Finding a Portfolio of Near-Optimal Aggregated Solutions to Capacity Expansion Energy System Models

Abstract

Energy system models are frequently being influenced by simplifications, assumption errors, uncertainties, incompleteness, and soft constraints which are challenging to model in a good way. In capacity expansion modeling, also the long time horizon and the high shares of renewable energies feed into the uncertainties. Consequently, a single optimal solution might not provide enough information to stand alone. Contrarily, a portfolio of different solutions, all being within an acceptance span of the system costs, would create more valuable decision support tool. This idea is known from the literature where a near-optimal solution space typically is explored by introducing integer cuts that iteratively cut off solutions as they are found. Generalizing this idea, we suggest an approach that explores the near-optimal solution space by iteratively finding new solutions which are as different as possible from earlier solutions with respect to investment decisions. Our method deviates from the literature since it maximizes the difference of the found solutions rather than finding k similar solutions. An advantage of this approach is that the resulting portfolio holds high diversity which creates a better basis for good decision-making. Moreover, it overcomes a potential struggle of getting symmetric solutions and it strengthens the robustness arguments of the different investment decisions. Furthermore, we suggest to search for alternative solutions in an aggregated solution space whereas the original solution space typically has been used for the search in previous work. We hereby exploit the speedup achieved through aggregation to find more solutions, and we observe that these solutions might indicate must have investments of the non-aggregated problem. The suggested approach is tested on a case study for three different limitations on the system costs. Results show that our approach by far outperforms the approach known from the literature when the neighborhood size exceeds 0.7%. Furthermore, using our approach, a portfolio of eight solutions with high diversity is found within the same time as the corresponding non-aggregated optimal solution. By looking into the different solutions, the relative importance of each unit investment is clearly identified, which potentially could be used to limit the gap between aggregated and non-aggregated solutions. Also, the portfolio in itself compensates for errors introduced by aggregation.

Appendices

Appendix A. Mathematical Formulation

Equations (6)–(19) cover the mathematical formulation of the capacity expansion model, where expansion costs are minimized subject to technical and operational constraints. A detailed description of the constraints is provided right after the mathematical model. Sets, parameters, and variables are listed in Table 4 while Table 5 provides an overview of the specific values applied in our case study. Notice that, for simplicity reasons, it is assumed that the variable costs are equal for all hours.

$$ \begin{array}{@{}rcl@{}} \text{Minimize} &{\sum}_{i\in I} \!\left( C_{i}^{INV} + C_{i}^{FOM}\right)\!y_{i} + {\sum}_{i\in I}{\sum}_{j\in J} &\!\!\!\left( C_{i}^{VAR} + C_{i}^{VOM} + P_{i}^{FUEL} HR_{i}\right)\\ &{}\times x_{ij} + C_{i}^{STUP} z_{ij} \end{array} $$

(6)

$$ \begin{array}{@{}rcl@{}} \text{Subject to}&{\sum}_{i\in I} x_{ij} \geq D_{j} & \forall j \in J \end{array} $$

(7)

$$ \begin{array}{@{}rcl@{}} &x_{ij} \leq \overline{P}_{i} y_{i} &\forall i\in I,\forall j \in J \end{array} $$

(8)

$$ \begin{array}{@{}rcl@{}} &x_{ij} \leq \overline{P}_{i} CF_{j}^{WIND} &\forall i\in I^{W},\forall j \in J \end{array} $$

(9)

$$ \begin{array}{@{}rcl@{}} &x_{ij} \leq \overline{P}_{i} CF_{j}^{SOLAR} &\forall i\in I^{S},\forall j \in J \end{array} $$

(10)

$$ \begin{array}{@{}rcl@{}} &u_{ij} - u_{ij-1} = z_{ij}-v_{i,j} &\forall i\in I^{T},\forall j \in J\setminus \{1\} \end{array} $$

(11)

$$ \begin{array}{@{}rcl@{}} &w_{ij} = x_{ij}-u_{ij}\underline{P}_{i} &\forall i\in I^{T},\forall j \in J \end{array} $$

(12)

$$ \begin{array}{@{}rcl@{}} &w_{ij} \leq u_{ij}(\overline{P}_{i}-\underline{P}_{i}) &\forall i\in I^{T},\forall j \in J \end{array} $$

(13)

$$ \begin{array}{@{}rcl@{}} &w_{ij} - w_{ij-1} \leq \overline{R}_{i}^{U} &\forall i\in I^{T},\forall j \in J\setminus \{1\} \end{array} $$

(14)

$$ \begin{array}{@{}rcl@{}} &w_{ij-1} - w_{ij} \leq \overline{R}_{i}^{D} &\forall i\in I^{T},\forall j \in J\setminus \{1\} \end{array} $$

(15)

$$ \begin{array}{@{}rcl@{}} &u_{ij} \geq {\sum}_{j^{\prime} > j-\overline{M}_{i}^{U}}^{j} z_{ij^{\prime}} &\forall i\in I^{T},\forall j \in J \end{array} $$

(16)

$$ \begin{array}{@{}rcl@{}} &1- u_{ij} \geq {\sum}_{j^{\prime} > j-\overline{M}_{i}^{D}}^{j} v_{ij^{\prime}} &\forall i\in I^{T},\forall j \in J \end{array} $$

(17)

$$ \begin{array}{@{}rcl@{}} &y_{i}, u_{ij}, v_{ij}, z_{ij} \in \{0,1\} &\forall i\in I, \forall j \in J \end{array} $$

(18)

$$ \begin{array}{@{}rcl@{}} &x_{ij}, w_{ij} \geq 0 &\forall i\in I, \forall j \in J \end{array} $$

(19)

Table 4 Sets, parameters, and variables of the capacity expansion model with unit commitment constraints

Table 5 Technological parameter values assumed for the case study of this paper. We assume $\overline {R}_{i}^{U} = \overline {R}_{i}^{D}$ wherefore only $\overline {R}_{i}^{U}$ is listed

The objective function (6) minimizes fixed and variable costs of investments and operations. The fixed costs $C_{i}^{FOM}$ cover investment costs and fixed O&M costs, while the variable costs $C_{ij}^{VOM}$ consist of fuel costs, variable O&M costs and variable operational costs. Constraint (7) ensures energy balance while constraints (8)–(10) handle the capacities. From these, it is seen that curtailment is allowed at no extra cost. Constraints (11)–(17) represent the unit commitment, meaning that these account for the commitment state, updating of shut-down and start-ups, ramping restrictions and minimum up and down times. Constraint (12) defines the auxiliary variable w_ij as the power generated above the minimum level of the unit. In Eqs. 18–19, the domain of the variables are defined. Note, that constraints (8) and (12) implicitly secure zero commitment state for non-built units.

Appendix B. Aggregation Technique and Aggregated Problem

This section provides further details on the exhaustive search (ES) aggregation technique which is applied in the case study. An outline of the algorithm is provided in Procedure 3. Recall from Section 4.1 that $L_{res}^{h} = D^{h} - W^{h} \cdot C_{W} - PV^{h} \cdot C_{P}$, where D is demand, W is wind, and C_W and C_P are the assumed maximal capacities of wind and PV in the system, respectively. Moreover, the algorithm introduces the normalized root mean square error (NRMSE) which is calculated as follows:

$$ \texttt{NRMSE} = \frac{\sqrt{\underset{t\in T}{\sum} \left( RLDC_{t} - \overline{RLDC}_{t}\right)^{2}}}{\vert T \vert}, $$

(20)

where RLDC_t is the original RLDC, $\overline {RLDC}_{t}$ is the approximated RLDC, and |T| is the amount of hours in the original instance (8736 h in our case study). We furthermore introduce χ ∈ X as a week χ belonging to the set of all weeks X. In our case study, we selects 4 weeks out of 52 weeks, corresponding to a 92% data reduction. How this affects the mathematical problem formulation is seen in Table 6. Some statistical relations between the non-aggregated and aggregated input time series are seen in Table 7. Figure 12 illustrates how the input time series differ among the three problem instances P₂₀₁₄, P₂₀₁₅, and P₂₀₁₆, which similarly is seen for the aggregated time series in Fig. 13.

Table 6 Relation between non-aggregated and aggregated problem sizes (notice that these relate to P and not to Q problems). The mathematical model size is equal for the three different problems P₂₀₁₄, P₂₀₁₅, and P₂₀₁₆, wherefore only one non-aggregated and one aggregated problem size are seen. The solution time, however, differs for the three instances and the listed solution times therefore relate to the average values

Table 7 Statistical measures of the three types of input time series: demand, wind, and PV both in non-aggregated and aggregated forms

Appendix C. Input Data

The non-aggregated input time series consist of hourly demand profiles and hourly wind and PV availability profiles. Each problem (P₂₀₁₄, P₂₀₁₅, P₂₀₁₆) covers a single region and hence is associated to a single input time series of each type. Graphical illustrations of the non-aggregated input time series are seen in Fig. 12, while the corresponding aggregated profiles are illustrated in Fig. 13. To further illustrate differences among time series of the different years and differences among aggregated and non-aggregated time series, a selection of statistical measures for each profile are seen in Table 7.

Fig. 12

Graphical illustration of how the input time series differ for the 3 years 2014, 2015, and 2016. Wind and PV are illustrated as predicted capacities throughout the year. This means that the graphs illustrate the wind and PV capacities arising from an investment in all candidate units of the respective type

Fig. 13

Graphical illustration of how the aggregated input time series differ for the 3 years 2014, 2015, and 2016. Wind and PV are illustrated as predicted capacities throughout the aggregated period. This means that the graphs illustrate the wind and PV capacities arising from an investment in all candidate units of the respective type

Appendix D. Supplementary Graphs for the Result Section

Fig. 14

The evolution of the deviation in investment strategies across the iterations. As the method prevents the occurrence of identical solutions, missing columns are caused by non-existing solutions

Fig. 15

Investment decisions for each iteration of the PoMDS algorithm solving the Q_maxmin problem. The solutions are related to P₂₀₁₄ and a maximum deviation in system costs of 5%

Fig. 16

Investment decisions for each iteration of the PoMDS algorithm solving the Q_sum problem. The solutions are related to P₂₀₁₄ and a maximum deviation in system costs of 5%

Fig. 17

Illustration of the solutions being the most different from the optimal one with respect to investment decisions. The relation between the degree of difference and the associated solution time is seen