLES
\(F_{\ell,t}\) – active power flow through line \(\ell\) (MW)
\(\theta_{b,t}\) – phase angle at bus \(b\) (radians)
DC FLOW APPROXIMATION
For each line \(\ell = (i,j)\) and time \(t\):
TRANSMISSION CAPACITY LIMITS
POWER BALANCE AT EACH BUS
Each bar \(b\) satisfies Kirchhoff’s Current Law (KCL) in the DC approximation:
where \(\delta_{b,\ell} = +1\) if bar \(b\) is the sending end of \(\ell\), \(-1\) if the receiving end, and \(0\) otherwise.
NETWORK REFERENCE (SLACK BUS)
One bar must be defined as the phase-angle reference:
BUS ANGLE LIMITS
For non-slack buses, the phase-angle is restricted as follows:
OBJECTIVE FUNCTION
The hydrothermal dispatch problem is formulated as a minimization of total generation and deficit costs, including:
Thermal generation costs;
Deficit costs (represented by a thermal deficit unit with fixed cost, as per Unsihuay, 2023);
Penalty for reservoir spillage.
TECHNICAL APPENDIX: SUMMARY OF THE DECOMP MANUAL
STRATEGIC PLANNING
The strategic planning of hydrothermal power system operation involves defining generation targets for hydro and thermal plants over a multi-year horizon, while respecting electrical and operational constraints and seeking to minimize total operating costs. The complexity of this problem stems from two main factors:
Temporal coupling, since current decisions affect future costs due to hydrological uncertainty;
Spatial coupling, arising from interdependent operation of cascade reservoirs.
This leads to the decision-maker’s dilemma: intensive use of water today may cause high thermal costs in future dry periods, whereas excessive thermal dispatch may lead to spillage and wasted energy during wet periods.
The formulation of the problem considers Immediate Cost (FCI) and Future Cost (FCF) functions, whose sum defines the Total Cost Function (FCT). The derivative of the FCF with respect to stored volume defines the value of water, i.e., the marginal cost of using water stored in reservoirs. The balance between the marginal thermal generation cost and the value of water determines the optimal dispatch policy.
Given the scale of the Brazilian National Interconnected System (SIN) and the hydrological uncertainty, the planning process is decomposed into multiple time horizons:
Medium term (NEWAVE): up to five years, with energy-equivalent reservoirs;
Short term (DECOMP): two months, with individual plant representation;
Very short term (DESSEM): hourly or sub-hourly, covering up to two weeks.
This hierarchical modeling chain, developed by CEPEL, ensures consistency across planning horizons.
IMMEDIATE, FUTURE, AND TOTAL COST FUNCTIONS IN STRATEGIC PLANNING
IMMEDIATE COST FUNCTION (FCI)
The FCI represents the cost arising from decisions at the current stage.
If demand is met by hydropower, the cost is nearly zero.
If thermal plants are dispatched, the FCI reflects fuel costs.
If demand is unmet, the deficit cost (penalty) is included.
Formally, for a stage ( t ):
where ( C_T ) is the thermal generation cost, ( g_t ) the thermal generation, ( C_D ) the deficit cost, and ( DEF_t ) the unserved energy.
FUTURE COST FUNCTION (FCF)
The FCF represents the expected cost of current decisions on future stages. When reservoirs are low, future costs rise due to increased thermal dispatch; when full, future costs drop.
The Bellman equation expresses this recursive relationship:
subject to water balance and demand constraints.
The slope of the FCF defines the value of water, representing the marginal opportunity cost of stored energy.
TOTAL COST FUNCTION (FCT)
The total cost is:
The optimal policy satisfies:
ensuring equilibrium between immediate and future costs — the marginal water value equals the marginal thermal cost.
APPLICATION
The joint use of FCI, FCF, and FCT enables robust multi-period water management. These functions form the conceptual foundation of NEWAVE, DECOMP, and DESSEM, where the FCF links successive planning horizons.
MARGINAL OPERATING COST (CMO)
The Marginal Operating Cost (CMO) quantifies the incremental cost of supplying one additional megawatt-hour.
It reflects the marginal opportunity cost of water or the benefit of retaining water for future use.
DEFINITION
According to DESSEM’s methodology, the CMO is determined in two steps:
Marginal Cost per Bus (CMB): obtained from the dual multipliers of bus power balance constraints.
Submarket CMO: the weighted average of bus CMOs, with weights based on bus demand.
RELATIONSHIP WITH COST FUNCTIONS
FCI: immediate thermal and deficit costs → CMO equals current marginal thermal cost.
FCF: captures future water value → implicit marginal hydro cost.
FCT: derivative equals total system CMO, balancing present and future costs.
Thus, CMO reflects the optimal equilibrium between water usage and conservation.
ROLE IN PLANNING
The CMO guides hydrothermal dispatch, market price formation (short-term PLD), and ensures inter-horizon consistency across NEWAVE, DECOMP, and DESSEM.
ADVANCED CONCEPTS IN STRATEGIC OPERATION PLANNING
RISK AVERSION
Traditional planning minimizes expected cost, assuming risk neutrality. To address hydrological uncertainty, risk-averse measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are introduced, penalizing extreme deficit scenarios and promoting more secure operational policies.
SPATIAL COUPLING
Spatial coupling arises from hydraulic interdependence among cascade plants and inter-area energy exchanges. It makes the FCF multidimensional, as it depends on multiple reservoir volumes simultaneously, greatly increasing model complexity.
CURSE OF DIMENSIONALITY AND DECOMPOSITION
Classical Dynamic Programming (DP) suffers from exponential state-space growth. The Stochastic Dual Dynamic Programming (SDDP) approach mitigates this through Benders decomposition, approximating the FCF with piecewise-linear cuts, enabling multi-year tractable planning.
INTEGRATION WITH PRICE FORMATION
The CMO, derived from optimization duals, is the foundation for Brazil’s Short-Term Settlement Price (PLD). Thus, planning results not only guide dispatch but also transmit economic signals, ensuring efficient and stable market operation.
SOLUTION STRATEGIES FOR STRATEGIC OPERATION PLANNING
The Strategic Operation Planning Problem of the SIN is complex, involving hydrological uncertainty, coupling effects, and multiple decision layers. The main approaches include:
Linear Programming (LP)
Dynamic Programming (DP)
Dual and Stochastic Extensions (PDD, PDDE)
Decomposition-based methods (as in CEPEL’s models).
MULTISTAGE LINEAR PROGRAMMING (LP)
Represents the entire planning horizon as a single linear model, with hydraulic and operational constraints. Provides global optimality but is computationally intractable for large systems.
DETERMINISTIC DYNAMIC PROGRAMMING (PDD)
Solves the problem recursively via Bellman’s principle, assuming known inflows. Pedagogically useful but limited by dimensionality.
DETERMINISTIC DUAL DYNAMIC PROGRAMMING (PDDD)
Approximates the FCF via Benders cuts, iteratively refined through forward and backward passes until convergence. Foundation of DECOMP.
STOCHASTIC DUAL DYNAMIC PROGRAMMING (PDDE)
Extends the dual formulation to stochastic inflows (NEWAVE model), evaluating expected future costs under probability distributions.
VERY SHORT-TERM DETERMINISTIC FORMULATIONS
Used in DESSEM, which employs a Mixed-Integer Linear Program (MILP) to represent hourly hydrothermal dispatch and network constraints, coupled with DECOMP via the Future Cost Function.
COMPARATIVE OVERVIEW OF STRATEGIES
Aspect |
Multistage LP |
PDD / PDDD |
PDDE (NEWAVE) |
DESSEM |
|---|---|---|---|---|
Typical Horizon |
Medium/short (small systems) |
Medium (deterministic) |
Long-term (stochastic) |
Very short-term (hourly) |
Uncertainty Handling |
None |
Deterministic |
Stochastic scenarios |
Deterministic |
Future Cost (FCF) |
Implicit |
Approximated via Benders cuts |
Stochastic Benders cuts |
Coupled via DECOMP |
Coupling |
Temporal & spatial (explicit) |
Temporal via recursion |
Temporal & spatial (aggregated) |
Temporal & spatial (detailed) |
Advantages |
Exact global optimum |
Conceptually clear; didactic |
Treats uncertainty efficiently |
High temporal resolution |
Limitations |
Infeasible for large systems |
Suffers from dimensionality |
May yield conservative policies |
Deterministic only |
Practical Use |
Academic |
Basis of DECOMP |
Basis of NEWAVE |
Basis of DESSEM |
These methods form a hierarchical chain:
NEWAVE (PDDE) defines mid-term water value policies,
DECOMP (PDDD) refines monthly dispatch and inter-area exchanges,
DESSEM (MILP) determines hourly operation and market pricing.
Together, they ensure temporal and economic consistency across Brazil’s entire energy planning framework.
MATHEMATICAL MODELING OF DECOMP
HYDROPOWER PLANTS
SETS AND INDICES
\(\mathcal{T} = \{1,\dots,T\}\) – periods
\(\mathcal{H} = \{\text{UHE}_1, \dots, \text{UHE}_{n_h}\}\) – hydropower plants
\(\mathcal{U}(h) \subseteq \mathcal{H}\) – set of upstream hydropower plants of \(h\)
PARAMETERS (DATA)
\(a_{h,t}\) – natural inflow to plant \(h\) in period \(t\) (hm\(^3\))
\(d_t\) – demand in period \(t\) (MWh)
\(V_{{\min}_h},\,V_{{\max}_h}\) – storage limits (hm\(^3\))
\(Q_{{\min}_h},\,Q_{{\max}_h}\) – turbined flow limits (hm\(^3\))
\(V_{{\text{ini}}_h}\) – initial reservoir volume (hm\(^3\))
\(V_{{\text{meta}}_h}\) – target terminal volume (hm\(^3\))
\(C_{{\text{def}_b}}\) – unitary cost of deficit ($/MWh)
DECISION VARIABLES
\(Q_{h,t} \ge 0\) – turbined flow (hm\(^3\))
\(S_{h,t} \ge 0\) – spillage (m\(^3\))
\(V_{h,t} \ge 0\) – stored volume (hm\(^3\))
\(G_{h,t} \ge 0\) – hydropower generation (MWmed)
\(D_{b,t} \ge 0\) – energy deficit (MWh/h)
HYDROPOWER PRODUCTION FUNCTION (HPF)
CONSTRAINTS
TOTAL INSTANTANEOUS INFLOW (CASCADE WITHOUT DELAY)
RESERVOIR MASS BALANCE
THERMAL UNITS
SETS AND INDICES
PARAMETERS
DECISION VARIABLES
CONSTRAINTS
CAPACITY
RENEWABLE ENERGIES AND STORAGE
SETS AND INDICES
\(\mathcal{T} = \{1,2,\dots,T\} \quad\) — periods
\(\mathcal{R} = \{1,2,\dots,N_R\} \quad\) — set of renewable units (wind/solar)
\(\mathcal{S} = \{1,2,\dots,N_S\} \quad\) — set of storage units (batteries)
PARAMETERS
For each \(t \in \mathcal{T}\), \(r \in \mathcal{R}\), and \(s \in \mathcal{S}\):
\(\overline{g}_{r,t} \text{ — exogenous renewable availability profile (MWmed)}\)
\(\Delta t \text{ — time step (T), typically } \Delta t = 1\)
\(E_{\min_s}, E_{\max_s} \text{ — min/max stored energy limits (MWh)}\)
\(E_{\text{ini}_s} \text{ — initial energy (MWh)}\)
\(\overline{P}_{\text{ch}_s}, \overline{P}_{\text{dis}_s} \text{ — maximum charge/discharge power (MW)}\)
\(\eta_{\text{c}_s}, \eta_{\text{d}_s} \in (0,1] – \text{ — charging and discharging efficiencies}\)
DECISION VARIABLES
\(g_{\text{ren}_{r,t}} \ge 0 \text{ — dispatched renewable generation of unit } r \text{ at } t \text{ (MWavg)}\)
\(E_{s,t} \ge 0 \text{ — stored energy (SoC) of battery } s \text{ at } t \text{ (MWh)}\)
\(P_{\text{ch}_{s,t}}, P_{\text{dis}_{s,t}} \ge 0 \text{ — charge/discharge powers (MW)}\)
CONSTRAINTS
RENEWABLE SOURCES — AVAILABILITY LIMIT
STORAGE — ENERGY BALANCE (SoC)
STORAGE - STATE OF CHARGE (SoC) LIMITS
STORAGE — POWER LIMITS
TRANSMISSION LINES AND CONNECTION BARS
SETS AND INDICES
\(\mathcal{T} = \{1,\dots,T\}\) – time periods (typically hourly)
\(\mathcal{L} = \{\text{LINE}_{1}, \dots, \text{LINE}_{n_\ell}\}\) – transmission lines
\(\mathcal{CB} = \{\text{BAR}_{1}, \dots, \text{BAR}_{n_b}\}\) – connection bars (buses)
\((i,j) \in \mathcal{E} \subseteq \mathcal{CB}\times\mathcal{CB}\) – ordered pair of bars defining endpoints of line \(\ell\)
\(\mathcal{L}(b)\) – set of lines incident to bar \(b\)
\(\mathcal{H}(b)\), \(\mathcal{G}(b)\), \(\mathcal{R}(b)\), \(\mathcal{S}(b)\) - generator attached to bar \(b\)
PARAMETERS
\(b_{\ell}\) – susceptance of line \(\ell\) (p.u. or 1/x)
\(\overline{F}_{\ell}\) – transmission capacity limit (MW)
\(\theta_{b,0}\) – reference (slack) bus angle (rad)
\(p_{\text{base}}\) – system base power (MW)
\(b \in \mathcal{CB}\) – bar with associated demand \(d_{b,t}\) and deficit \(D_{b, t}\)
DECISION VARIABLES
\(F_{\ell,t}\) – active power flow through line \(\ell\) (MW)
\(\theta_{b,t}\) – phase angle at bus \(b\) (radians)
DC FLOW APPROXIMATION
For each line \(\ell = (i,j)\) and time \(t\):
TRANSMISSION CAPACITY LIMITS
POWER BALANCE AT EACH BUS
Each bar \(b\) satisfies Kirchhoff’s Current Law (KCL) in the DC approximation:
where \(\delta_{b,\ell} = +1\) if bar \(b\) is the sending end of \(\ell\), \(-1\) if the receiving end, and \(0\) otherwise.
NETWORK REFERENCE (SLACK BUS)
One bar must be defined as the phase-angle reference:
BUS ANGLE LIMITS
For non-slack buses, the phase-angle is restricted as follows:
OBJECTIVE FUNCTION
The hydrothermal dispatch problem is formulated as the minimization of total generation and deficit costs, namely:
Thermal generation costs
Deficit cost (represented by a thermal unit with fixed cost, as in Unsihuay, 2023)
Spillage penalty (energy waste)
For the PDDD case, the future cost term \(\alpha\) (FCF) is included
SINGLE LP OBJECTIVE FUNCTION
OBJECTIVE FUNCTION FOR PDDD
TECHNICAL APPENDIX: SUMMARY OF EPE MANUALS ON THE MDI
CENTRAL CONCEPT
The Investment Decision Model (MDI) is the analytical tool developed by the Energy Research Office (EPE) to determine the optimal expansion plan for Brazil’s electricity generation system, as used in the Ten-Year Energy Expansion Plan (PDE 2030).
The model minimizes the expected total cost of expansion, given by:
It is formulated as a Mixed-Integer Linear Programming (MILP) problem, implemented in Pyomo and solved using CPLEX.
MODEL STRUCTURE
DEMAND REPRESENTATION
Electricity demand is segmented into load levels—light, medium, heavy, and peak—each with its own duration and intensity. The total energy across levels equals the monthly average demand:
This representation allows a more accurate assessment of how generation technologies meet the system’s load profile.
SYSTEM COMPONENTS
The MDI explicitly models:
Existing, planned, and candidate generation units;
Interconnections between subsystems (represented as a capacitated graph);
Operational reserves and capacity constraints.
OBJECTIVE FUNCTION
The objective is to minimize the expected total discounted cost of expansion, with or without perpetuity:
The function includes fixed costs (investment, O&M, capital charges) and variable costs (thermal dispatch, deficits), as well as penalty terms.
FUNDAMENTAL CONSTRAINTS
ENERGY BALANCE
The sum of generation, interchanges, and deficits, minus stored energy, must meet or exceed the demand in each subsystem, scenario, and load level.
CAPACITY ADEQUACY
Ensures that peak demand plus reserve margin (typically 104% in PDE 2030) is met across all subsystems.
SOURCE AVAILABILITY
Establishes lower and upper bounds for generation and prevents disinvestment, ensuring that installed capacity does not decline over time.
SYSTEM REPRESENTATION
Limits energy interchange between subsystems:
incorporating losses, regional aggregation, and dynamic limits.
INVESTMENT RESTRICTIONS
Each project is represented by a binary decision variable (\(x_{i,t} \in \{0,1\}\)) indicating the period of investment. A project can only be built once within the planning horizon.
ADDITIONAL POLICY CONSTRAINTS
Used to enforce regulatory or strategic rules, such as:
Uniform or stepwise expansion;
Annual and incremental limits;
Minimum self-sufficiency by subsystem;
Expansion restricted to specific months (e.g., January/July).
GENERATION SOURCE MODELING
HYDROELECTRIC PLANTS
Hydropower generation is represented using energy and capacity time series obtained from SUISHI and NEWAVE simulations. The investment decision is binary (installed or not), and capacity is assumed to increase linearly after commissioning.
THERMAL PLANTS
Thermal generation is dispatched by load level. Projects can be continuous (generic expansion) or integer (specific projects). The Variable Operating Cost (CVU) is adjusted annually based on EIA projections.
RENEWABLE SOURCES
Includes wind, solar, and small hydro (PCH) plants. They are modeled using monthly seasonality and average contribution factors by load level.
ENERGY STORAGE PROJECTs
Models both charging and discharging in the same period, considering round-trip efficiency losses and energy purchase costs for recharging.
UNCERTAINTY SCENARIOS
The MDI accounts for hydrological and wind uncertainties through a scenario tree structure (deterministic equivalent formulation).
The goal is to minimize the expected value of total cost across all scenarios:
Each scenario branch represents possible outcomes of future conditions, with corresponding probabilities.
MARGINAL EXPANSION COST (CME)
DEFINITION
The Marginal Expansion Cost (CME) quantifies the incremental cost of meeting an additional unit of demand:
CALCULATION
CME is derived from the dual variables of the energy and capacity constraints. The procedure adds one unit of demand per load level and measures the incremental cost impact.
To preserve the load shape:
+1 unit to energy demand,
+(1 + Reserve Margin) to power demand.
This yields the energy and capacity components of the CME.
KEY CONCEPTS
Concept |
Description |
|---|---|
CME |
Marginal Expansion Cost (sum of energy and capacity components). |
MDI |
Investment Decision Model for expansion planning. |
CMO |
Marginal Operating Cost (short-term). |
Operational Reserve |
Safety margin above maximum demand (≈ 4%). |
Perpetuity |
Ensures end-of-horizon consistency by valuing long-term investments. |
Solver |
Optimization engine: CPLEX via Pyomo. |
Simulation Models |
SUISHI and NEWAVE provide hydro and energy series inputs. |
In summary, the MDI integrates economic, technical, and stochastic modeling to produce a consistent, cost-optimal expansion plan for Brazil’s power system, bridging the gap between hydrothermal simulation and long-term investment strategy.
MATHEMATICAL MODELING OF MDI
The objective of this model is to determine the optimal expansion plan of the electric generation system, minimizing the total investment and operation cost, considering the availability of multiple technologies (hydroelectric, thermal, solar, wind, and battery storage), the demand satisfaction in two load levels (peak and off-peak), and the distinction between existing and candidate units.
This constitutes a Mixed-Integer Linear Programming (MILP) problem, solved by decomposition methods or Branch and Bound, as discussed in the theoretical overview.
SETS
\(\mathcal{G}\) – total set of generating units (hydroelectric, thermal, solar, and wind)
\(\mathcal{G}_E \subset \mathcal{G}\) – subset of existing plants
\(\mathcal{G}_C \subset \mathcal{G}\) – subset of candidate plants
\(\mathcal{S}\) – total set of storage units (batteries)
\(\mathcal{S}_E \subset \mathcal{S}\) – subset of existing batteries
\(\mathcal{S}_C \subset \mathcal{S}\) – subset of candidate batteries
\(\mathcal{T}\) – planning periods
\(\mathcal{P}\) – load levels
\(\mathcal{L} = \{\text{LINE}_{1}, \dots, \text{LINE}_{n_\ell}\}\) – transmission lines
\(\mathcal{L}_E \subset \mathcal{L}\) – subset of existing transmission lines
\(\mathcal{L}_C \subset \mathcal{L}\) – subset of candidate transmission lines
\(\mathcal{CB} = \{\text{BAR}_{1}, \dots, \text{BAR}_{n_b}\}\) – connection bars (buses)
\((i,j) \in \mathcal{E} \subseteq \mathcal{CB}\times\mathcal{CB}\) – ordered pair of bars defining endpoints of line \(\ell\)
\(\mathcal{L}(b)\) – set of lines incident to bar \(b\)
\(\mathcal{G}(b)\) – set of generators attached to bar \(b\)
\(\mathcal{S}(b)\) – set of batteries attached to bar \(b\)
PARAMETERS
\(C_{\text{inv}_g}\) – investment cost of generator \(g\)
\(C_{\text{op}_g}\) – operating cost of generator \(g\) [$ /MWh]
\(C_{\text{inv}_s}\) – investment cost of battery \(s\)
\(C_{\text{op}_s}\) – operating cost of battery \(s\) [$ /MWh]
\(C_{\text{inv},\ell}\) – investment cost of line \(\ell\)
\(C_{\text{op},\ell}\) – operation cost of line \(\ell\) [$ /MWh]
\(P_{\text{max}_g}\) – maximum capacity of generator \(g\) [MW]
\(E_{\text{max}_s}\), \(E_{\text{min}_s}\) – state-of-charge limits for battery \(s\) [MWh]
\(P_{\text{ch}_{\text{max}_s}}\), \(P_{\text{dis}_{\text{max}_s}}\) – maximum charge/discharge power of battery \(s\) [MW]
\(E_{0_s}\) – initial state of charge of battery \(s\) [MWh]
\(\eta_{c_s}\), \(\eta_{d_s}\) – charge/discharge efficiencies (0.95)
\(D_{b,p,t}\) – demand in load level \(p\) and period \(t\) [MW] per bus
\(h_p\) – duration of load level \(p\) [h/year]
\(x_{g,0}\) – 1 if generator \(g\) exists at the beginning of the horizon, 0 otherwise
\(x_{s,0}\) – 1 if battery \(s\) exists at the beginning of the horizon, 0 otherwise
\(x_{\ell,0}\) – 1 if line \(\ell\) exists at the beginning of the horizon, 0 otherwise
\(b_{\ell}\) – susceptance of line \(\ell\) (p.u. or 1/x)
\(\overline{F}_{\ell}\) – transmission capacity limit (MW)
\(\theta_{b,0,p}\) – reference (slack) bus angle (rad)
\(p_{\text{base}}\) – system base power (MW) (only for line transmission power conversion - optional if the system has no explicit topology)
DECISION VARIABLES
\(y_{g,t} \in \{0,1\}\) – construction (1) or not (0) of candidate generator \(g\) in period \(t\)
\(y_{s,t} \in \{0,1\}\) – construction (1) or not (0) of candidate battery \(s\) in period \(t\)
\(y_{\ell,t} \in \{0,1\}\) – construction (1) or not (0) of candidate transmission line \(\ell\) in period \(t\)
\(x_{g,t} \in \{0,1\}\) – existence of generator \(g\) in period \(t\) (1 if built up to \(t\))
\(x_{s,t} \in \{0,1\}\) – existence of battery \(s\) in period \(t\) (1 if built up to \(t\)) _
\(x_{\ell,t} \in \{0,1\}\) – existence of transmission line \(\ell\) in period \(t\) (1 if built up to \(t\))
\(P_{g,t,p} \ge 0\) – generation of unit \(g\) in load level \(p\) and period \(t\) [MW]
\(P^{c}_{s,t,p} \ge 0\) – charging power of battery \(s\) in load level \(p\) and period \(t\) [MW/level]
\(P^{d}_{s,t,p} \ge 0\) – discharging power of battery \(s\) in load level \(p\) and period \(t\) [MW/level]
\(E_{s,t,p}\) – state of charge (SoC) of battery \(s\) [MWh]
\(F_{\ell,t,p}\) – active power flow through line \(\ell\) (MW)
\(\theta_{b,t,p}\) – phase angle at bus \(b\) (radians) and level \(p\)
CONSTRAINTS
DEMAND REQUIREMENT PER BUS
This ensures power balance at each period and load level: total net generation (generation plus storage balance) equals system demand.
CAPACITY ADEQUACY
Guarantees that total available capacity (generation + discharge) is sufficient to meet demand in all time steps.
GENERATION LIMITS
Ensures that generation of each unit does not exceed its maximum capacity and is zero when inactive.
STORAGE DYNAMICS
Ensures energy continuity across load levels and periods, applying charge/discharge efficiencies and introducing initial energy only when the unit is built.
STATE OF CHARGE LIMITS
Keeps the state of charge within the operational range, proportional to the unit’s availability.
CHARGE/DISCHARGE POWER LIMITS
Restricts charging and discharging powers according to the installed battery capacity and availability.
EXPANSION DYNAMICS (EXISTENCE ACCUMULATION)
Guarantees temporal coherence of the expansion plan: units exist only if previously constructed.
DC FLOW APPROXIMATION
For the DC model, this constraint ensures that the power flow is proportional to the angular difference between the buses connected by the transmission line. In the transportation model, this constraint is disregarded. It is worth noting that the formulation of this constraint is valid only in cases where \(\sin(\theta_{i,t,p} - \theta_{j,t,p}) \approx \theta_{i,t,p} - \theta_{j,t,p}\), that is, for angular differences up to \(\frac{\pi}{6}\).
TRANSMISSION CAPACITY LIMITS
Ensures that transmission power flow of each transmission line unit does not exceed its maximum capacity and is zero when inactive.
NETWORK REFERENCE (SLACK BUS)
One bar must be defined as the phase-angle reference:
BUS ANGLE LIMITS
For non-slack buses, the phase-angle is restricted as follows:
UNIQUE CONSTRUCTION
Prevents multiple constructions of the same unit within the planning horizon.
MONOTONIC GROWTH
Ensures that the set of existing units grows monotonically, avoiding deactivation after construction and maintaining temporal consistency in expansion.
OBJECTIVE FUNCTION
Where:
\(Z\) — total objective function value (minimum system cost)
\(t_x\) — discount rate (\(t_x \in [0,1)\))
\(C^{\text{inv}}_{g}, C^{\text{inv}}_{s} , C^{\text{inv}}_{\ell}\) — investment costs for generation and storage units [$]
\(C^{\text{op}}_{g}, C^{\text{op}}_{s}, C^{\text{op}}_{\ell}\) — operating costs for generation and storage units [$/MWh]
\(x_{g,t}, x_{s,t}, x_{\ell,t}\) — binary variables for existence of generation/storage units
\(P_{g,t,p}\) — generated power [MW]
\(P^{d}_{s,t,p}, P^{c}_{s,t,p}\) — discharging and charging powers [MW]
\(F_{l,t,p}\) - transmission line power flux [MW]
\(h_p\) — duration of load level \(p\) [h]
This objective minimizes total system cost across the planning horizon, combining investment and operating costs weighted by the duration of each load level. The formulation captures the trade-off between capacity expansion and operation, ensuring an economically optimal solution under technical and energy constraints.
TECHNICAL APPENDIX: OPTIMIZATION METHODS
LP (LINEAR PROGRAMMING – CONTINUOUS LINEAR PROGRAMMING)
Linear Programming (LP) solves problems where both the objective function and all constraints are linear, and all variables are continuous. LP models are highly efficient and robust, supported by commercial solvers such as CPLEX and Gurobi, as well as open-source alternatives like HiGHS and CBC.
Common applications include relaxations of integer problems, feasibility verification, or piecewise linear cost formulations (PWL) without binary variables. LP serves as the foundation for most advanced optimization formulations.
MILP (MIXED-INTEGER LINEAR PROGRAMMING)
MILP extends LP by introducing integer or binary variables, enabling the modeling of discrete decisions such as the on/off status of power plants. It is widely used in Unit Commitment (UC) problems, incorporating:
Piecewise linear costs (PWL);
Minimum up/down time constraints;
Ramp-up and ramp-down limits;
Hot and cold start-up costs.
The most efficient solvers are Gurobi and CPLEX, with HiGHS emerging as a strong open-source alternative.
MILP is the workhorse formulation for real-world power system operations.
QP (QUADRATIC PROGRAMMING – CONVEX CONTINUOUS QUADRATIC PROGRAMMING)
Quadratic Programming (QP) deals with problems featuring a convex quadratic objective and linear constraints. It is well-suited for representing smooth quadratic generation cost curves without discrete decisions.
Typical applications include continuous economic dispatch problems, where fuel cost functions are quadratic.
Solvers such as Gurobi, CPLEX, and SCIP efficiently handle convex QP formulations.
QP models strike a balance between realism and computational simplicity.
MIQP (MIXED-INTEGER QUADRATIC PROGRAMMING – CONVEX FORMULATION)
MIQP combines integer variables with a convex quadratic objective function. It is useful for Unit Commitment models that maintain quadratic fuel cost representations while still accounting for binary on/off decisions.
Although more computationally demanding than MILP, solvers like Gurobi, CPLEX, and SCIP deliver robust performance.
MIQP is frequently employed in academic and research applications seeking higher cost fidelity.
NLP (NONLINEAR PROGRAMMING – CONTINUOUS NONLINEAR PROGRAMMING)
Nonlinear Programming (NLP) addresses problems where the objective and/or constraints are nonlinear, with all variables continuous.
A classic example is the exact Hydropower Production Function (HPF), which is nonlinear due to the interaction between flow, head, and efficiency.
Other examples include nonlinear hydraulic balance equations and smooth operational constraints.
The Ipopt solver is widely used for this class of problems.
While NLP allows greater physical realism, it relies on convexity assumptions or heuristics to avoid local minima.
MINLP (MIXED-INTEGER NONLINEAR PROGRAMMING)
MINLP represents the most general class of optimization problems, combining discrete and continuous variables under nonlinear relationships.
It is applied in hybrid cases such as:
Unit Commitment with nonlinear hydro constraints;
Joint cost–operation formulations;
Complex multi-energy system integration.
In the Pyomo ecosystem, the MindtPy solver is a key tool for solving MINLPs.
Despite their flexibility, MINLPs are computationally intensive and often require relaxation techniques, Benders decomposition, or outer approximation to become tractable in large-scale systems.
They provide maximum modeling realism at the expense of computational complexity.
SUMMARY TABLE
Method |
Variable Type |
Objective Type |
Typical Application |
Main Solvers |
|---|---|---|---|---|
LP |
Continuous |
Linear |
Relaxations, feasibility, PWL models |
CPLEX, Gurobi, HiGHS, CBC |
MILP |
Continuous + Integer |
Linear |
Unit Commitment, expansion planning |
CPLEX, Gurobi, HiGHS |
QP |
Continuous |
Quadratic (Convex) |
Economic Dispatch (continuous) |
CPLEX, Gurobi, SCIP |
MIQP |
Continuous + Integer |
Quadratic (Convex) |
UC with quadratic costs |
CPLEX, Gurobi, SCIP |
NLP |
Continuous |
Nonlinear |
Exact HPF, hydraulic modeling |
Ipopt |
MINLP |
Continuous + Integer |
Nonlinear |
Hybrid UC + hydro problems |
MindtPy, SCIP, CPLEX-NLP |
In conclusion, these optimization paradigms form the computational backbone of modern energy system modeling. Each class trades off mathematical precision, computational effort, and interpretability, allowing planners and researchers to select the most suitable formulation for the scale, realism, and temporal scope of the problem under study.
SUPPLEMENTARY APPENDIX: UNCERTAINTIES AND D-OPF
OPTIMIZATION UNDER UNCERTAINTY
The operation of modern electric power systems is subject to significant uncertainties, primarily related to hydrological inflows, renewable generation, and load variability.
To address these sources of uncertainty, several mathematical paradigms of optimization have been developed, each with different trade-offs between robustness, complexity, and economic efficiency.
STOCHASTIC PROGRAMMING (SP)
Stochastic Programming explicitly models uncertainty using probabilistic scenarios, each associated with a likelihood of occurrence.
The objective is to minimize the expected operational cost across all scenarios:
where ( p_s ) is the probability of scenario ( s ), ( C(x_s, \xi_s) ) the corresponding cost, and ( x_s ) the decision variables.
SP captures the statistical nature of uncertainty but can become computationally intractable as the number of scenarios grows, due to the curse of dimensionality.
ROBUST OPTIMIZATION (RO)
Robust Optimization takes a different stance: instead of relying on probabilistic information, it defines an uncertainty set ( \mathcal{U} ) within which all possible realizations of uncertain parameters must lie.
The optimization then seeks the best worst-case solution:
This approach sacrifices part of optimality under nominal conditions in exchange for guaranteed feasibility across all admissible realizations.
RO is particularly useful in operational contexts requiring high reliability, such as critical infrastructure and distribution network operation.
DATA-DRIVEN DISTRIBUTIONALLY ROBUST OPTIMIZATION (DDDRO)
A more recent and advanced approach, Data-Driven Distributionally Robust Optimization (DDDRO), combines the strengths of SP and RO.
It constructs ambiguity sets of probability distributions based on historical data, typically using Wasserstein distances or moment-based metrics.
The objective is to ensure robustness against unknown or misspecified distributions, minimizing the worst-case expected cost across all plausible distributions:
DDDRO thus provides a balance between stochastic realism and robust guarantees, making it highly suitable for systems with intermittent renewables, where statistical patterns evolve dynamically.
These methods enable resilient and adaptive energy planning under deep uncertainty.
D-OPF FORMULATIONS (OPTIMAL POWER FLOW IN DISTRIBUTION NETWORKS)
The Distribution Optimal Power Flow (D-OPF) problem determines the optimal power dispatch and network configuration in distribution systems that include Distributed Energy Resources (DERs), energy storage, and ancillary services.
Unlike transmission-level OPF, the D-OPF must handle radial or weakly meshed topologies, voltage-dependent losses, and bidirectional power flows.
EXACT AC FORMULATION
The AC D-OPF formulation is based on nonlinear and non-convex power flow equations derived from Kirchhoff’s laws:
where ( P_i, Q_i ) are active and reactive powers, ( V_i ) the voltage magnitude, and ( \theta_{ij} ) the phase angle difference.
Due to their non-convexity, these equations make the AC-OPF problem NP-hard.
CONVEX RELAXATIONS
To make D-OPF computationally tractable, convex relaxations are often applied:
Second-Order Cone Programming (SOCP)
Approximates nonlinearities via convex cones, providing near-exact results for radial networks under moderate loading conditions.
It ensures polynomial-time solvability while preserving reasonable physical fidelity.Semidefinite Programming (SDP)
Relaxes the voltage products into a positive semidefinite matrix constraint.
SDP formulations offer tight approximations and can recover globally optimal solutions under certain network conditions, though at higher computational cost.
These relaxations transform the D-OPF into a convex optimization problem, solvable by efficient algorithms and modern solvers.
MIXED-INTEGER EXTENSIONS
Real-world distribution systems include discrete operational decisions—such as switching configurations, capacitor banks, or DER activation.
These are modeled using Mixed-Integer Second-Order Cone Programming (MISOCP), which combines the tractability of SOCP with the binary decision modeling of MILP:
MISOCP formulations are essential for Active Distribution Networks (ADNs) and microgrids, enabling optimal scheduling, network reconfiguration, and participation in ancillary service markets.
SUMMARY OF D-OPF FORMULATIONS
Formulation |
Nature |
Convexity |
Typical Application |
Solver Examples |
|---|---|---|---|---|
AC D-OPF |
Nonlinear, nonconvex |
✗ |
Accurate modeling, research benchmarks |
Ipopt, Knitro |
SOCP D-OPF |
Convex relaxation |
✓ |
Radial networks, fast operation |
Gurobi, CPLEX, Mosek |
SDP D-OPF |
Semidefinite relaxation |
✓ (tight) |
Precise analysis, voltage optimization |
CVX, SDPT3, SeDuMi |
MISOCP D-OPF |
Mixed-integer convex |
✓ |
DER dispatch, topology switching |
Gurobi, CPLEX |
In conclusion, the integration of uncertainty-aware optimization with advanced D-OPF formulations provides the methodological foundation for smart, resilient, and data-driven distribution systems.
These approaches enable operators to make risk-informed decisions, optimize distributed generation and flexibility, and ensure secure operation even under volatile renewable and demand conditions.