Technical Background of Optimization Illustrated Through Energy Market Applications

Optimization is a mathematical modeling methodology used to find the best solutions to complex problems subject to user-specified criteria and system constraints. This article provides a basic technical background on the application of optimization modeling techniques in energy market analysis, trying to demystify the technical terminology and make it accessible to practitioners.

Key Takeaways

• Mathematical optimization is vital in energy market analysis as it helps navigate multi-dimensional decisions and trade-offs between objectives.
• Optimization models are built on key components: decision variables, parameters, objective functions, and constraints, which collaboratively guide decision-making processes toward optimal solutions.
• There are various categories of optimization models that serve distinct purposes in energy market applications, facilitate various degrees of complexity, and address uncertainties in data.

Introduction

As the previous articles in this series have demonstrated, mathematical optimization is an indispensable tool for the analysis of energy markets and, specifically, electric power markets. Its value stems from its ability to consider multi-dimensional decisions and account for tradeoffs between goals and requirements. But how does it work? With the ubiquity of AI, any computer-generated analytical solution might seem like “AI magic” to a novice user, making optimization appear like a black box. Although optimization serves as a building block for some AI methods and is an AI methodology itself, its unique structure and algorithms set it apart.

Optimization is a prescriptive methodology, offering outputs that aid in decision-making. Grasping the technical background of this framework enhances appreciation for the recommendations produced by optimization models. This knowledge is beneficial for consumers of modeling results and essential for analysts to derive insights from the model’s solutions.

While this article deals with the technical aspects of optimization, it provides practical examples from applications to electric power markets and is accessible to a wide audience.

Building Blocks of Optimization Models

Optimization is useful in cases where there is some set of available courses of action to choose from in order to achieve an objective subject to a set of constraints. Accordingly, models consist of several elements that work together to represent the problem, explore possible solutions, and achieve specific objectives.

These components, which are standard across all models, are described below and include decision variables, parameters, an objective function, and constraints. Each plays a crucial role in shaping the model and guiding it towards an optimal solution.

Decision Variables

Decision variables represent the available options or actions that the model seeks to optimize. In other words, they represent the choices that are under the decision maker’s control. When optimization algorithms run to solve optimization problems, they search for the optimal values of these decision variables. Here are some examples of decision variables in the realm of power markets:

• • The capacity to build of a specific resource type, such as a solar resource or a combined cycle, in a given year
  • The optimal timing of a coal plant retirement
  • Dispatch levels for each generator in a utility’s fleet in a given hour
  • Whether a generator is synchronized to the grid or offline
  • The amount of energy to store or release from energy storage in a given hour

These variables are crucial for decision making as they directly influence the outcomes of the model. Decision variables are usually “free” variables chosen by the model in trying to optimize some objective which depends on these variables, but some of the variables can also be fixed at specific values. By manipulating these variables, one can simulate various strategies and identify the most effective approach for achieving the desired objectives.

Defining and understanding decision variables is essential because they effectively represent specific levers that modelers and stakeholders can pull to explore the trade-offs and gauge their influence on overall profits or costs.

Parameters

Parameters are fixed and exogenous inputs to the model that describe the environment or constraints within which decisions are made. These values are typically derived from physical system characteristics, historical data, forecasts, or policy requirements. They are assumed to be beyond the control of the decision maker. In electric power markets, common parameters include:

• • Natural gas, coal, uranium, and other generation fuel prices
  • Hourly wind and solar generation availability as a percentage of maximum rated capacity
  • Electricity demand forecasts, their hourly shape, and their year-to-year growth
  • Capital investment costs for a new power plant
  • Physical constraints such as generation capacity limits, ramping abilities, or transmission line ratings
  • Regulatory requirements like emissions caps or renewable portfolio standards

The role of parameters is to set the stage for the optimization process. They provide the context within which decision variables operate, and their values significantly influence the outcomes of the model. By accurately defining parameter values, modelers ensure that the optimization models reflect real-world conditions.

Objective Function

The objective function defines the model’s goal, whether it is minimizing costs, maximizing profits, or achieving an optimal balance of multiple goals. This function is the driving force behind the optimization process, guiding the model towards its desired outcome.

In electric power markets, common objective functions include:

• • Cost minimization, which focuses on reducing the total cost of electricity generation and delivery
  • Profit maximization, which aims to maximize revenues for power producers while considering market prices and costs of production
  • Environmental goals, such as minimizing greenhouse gas emissions or renewable curtailment

Formulating the objective function is a critical step in mathematical optimization. It requires a deep understanding of the problem at hand and the ability to translate complex goals into mathematical terms. This process may involve balancing multiple objectives and making tradeoffs to achieve the best possible outcome.

Constraints

Constraints are the rules, limits, or requirements that the solution must adhere to. They ensure the feasibility and practical applicability of the model’s outputs. Constraints are represented by mathematical equations that define the feasible space of the optimization problem. Following are some examples of constraints in electric power markets:

• • Physical constraints include power plants’ capacity and ramping limits (e.g., generating up to but not above the rated capacity) and energy limits (e.g., storage systems being able to discharge only the energy they have stored). Transmission line ratings are another example of physical constraints as they are used to specify limits on transmission flows between regions.
  • Market constraints involve adherence to market rules, such as ensuring that demand is met by generation and net imports or that the frequency regulation requirements are met at every time step.
  • Policy constraints include compliance with emissions limits or renewable energy mandates. In a capacity expansion context, planning reserve margins or expected peak energy requirements expressed in terms of effective load carrying capability or a similar metric are also policy constraints.
  • Logical constraints are used in cases where certain decisions depend on others. They enforce logical relationships, such as an “if-then” or an “either-or” dependency. For example, a requirement to build a storage resource if more than a certain amount of solar capacity is added to a system can be encoded as a logical constraint. Another example could be a choice between two mutually exclusive options, such as building either a new peaker or a new combined cycle.

Defining the decision variables, identifying parameters, and formulating an optimization model by writing out the objective function and constraints are crucial steps of the optimization framework. A skilled modeler can translate the important aspects of a real-world process into mathematical equations which allows for detailed analysis of the process and robust decision-making.

Categories of Optimization Models in Energy Market Applications

Optimization models come in various forms, each tailored to specific types of problems and challenges. Some of the aspects that affect the choice of model are the assumptions that the stakeholders make, the complexity of the problem, and the level of precision required. The classification will affect the ease with which algorithms are able to solve the problem.

We will go over three common categories of optimization models: linear optimization, integer optimization, and stochastic optimization. Each of these types of models offers unique capabilities and is suited to different types of problems.

Linear Optimization

Linear programming (LP) models are used when the relationships between variables are linear. In the context of an economic dispatch problem, this means that, for example, we assume the rates of increase of variable O&M costs and fuel costs are constant with respect to generation. LP models also assume that decision variables are continuous, which means that a decision variable, such as the amount of generation or the capacity to build, can take on any value between its minimum and maximum bounds.

One important property of LP models is that they can be solved to global optimality. In other words, an LP solution algorithm is able to locate a solution such that there is no alternative set of values that satisfy all constraints and that can produce a better outcome. This is often very insightful for decision makers as the optimal course of action can be something they never even considered. This property sets LPs apart from non-linear programming (NLP) models where global optimality of solutions cannot be guaranteed.

Applications of LP models in the energy market include real-time market clearing, determining least-cost generation dispatch, and optimizing fuel procurement decisions. The ease of implementation and the ability to solve large-scale problems efficiently make LP models widely used in these applications. LP models also have some limitations, chief of which is their inability to capture non-linear relationships, such as elasticity of demand or fixed costs. In spite of this, LP models remain a powerful tool for optimization in energy markets, providing optimal solutions when assumptions hold.

Integer Optimization

What sets integer programming (IP) models apart is that the decision variables are restricted to take on discrete integer values.  In some generation capacity expansion models, capacity can only be added in blocks. For example, if there is a standard size for a peaker, the integer variable can represent the number of peaker units to build. It can take on values of 0, 1, 2, and so forth, but not fractional values such as 1.7.

Further, a special case of an integer variable is a binary variable that is restricted to take values of 0 or 1, which is useful for handling logical constraints. In a unit commitment model, binary variables are used to represent whether or not a specific generator is synchronized to the grid and to track startup and shutdown decisions and account for the associated costs. These variables are essential for decisions that involve discrete choices.

While IP models offer flexibility to model complex operational constraints and capture real-world discrete decisions, they are computationally intensive for large-scale problems. The increased computational demands can be a significant drawback, but the ability to represent discrete decisions makes IP models invaluable for certain types of optimization tasks.

Stochastic Optimization

Stochastic optimization models incorporate uncertainty into the optimization process, making them ideal for problems with variable inputs such as fuel prices or weather-dependent generation availability. Unlike scenario analysis, which solves multiple deterministic models separately, stochastic programming endogenizes the uncertainty and produces a single solution that accounts for the fact that the exact values of some high impact parameters are not known with certainty at the time stakeholders have to make the decision.

A useful feature of stochastic models is that their objective can be expressed in terms of a risk metric (e.g., conditional value at risk). Optimizing with regards to expected value of costs or revenues assumes that the stakeholders are risk neutral. With stochastic models, modelers are able to optimize the decisions with regards to a risk metric, which aligns the solution and the recommendations with the stakeholders’ risk preferences.

Applications of stochastic optimization in the energy market include capacity expansion under uncertain demand growth, fuel costs, or regulatory requirements or scheduling of maintenance outages facing uncertain demand and renewable resource availability profiles.

The increased complexity and computational demands of stochastic optimization models are notable limitations. However, the robustness of solutions and the ability to account for uncertainty make stochastic optimization a powerful tool for strategic planning in energy markets.

Optimization Solvers

Optimization solvers are the engines that drive the optimization process. They transform mathematical models into actionable insights. Optimization solvers use algorithms that are designed to find the best possible solution to an optimization problem. Given a mathematical model, an optimization solver systematically explores the feasible solution space defined by the constraints to identify an optimal solution for the specified objective. If you are in a room, think of the feasible solution space as that room. An optimal solution to the optimization problem can be anywhere in the room but not outside of it. The floor, ceiling, and walls are the constraints. A solver explores the room to find an optimal solution. It does so in a methodical way and efficiently, so that it does not have to test every single point in the room. 

When the model is feasible and bounded (i.e., when there exists at least one solution that satisfies all the constraints and there is at least one constraint that prevents the objective function from improving indefinitely), the algorithm will yield an optimal solution. A solution to an optimization problem consists of a set of values for the decision variables that produces the best objective function value and can be used to guide the stakeholders in their decision-making process.

Summary

Throughout this guide, we have explored the technical background of optimization and its applications in energy markets. Understanding the building blocks of optimization models, including decision variables, parameters, objective functions, and constraints, is essential for effectively applying optimization methods in the energy sector. We have also examined different categories of optimization models. Each of the models offers unique capabilities and is suited to different types of applications. Lastly, optimization solvers play a critical role in producing outputs that can be used by analysts to derive actionable insights and recommendations.

As the energy industry continues to evolve, the importance of optimization will only grow with it. The purpose of this article is to expose stakeholders to the basic technical background of optimization and to demystify the optimization process so that it is more accessible and more widely used by practitioners.

Frequently Asked Questions

What are decision variables in optimization models?

Decision variables in optimization models are the specific actions or choices that the decision maker can control, such as capacity builds or resource allocations, which the model aims to optimize for better outcomes.

How do parameters influence optimization models?

Parameters significantly influence optimization models by providing the necessary fixed inputs, such as fuel prices and demand forecasts, that define the maximum availability of resources, the minimum requirements, and the environment for decision-making. Their values can directly impact the effectiveness and outcomes of the optimization process.

What is the role of the objective function in optimization models?

The objective function plays a crucial role in optimization models by defining the specific goal of the model, such as minimizing costs or maximizing profits, and guiding the process toward achieving that desired outcome. Without a clear objective function, the optimization model would lack direction and purpose and any feasible solution would be an optimal solution.

Why are constraints important in optimization models?

Constraints are essential in optimization models as they establish the rules and limits that ensure the feasibility and practical applicability of the solutions. By defining the feasible space, they guide the optimization process towards realistic and implementable outcomes.

What are the advantages of stochastic optimization in energy markets?

Stochastic optimization enhances decision-making in energy markets by explicitly addressing uncertainty, thereby improving solution robustness and enabling the management of risk exposure. This approach accommodates the risk preferences of stakeholders, making it a valuable tool in fluctuating environments.