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

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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.

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Abbreviations

CEP :

capacity expansion problems

GEP :

generation expansion problems

MID :

maximized investment different solution

MDS :

maximized diversity solution

ATQ :

aggregation technique

Agg.:

aggregated

non − Agg.:

non-aggregated

DS :

data series

D S a g g :

aggregated data series

P o M D S :

portfolio of maximized diversity solutions

NOS :

near-optimal solution

VRE :

variable renewable energy

PV :

photovoltaics

CCGT :

combined cycle gas turbine

OCBT :

open cycle gas turbine

ES :

exhaustive search

RLDC :

residual load duration curve

I R x :

irregular run, type x

MH :

must haves

RC :

real choices

R C W :

weak real choices

R C S :

strong real choices

MA :

much avoids

Q s u m :

problem where the objective maximizes the sum of distances to all previous solutions

Q m a x m i n :

problem where the objective maximizes the minimum distance to the previous solutions

T m a x :

time limit for each iteration

𝜖 :

maximum change in system cots (neighborhood size)

P x :

Capacity expansion problem based on data series x

References

  1. Alvarez GE, Marcovecchio MG, Aguirre PA (2020) Optimization of the integration among traditional fossil fuels, clean energies, renewable sources, and energy storages: an milp model for the coupled electric power, hydraulic, and natural gas systems. Computers & Industrial Engineering 139:106–141

    Google Scholar 

  2. Babatunde OM, Munda JL, Hamam Y (2019) A comprehensive state-of-the-art survey on power generation expansion planning with intermittent renewable energy source and energy storage. Energy Research

  3. Buchholz S, Gamst M, Pisinger D (2019) A comparative study of aggregation techniques in relation to capacity expansion energy system modeling. TOP Issue 3:2019

    Google Scholar 

  4. Bylling HC, Pineda S, Boomsma TK (2017) The impact of short-term variability and uncertainty on long-term power planning problems. Ann Oper Res 239:1–27

    Google Scholar 

  5. Cao KK, Metzdorf J, Birbalta S (2018) Incorporating power transmission bottlenecks into aggregated energy system models. Sustainability (Switzerland) 10:1916

    Article  Google Scholar 

  6. Chung TS, Li YZ, Wang ZY (2004) Optimal generation expansion planning via improved genetic algorithm approach. International Journal of Electrical Power and Energy System 26:655–659

    Article  Google Scholar 

  7. De Jonghe C, Delarue E, Belmans R, D’haeseleer W (2011) Determining optimal electricity technology mix with high level of wind power penetration. Energy Systems 88:2231–2238

    Google Scholar 

  8. de Sisternes FJ, Webster M (2013) Optimal selection of sample weeks for approximating the net load in generation planning. problems Massachusetts Institute of Technology Engineering Systems Division

  9. de Sisternes FJ, Webster MD, Pérez-Arriaga I.J. (2015) The impact of bidding rules on electricity markets with intermittens renewables. IEEE Trans Power Sys 30(3):1603–1613

    Article  Google Scholar 

  10. Energinet.dk. Markedsdata

  11. Fazlollahi S, Mandel P, Becker G, Maréchal F. (2012) Methods for multi-objective investment and operating optimization of complex energy systems. Energy 45:12–22

    Article  Google Scholar 

  12. Fischetti M, Monaci M (2014) Proximity search for 0-1 mixed-integer convex programming. J Heuristics 20:709–731

    Article  Google Scholar 

  13. Gamst M Sifre: Simulation of flexible and renewable energy sources

  14. Gebreslassie Berhane H, Gonzalo GG, Laureano J, Dieter B (2009) Design of environmentally conscious absorption cooling systems via multi-objective optimization and life cycle assessment. Appl Energy 86:1712–1722

    Article  Google Scholar 

  15. Heuberger CF, Rubind ES, Staffell I, Shah N, Dowell NM (2017) Power capacity expansion planning considering endogenous technology cost learning. Appl Energy 204:831–845

    Article  Google Scholar 

  16. Hinojosa V, Gil E, Calle I (2018) A stochastic generation capacity expansion planning methodology using linear distribution factors and hourly load modeling. 2018 International Conference on Probabilistic Methods Applied To Power Systems, Pmaps 2018 - Proceedings 88:8440244

    Google Scholar 

  17. IRENA (2017) Planning for the renewable future: long-term modelling and tools to expand variable renewable power in emerging economies. International Renewable Energy Agency, Abu Dhabit

  18. Kåberger T. (2018) Progress of renewable electricity replacing fossil fuels. Global Energy Interconnection 1:48–52

    Google Scholar 

  19. Kirschen DS, Ma J, Silva V, Belhomme R (2011) Optimizing the flexibility of a portfolio of generating plants to deal with wind generation. IEEE Power and Energy Society General Meeting

  20. Koltsaklis NE, Dagoumas AS (2018) State-of-the-art generation expansion planning: a review. Applied Energy 230:563–589

    Article  Google Scholar 

  21. Krishnan V, Cole W (2016) Methods for multi-objective investment and operating optimization of complex energy systems. IEEE Power and Energy Society General Meeting: 7741996

  22. Liu P, Pistikopoulos EN, Li Z (2010) An energy systems engineering approach to the optimal design of energy systems in commercial buildings. Energy Policy 38:4224–4231

    Article  Google Scholar 

  23. Matthew D, Efstathios E, Dimitrios N (2019) Michaelidesr Michaelides Leonard Energy storage needs for the substitution of fossil fuel power plants with renewables. Renew Energy 145:951–962

    Google Scholar 

  24. Merrick JH (2016) On representation of temporal variability in electricity capacity planning models. Energy Economics 59(19):261–274

    Article  Google Scholar 

  25. Munoz FD, Mills AD (2015) Endogenous assessment of the capacity value of solar pv in generation investment planning studies. Transactions on Sustainable Energy 6(4):1574–1585

    Article  Google Scholar 

  26. Müller B., Gardumi F, Hülk L. (2018) Comprehensive representation of models for energy system analyses Insights from the energy modelling platform for europe (emp-e) 2017. Energy Strategy Reviews 21:82–87

    Article  Google Scholar 

  27. Oree V, Sayed Hassen SZ, Fleming PJ (2017) Generation expansion planning optimisation with renewable energy Sayed a review. Renewable and Sustainable Energy Reviews 69:1369–1394

    Article  Google Scholar 

  28. Palmintier B (2013) Incorporating operational flexibility into electric generation planning : impacts and methods for system design and policy analysis. Massachusetts Institute of Technology

  29. Palmintier B, Webster M (2011) Impact of unit commitment constraints on generation expansion planning with renewables. IEEE, Power and Energy Society General Meeting

  30. Pereira AJC, Saraiva JT (2011) Generation expansion planning (gep) - a long-term approach using system dynamics and genetic algorithms (gas). International Journal of Electrical Power and Energy System 65:5180–5199

    Google Scholar 

  31. Poncelet K, Delarue E, Six D, Dueinck J, D’haeseleer W (2015) Impact of the level of temporal and operational detail in energy-system planning models. Appl Energy 162(58):631–643

    Google Scholar 

  32. Poncelet K, Hoschle H, Delarue E, Virag A, D’haeseleer W (2016) Selecting representative days for capturing the implications of integrating intermittent renewables in generation expansion problems. IEEE Transactions on Power Systems PP(99):1–1

    Google Scholar 

  33. Ravn H The balmorel model structure

  34. Ringkjøb H.-K., Haugan PM, MarieSolbrekke I (2018) A review of modelling tools for energy and electricity systems with large shares of variable renewables. Renew Sustain Energy Rev 96:440–459

    Article  Google Scholar 

  35. Safari S, Ardehali MM, Siriz MJ (2013) Particle swarm optimization based fuzzy logic controller for autonomous green power energy system with hydrogen storage. Energy Convers Manag 36:41–49

    Article  Google Scholar 

  36. Schwele A, Kazempour J, Pinson P (2018) Do unit commitment constraints affect generation expansion planning? a scalable stochastic model. Energy Sys. https://doi.org/10.1007/s12667-018-00321-z

  37. Taghavi M, Huang K (2014) Stochastic capacity expansion with multiple sources of capacity. Oper Res Lett 14:263–267

    Article  Google Scholar 

  38. Teichgraeber H, Brandt AR (2019) Clustering methods to find representative periods for the optimization of energy systems: an initial framework and comparison. Appl Energy 239:1283–1293

    Article  Google Scholar 

  39. Ueckerdt F, Brecha R, Luderer G (2015) Analyzing major challenges of wind and solar variability in power systems. Renew Energy 81(1):1–10

    Article  Google Scholar 

  40. Voll P, Jennings M, Hennen M, Shah N, Bardow A (2015) The optimum is not enough: a near-optimal solution paradigm for energy systems synthesis. Energy 82:446–456

    Article  Google Scholar 

Download references

Funding

This study was funded by the FUTUREGAS research project.

Author information

Authors and Affiliations

  1. DTU Management Engineering, Technical University of Denmark, Produktionstorvet, 424, 2800, Kgs. Lyngby, Denmark

    Stefanie Buchholz & David Pisinger

  2. Energinet.dk, Tonne Kjærsvej, 65, 7000, Fredericia, Denmark

    Mette Gamst

Authors
  1. Stefanie Buchholz
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Correspondence to Stefanie Buchholz.

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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
Full size table
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
Full size table

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 wij 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 CW and CP 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 RLDCt 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.

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

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
figure 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

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Fig. 13
figure 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

Full size image

Appendix D. Supplementary Graphs for the Result Section

Fig. 14
figure 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

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Fig. 15
figure 15figure 15

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

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Fig. 16
figure 16figure 16

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

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Fig. 17
figure 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

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Buchholz, S., Gamst, M. & Pisinger, D. Finding a Portfolio of Near-Optimal Aggregated Solutions to Capacity Expansion Energy System Models. SN Oper. Res. Forum 1, 7 (2020). https://doi.org/10.1007/s43069-020-0004-y

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