*We had a slightly different speaker than originally planned at this week’s seminar but it was great to be able to host the wonderful Maike Hennen. Maike is a PhD student from RWTH Aachen; she has been visiting Imperial to work in Prof. Nilay Shah’s group for the last few months. Her talk on designing energy supply systems was really interesting and she has written this blog on the same topic and you can also download her slides[PDF].*

How we move energy around, whether that is in a building, city or country is a very complex problem. A well planned energy supply system will save you time, money and headaches in the long run. This is not to say it is easy but is vital if we want to be sustainable: The design of an energy system fixes already the major part of costs and life-cycle emissions of the energy system even before it is built. While my research focusses on how these systems can be best designed at medium scale, like an industrial park or a city district, the framework is scalable from a single office to an entire country.

To support the decision maker in the design of energy systems, mathematical optimisation can be used. For this purpose, a model of the energy system has to be developed.

This model contains three levels (Frangopoulos et al., 2002) as shown in Figure 1:

- On the upper level, it is decided which components are part of the optimal system transforming input energy (gas, electricity) into energy to satisfy demands (heat, cold, electricity)
- On the second level, the components chosen on the synthesis level are sized, i.e., the optimal capacity is selected.
- On the lowest level, the operation schedule for each component is defined

Solving the design problem requires to make all these decisions at once.

## What is your objective?

Normally, the aim of an optimisation problem in the design of energy systems is finding the cost-optimal solution. However, recently other objectives such as environmental impact can play an important role. Thus, one single objective function is insufficient to capture the goals of the designer. Considering multiple objective functions at once leads to a large set of solutions. Each solution performs best regarding a specific combination of all objective functions. The designer has the complicated task of selecting a single compromise solution. To simplify this task, my team has recently proposed a method to reduce the number of solutions (Hennen et al., 2016).

In Figure 2, this new method is illustrated. In the first step, the number of objectives is reduced. Only objective functions that show the inherent trade-offs best are retained using the method by Guillén-Gosálbez (2011). This reduced set of objectives corresponds to a dimensionality reduction of the problem and already simplifies the selection task for the designer.

However, objective functions which show best the inherent trade-offs usually do not include aggregated objective functions such as total annual costs. Still, total annual costs or net present value are the most important decision criterion in investment planning projects.

So in the second step, we reduce the solution space by only taking into account solutions which perform close to the minimal total annual costs. The resulting set of solutions only includes solutions which are relevant in the final decision and show the most important trade-offs.

In Figure 3, the results of a case study of an industrial site are shown. On the industrial site, cooling, heating, and electricity demands have to be satisfied. The first step of the proposed method (Figure 2) identifies investment costs and global warming impact as the two objective functions which capture the most important trade-offs. The results of a bi-objective optimization regarding investment costs and global warming impact are shown in Figure 3 in black. The second step of the proposed reduction method reduces the solution space to solutions with total annual costs lower than 110 % of the minimal costs. With this additional constraint on total annual costs, the red curve in Figure 3 is the result of the optimization. These final solutions are concentrated in an area of good compromise solutions. Thus, the decision maker has to analyse and evaluate only a small set of solutions that all are relevant in terms of total annual costs.

## What to do if “all models are wrong”?

The optimal solution of the mathematical optimization problem is the best solution – but only for the chosen model of the energy system. Besides modelling errors, the transfer of this solution into practice also usually requires additional soft constraints which cannot be modelled. Thus, the optimal solution has to be adapted. (Box and Draper, 1987)

However, having only the optimal solution at hand, the design engineer does not know which components of the solution are must-haves of a good solution and which changes of the structure are must-avoids (Voll et al., 2015). To provide additional solutions, the near-optimal decision space can be explored. The exploration of the decision space includes the calculation of the optimal solution and near-optimal solutions. To generate the near-optimal solutions, commonly integer-cuts are applied which exclude the already known solutions and resolve the optimization problem (Fazlollahi et al., 2012). The additional generated solutions provide knowledge to the decision maker: Which parts of the solution can be changed without worsening it? And what impact does this change have on the remaining components? With the answers to these questions, the design engineer can adapt the solution to his/her needs.

## Is optimal always the best?

In the design of energy systems, it is difficult to define what is optimal as discussed above for multi-objective optimization. Furthermore, the optimal solution of the model often has to be adapted to the real world due to soft constraints and modelling accuracy. Thus, optimal solutions of an optimization problem in the design of energy systems are the best in terms of mathematical optimality. However, the final choice of the decision maker probably involves changes to the mathematical optimal solution to find the practical feasible best solution. This adaptation of the optimal solution can be supported by the generation and analysis of near-optimal solutions.