Optimization models

Optimization models focus on identifying the best option out of all the possible ones by defining a single target that is either to be minimized or maximized while accounting for side constraints.

An optimization problem has the general form of:

(1)   (min) (F(x))

(2) s.t. (G(x)=0)

(3)   (H(x)leq0)

Equation (1) represents the desired objective function, (F(x)). Note that minimizations (or maximizations) can be transformed into maximizations (minimizations) by multiplying the objective function by -1 (ie, (max) (F(x) = min) (–F(x))).
Equation (2) represents equality constraints covering all constraints that have to hold with equality (eg, flow conservation constraints or temporal balances) and definitions.
Equation (3) represents inequality constraints covering upper and lower limits (eg, production capacities). Lower equal and greater equal formulations can be transferred into each other by multiplying the constraint by -1.

Note that you will not necessarily need all types of constraints for all models. Even if the detailed design depends on the underlying focus of your model, the overall model layout will always follow the above-described structure. For example, if you want to design a market model you will need constraints for the supply side, the demand side, and their market interaction. The structure of optimization models makes it easy to add or withdraw elements from a model by simply changing the formulation of the side constraints (ie, adding a further technical restriction).

Optimization models are often used to represent benchmark market conditions. They can easily obtain least-cost or welfare maximizing solutions that correspond to a perfect competitive market environment. Many large scale, bottom-up energy market models follow an optimization approach and include several technical side constraints to capture the underlying energy conversion and transport mechanics.

Recommended readings

In the literature recommendation below, you will find a simple natural gas market model (Neumann et al. 2009) and an electricity network model (Leuthold et al. 2008) examples. Both models follow a welfare maximizing approach. In the gas model, the constraints capture the gas transport via pipelines, ship (both require a network topology), and intertemporal storage. In the electricity model, the physics of power flows have to be included (again requiring a network topology) as well as power plant characteristics (introducing binary variables, more on this in Week 5) and pumped storage dynamics. Market models following those two examples are typically relatively easy to design as they have a limited set of needed equations. Nevertheless, they allow us to analyse and evaluate market challenges and thereby provide a good starting point for numerical modelers.

Neumann, A. et al. (2009). InTraGas – A Stylized Model of the European Natural Gas Network. Dresden University of Technology.

Leuthold, F. et al. (2008). ELMOD – A Model of the European Electricity Market. Dresden University of Technology. (Journal Version: Leuthold, F. et al. (2012). A large-scale spatial optimization model of the European electricity market. Networks and spatial economics, 12(1), pp. 75-107.)

For those who want to get more familiar with GAMS as modeling software: the initial GAMS tutorial uses a simple transport optimization problem to introduce the different features of GAMS.