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PROCEEDINGS OF ECOS 2019 - THE 32ND INTERNATIONAL CONFERENCE ON

EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS

JUNE 23-28, 2019, WROCLAW, POLAND

RiSES4

Rigorous Synthesis of Energy Supply Systems

with Seasonal Storage by relaxation and time-

series aggregation to typical periods

Nils Baumgärtner a, Frederik Temme a, Björn Bahl a, Maike Hennen a, Dinah

Hollermann a, and André Bardow a,b

a Institute of Technical Thermodynamics, RWTH Aachen University, 52056 Aachen, Germany,

Nils.Baumgaertner@ltt.rwth-aachen.de, Frederik.Temme@rwth-aachen.de, Bjoern.Bahl@rwth-

aachen.de, Maike.Hennen@rwth-aachen.de, Dinah.Hollermann@rwth-aachen.de

b Institute of Energy and Climate Research - Energy Systems Engineering (IEK-10), Forschungszentrum

Jülich GmbH, 52425 Jülich, Germany, Andre.Bardow@ltt.rwth-aachen.de

Abstract:

The synthesis of energy systems is a complex task that requires the simultaneous

optimization of the design and operation of all energy conversion units and storage

systems. Typically, the synthesis depends on multiple large time series, e.g., demand

profiles, electricity prices, and renewable resources, leading to large-scale optimization

problems. Problem complexity increases further due to long-term time-coupling

constraints, e.g., due to seasonal storage. Consequently, the resulting synthesis problems

are computationally challenging, and thus, often not solvable within reasonable

computational time or memory limits. In practice, the problem size of synthesis problems

is therefore usually reduced by time-series aggregation. However, the solution of a

reduced synthesis problem is not the solution of the original synthesis problem. Thus, the

solution quality is unknown and the resulting design might even be infeasible for the full

time series. To obtain a feasible solution with known quality, exact solution strategies are

needed. Previously, we proposed an exact decomposition method to prove optimality and

feasibility of the resulting design. However, the previously proposed method does not

consider long-term time-coupling constraints, which, e.g., prohibits modelling of seasonal

storage. Here, we propose the method RiSES4 that allows the synthesis of energy systems

with long-term time-coupling constraints with known solution quality. RiSES4 provides

feasible solutions (upper bounds) based on restrictions and determines the solution quality

based on lower bounds. Lower bounds are provided by linear-programming relaxation

and relaxation based on time-series aggregation. We obtain feasible solutions by time-

series aggregation in the synthesis problem and subsequently we solve an operational

problem. The bounds are tightened by iteratively increasing the resolution of the time-

series aggregation. RiSES4 is applied to 2 complex synthesis problems considering large

time series and long-term time-coupling constraints. RiSES4 shows fast convergence

significantly outperforming a commercial state-of-the-art solver.

Keywords:

Large-scale MILP, Design optimization, Typical periods, Storage Systems, Decomposition

1. Introduction

Climate change mitigation in combination with growing energy demands forces a transition from

fossil-based towards renewable-based energy systems. However, the integration of renewable-based

energy is challenging, as their generation is difficult to predict, the output power is highly fluctuating

and rarely correlates with the given energy demand [1].

To overcome these challenges, energy storage can balance supply and demand. Zerahn et al. [2] show

by a literature review that the requirement for short-term energy storage systems is increasing with

the usage of renewable energy. Cebulla et al. [3] conclude that for high shares of renewable energy,

energy systems have to cope with longer periods of low renewable generation. Thus, seasonal energy

storage are necessary to balance longer periods of low renewable generation. Hence, both short-term

and long-term storage systems are needed for the successful synthesis of renewable-based energy

systems [4].

Synthesis of energy systems including renewable energies and storage systems is often realized by

mathematical optimization [5]. Synthesis optimization problems are typically formulated as linear

programming (LP) [6] or mixed-integer linear programming (MILP) problems [5,7]. Small-scale

(MI)LPs can be solved fast to global optimality. However, the synthesis of energy systems depends

on multiple large time series, e.g., hourly demand profiles, electricity prices, and renewable resources,

leading to large-scale optimization problems. Consequently, the resulting synthesis optimization

problems are computationally challenging. Recently, Goderbauer et al. [8] proofed that synthesis

optimization of energy systems are even strongly NP-hard (unless P=NP). Thus, synthesis

optimization problems are often not solvable within reasonable computational time or memory limits.

In particular, the integration of (seasonal) energy storage increases the complexity of the synthesis

optimization problem, because energy storage systems require a large and highly resolved time series

[9,10] and lead to time-coupling constraints [11]. To still solve the resulting synthesis problems, the

problem size is usually reduced by time-series aggregation [12]. However, state-of-the-art time-series

aggregation methods often consider independent typical days, e.g. [13], and thus are not able to

include long-term energy storage into the synthesis optimization. To consider seasonal storage with

time-series aggregation, Rager [14] and Samsatli et al. [15] select typical days for each month of the

year or per season resulting in coupled typical days. However, this approach does not represent the

original time series accurately, as diversity of days within a month or season might not be captured

by only one typical day [16]. Gabrielii et al. [10] as well as Renaldi et al. [17] therefore developed

synthesis methods for energy systems including seasonal storage systems based on time-series

aggregation. In Reference [10,16], typical periods are coupled. In reference [17], a second time grid

is introduced. Their methods enable integration of seasonal storage systems into energy systems

synthesis in reasonable computational time.

However, these synthesis methods only solve a reduced synthesis problem. Thus, the solution does

not correspond to the solution of the original synthesis problem employing the full time series. As a

result, the solution quality of the reduced synthesis problem for the full time series is unknown and

the resulting design might even be infeasible. In particular, storage optimization is more sensitive to

time-series aggregation than other technologies of energy systems [18]. This is one reason why the

amount of required storage systems in energy systems is an open research question and the reported

requirement of storage capacity varies strongly [2,19].

To obtain a feasible solution with known quality, exact solution strategies are needed. For this purpose

we build on our previously proposed exact decomposition methods [20]. The previously proposed

methods measure the solution quality of the reduced synthesis problem. Thus, the methods enable

rigorous optimization. However, the previously proposed decomposition methods are not applicable

for long-term storage cycles.

In this paper, we propose the synthesis method RiSES4 (Rigorous Synthesis of Energy Supply

Systems with Seasonal Storage). In RiSES4, we combine an aggregation method to typical days [10]

with a method to considering seasonal cycles [16] and employ a rigorous method for measuring the

solution quality [20]. By combining these methods, we solve such complex and coupled synthesis

problems including seasonal storage with known solution quality.

RiSES4 provides feasible solutions (upper bounds) with known solution quality based on lower

bounds. To obtain feasible solutions, we use time-series aggregation with coupled typical periods in

the synthesis problem considering seasonal storage systems yielding a design candidate of the energy

system. Subsequently, based on this design candidate, we solve an operational problem yielding an

upper bound, as the operational problem is a restricted synthesis problem. To provide lower bounds,

RiSES4 employs linear-programming relaxation and relaxation based on time-series aggregation. To

tighten the bounds, we iteratively increase the resolution of the time-series aggregation and tighten

the relaxation.

RiSES4 is applied to 2 complex synthesis problems considering large time series and long-term time-

coupling constraints. RiSES4 shows fast convergence, outperforming a commercial state-of-the-art

solver.

2. Generic synthesis problem

Industrial energy systems are often modelled as mixed-integer linear program (MILP) [5,7]. In

contrast, large-scale energy systems are often simplified to linear programs (LP) [6]. Thus, in Eq. (1),

we state a generic synthesis problem of energy systems as MILP, which results in a LP by removing

all binary terms.

(1)

We employ the total annualized costs TAC as objective function. The total annualized costs TAC

consists of 2 parts representing the two-stage character of the synthesis problem: the operational and

capital expenditures, OPEX and CAPEX. The OPEX are defined as the sum of the output power

of every component in every time step divided by the efficiency and multiplied by the

specific operation cost

and the duration of a time step. The operational expenditures OPEX

directly depend on the set of considered time steps , while the capital expenditures CAPEX only

depend on one-time investment decisions . The capital expenditures CAPEX are the nominal

capacity of each component multiplied by the specific investment costs

summed up for all

components . The objective function TAC is minimized subject to several constraints. The sum of

the components output power and the net energy output of the storage units

have to

meet the energy demand at every time step . The future storage level is calculated based

on the current storage level plus the net energy output of the storage units

multiplied

by the duration of a time step t. Further (in)equalities with the coefficient matrices

and the vectors determine the binary on/off status , the binary existence of components,

and the nominal power , the complete model formulation is given in the Appendix of [7].

To handle the complexity of large-scale energy system models, the binary on/off status and the

binary existence is often neglected [6]. In this case, minimal load, part-load behaviour, cost curves,

and minimal unit size cannot be modelled, however, these changes lead to a less complicated LP

formulation.

Only equations including the output power , the storage variables or the on/off status have to

be stated for each time step and thus depend on the size of the time series . All other variables of

the original synthesis problem are represented by the vector . Additional constraints are here

summarized in the surrogate equation . For large time series , the original synthesis problem

(Eq. (1)) is often not solvable in reasonable computational time or within memory limits.

To still solve such large-scale synthesis problems, we propose the rigorous synthesis method RiSES4

in the next section.

3. The RiSES4 method

The proposed method RiSES4 is suited to solve MILP and LP synthesis problems that depend on

large-scale time series. RiSES4 consists of up to 3 parallel branches, Fig. 1, to calculate upper bounds

and 2 competitive lower bounds. The upper bounds are feasible solutions resulting from an

Aggregated synthesis problem and a restricted Operational problem, A&O branch, Section 3.1.

RiSES4 employs up to 2 competitive relaxation methods to compute lower bounds. For MILP

problems, the B&C branch is based on linear-programming relaxation implemented as the Branch-

and-Cut procedure available in commercial solvers [21], here not further discussed. For MILP and

LP problems, the R&A branch is based on time-series Relaxation and Aggregation of input

parameters, Section 3.2. The tighter of the B&C and R&A relaxation serves as lower bound in the

RiSES4 method.

In RiSES4, the current upper and lower bound is compared to calculate the optimality gap ε of the

original synthesis problem, Section 3.3. If the optimality gap εRiSES is not satisfied, the restrictions

and relaxations are tightened and the branches continue, Section 3.3. The branches stop when the

resulting optimality gap ε satisfies the desired optimality gap εRiSES.

Figure 1. RiSES4: Rigorous Synthesis of Energy Supply and Seasonal Storage Systems

3.1. Upper bounds by the A&O branch

In the Aggregate and Operate (A&O) branch, feasible solutions (upper bounds) of the original

synthesis problem, Eq. (1), are calculated based on 3 steps (i-iii), Fig. 1.

In step (i), time-series aggregation reduces the complexity of the synthesis problem. We employ a

time-series aggregation method based on Bahl et al. [13] which has been extended for seasonal storage

by coupled typical periods as in Kotzur et al. [16]. First, the method identifies the length of typical

periods by looking for periodic patterns in the time-series data by using autocorrelation [22]. Second,

the identified period length is used to split the original time series into periods. Third, the periods are

aggregated to typical periods based on k-means clustering. Within each typical period, the time steps

are further aggregated to segments. To aggregate segments, we adapt the k-means idea [23] and

calculate the average of a set of randomly chosen consecutive time steps. We use the average value

with the lowest Euclidean distance to the original time steps for the entire aggregated segment. The

time-series aggregation thus aggregates in two dimensions: the number of typical periods and the

number of segments per typical period. Time-series aggregation by typical periods maintains the

chronology within each typical period, thereby, enables storage within each typical period (intra-

period).

However, as non-consecutive periods are clustered, only intra-period storage is directly possible. To

model seasonal storage, an inter-period storage difference has to be considered. This inter-period

storage difference is defined as difference of the storage level from the beginning to the end of

each typical period . Figure 2 shows schematically the inter-period storage difference for 3

typical periods.

Figure 2. Schematic representation of the intra-period storage difference for 3 typical periods.

To consider this inter-period storage differences for seasonal storage, a second inter-period time grid

couples the typical periods [16]. To couple the typical periods in the second time grid, we assign each

original unclustered period to its corresponding typical period. Thus, the second time grid contains

the information on the chronological order of the typical periods. Table 1 shows exemplary this

assignment for 3 typical periods and 365 unclustered periods. By this chronological order, the inter-

period storage level difference are then also ordered.

Table 1. Look-up table to assign each unclustered period to the corresponding typical period k. The

columns assigns unclustered periods to typical periods, leading to ordered typical periods in row 2.

unclustered period

1

2

3

4

5

6

…

362

363

364

365

typical period

2

3

3

2

1

1

…

2

2

3

1

The superposition of this inter-period storage level differences and the intra-period storage level

within each typical period results in the actual storage level, Fig. 3.

Figure 3. Schematic representation of the superposition of the intra-period and inter-period storage

difference.

intra-period storage level

inter-period storage level

Considering the actual storage level from the superposition enables RiSES4 to design seasonal storage

systems in the aggregated synthesis problem.

In step (ii), we use the aggregated time series instead of the original time series for the synthesis

optimization. The aggregated time series is much smaller, and thus the synthesis optimization can be

solved efficiently. The solution of the aggregated synthesis optimization yields a design candidate of

the energy system.

Subsequently, in step (iii), the design candidate of the energy system is fixed in the original synthesis

optimization. Fixing the variables of the design candidate, i.e., selection and sizing of units, reduces

the original synthesis optimization to an operational optimization with a reduced number of variables.

This operational optimization can be solved efficiently, even though the full original time series is

used. The solution of the operational optimization is an upper bound for the original synthesis

problem, as the operational problem is a restricted problem of the original synthesis problem, since

the design variables are fixed.

To evaluate the solution quality of the upper bound, lower bounds are calculated in RiSES4.

3.2. Lower bounds by the R&A branch

In the Relax and Aggregate (R&A) branch, lower bounds of the original synthesis problem, Eq. (1),

are calculated based on 3 steps (a-c), Fig. 1.

In step (a), we use the same time-series aggregation method as in step (i) of the A&O branch. The

time-series aggregation yields coupled typical periods with aggregated segments.

In step (b), the aggregated time series are relaxed. The relaxation of the aggregated time series

depends on the original synthesis problem studied. For the common LP formulations of energy

synthesis optimizations, the aggregated time series, with mean values as representative time steps,

are a relaxation of the original time series as shown by Teichgräber and Brandt [24]. The employed

k-means and segmentation methods are using mean values as representative time steps, and thus

steps (b) and step (c) directly yield the desired lower bound. For MILP (or more general LP)

problems, in step (b), we identify an underestimator and overestimator for all segments within each

typical period. The underestimator is the smallest value of all original time steps assigned to a

segment, and the overestimator the largest value, respectively, for more details see [20].

In step (c), we employ the over- and undererstimator to solve an aggregated and relaxed synthesis

optimization. To relax the synthesis optimization, we replace every time-dependent equation of the

original synthesis problem Eq. (1) by 2 constraints bounding the equation between the over- and

undererstimator. As in step (ii), in the A&O branch, the relaxed synthesis optimization can be solved

efficiently, as aggregated time series are used.

Using the lower and upper bounds, an optimality gap can be calculated.

3.3. Optimality Gap and increase of time resolution

Last, we compare the best resulting lower bound with the upper bound and check if the desired

optimality gap is satisfied.

(2)

We iteratively increase the time resolution in the A&O and R&A branch. We increase the number of

either periods or segments based on finite backwards differences, as in [13,20]. The heuristics selects

larger backward difference as most promising direction to increase the resolution of the aggregation.

The iterative increase of the time resolution for the time-series aggregation stops, as soon as the

optimality gap is satisfied, yielding a feasible solution of the original synthesis problem, Eq.

(1), with known solution quality.

4. Case studies

For validation, we apply RiSES4 to 2 complex synthesis problems including seasonal storage.

4.1. MILP synthesis of an industrial energy system

In this section, we apply RiSES4 to an MILP industrial synthesis problem including seasonal storage

based on Baumgärtner et al. [7]. The industrial energy system provides electricity, low-temperature

heat, steam, and cooling.

We use a superstructure with 3 units of each energy conversion technology (absorption chiller, boiler,

CHP engine, compression chiller, electric boiler, and heat pump), additional roof-top PV, an inverter

station, and a wind turbine. As storage systems, the superstructure includes a battery system and 1

storage tank for hot and 1 for cold water. The original time series consists of 1 year with 2 hourly

demand data for steam, hot and cold water, electricity, electricity grid prices, ambient temperature,

solar radiation, and wind speed. The detailed MILP formulation and model description is given in the

Appendix of [7].

After presolve, the original synthesis problem with full time series contains 9∙105 equations and 4∙105

variables (1.4∙105 binaries) with 2.4∙106 nonzero elements. The benchmark and the RiSES4

calculations are performed using 4 Intel-Xeon CPUs with 3.0 GHz and 64 GB RAM. All MILP

problems are solved using CPLEX 12.6.3.0. The optimality gap εRiSES is set to 2 % and all MILP

problems are solved with a gap of 0.5 %. The time limit of the synthesis optimizations with RiSES4

is set to 3 hours. The time limit of the operational optimizations is set to 20 minutes.

RiSES4 satisfies the required optimality gap εRiSES in 524 seconds, Fig. 4. To satisfy the required

optimality gap εRiSES, the aggregated synthesis in the A&O branch uses only 13 aggregated time steps.

Thereby, the aggregated synthesis problems consists of only 6698 equations, 2113 variables (493

binaries) and 19,000 nonzero elements. Thus, the size of the synthesis optimization is reduced by 2

orders of magnitude. For this purpose, the method is not stopped once the optimality gap εRiSES is

achieved but continued to the time limit of 3 hours. However, the accuracy increases only slightly by

a higher time resolution in the branches, Fig. 4.

As a benchmark, we directly try to solve the original synthesis employing the full time series with

CPLEX 12.6.3.0 within a time limit of 105 seconds (~28 hours). CPLEX finds the first feasible

solution after 4489 s and reaches the time limit still with a relative gap of 29.9 %.

For validation, we repeat RiSES4 and the benchmark with 5 instances generated by statistical noise

using Latin hypercube sampling with a variation by the time series of ±5 % [25].

Figure 4. Optimality gap ε of RiSES4 and the benchmark CPLEX as function of the computational

time for the industrial synthesis problem. The required optimality gap εRiSES is marked in red.

RiSES4 performs similarly in all instances, outperforming the benchmark in all instances. RiSES4

always provides a solution satisfying the required optimality gap εRiSES in under 1 hour. The

performance of the benchmark differs: in all instances, a feasible solution is found within the time

limit of 105 seconds; however, only in 2 instances, a solution satisfying the required optimality gap

εRiSES is found; whereas in the 3 other instances, the optimality gap ε remains between 5 and 31 %.

Thus, RiSES4 always satisfies the required optimality gap εRiSES before the benchmark provides any

feasible solution at all.

Figure 5 shows the lower bounds TACR&A and TACB&C of the parallel branches R&A and B&C

together with the upper bound TACA&O of the A&O branch for the original instance. Additionally,

the optimality gap ε of RiSES4 is plotted as function of the solution time.

Figure 5. Lower and Upper bounds of RiSES4 for the original instances. As secondary axes the

optimality gap ε is shown.

The R&A branch provides the first lower bound TACR&A within few seconds, based on the proposed

simultaneous under- and overestimation, Section 3.2. However, the lower bound TACR&A converges

slowly over many iterations, thus optimality could not been proven by lower bound TACR&A within

the time limit. In contrast, the linear-programming relaxation in the B&C branch provides the first

bound after 524 seconds. In this case study, this first bound is sufficient to proof optimality of the

found feasible solution of the A&O branch.

The first design candidate satisfying the optimality gap consists of 2 boilers, 2 CHP engines, 2

compression, 2 adsorption chillers, and 3 heat exchangers between the steam and low-temperature

demand. Neither renewables nor storage systems are necessary for solutions with excellent quality.

The other design candidates of RiSES4 with increased time resolution are similar to the first design

candidate, though small storage systems are added, improving the solution quality slightly.

However, storage systems are only used for short periods, thus seasonal storage is not optimal in this

industrial energy system. To further validate RiSES4 and show the application in a case study

including seasonal storage systems, we apply RiSES4 to a second case study in the next section.

4.2. LP synthesis of a national energy system

In this section, we apply RiSES4 to a LP national energy system synthesis problem under emission

restrictions for greenhouse gases. The national energy system provides electricity, residual as well as

industrial heating, and is coupled to the private vehicle sector. As case study, we model the German

energy system with 438 hubs and the electricity grid, based on the model ELMOD-DE [26] with the

existing infrastructure of the year 2016 and emission restrictions given by the political goal of the

year 2030. At each hub, we use a superstructure of conventional heat and power plants, diesel and

gasoline cars, as well as alternative technologies as PV, onshore and offshore wind turbines, heat

pumps, thermal isolation, electrical boilers, gas-driven - , electric - and hybrid vehicles, and power-

to-gas and power-to-fuel technologies. As storage systems, the superstructure includes battery

systems and hydrogen storage. In the given infrastructure, possible storage options are pump-hydro

reservoirs and the public gas grid.

The original time series consists of 1 year with hourly demand data for electricity and heating, solar

radiations and wind speed. The original synthesis problem with full time series is unexecutable due

to memory limits. Thus, to show the advantage of RiSES4 for different sizes of LP energy system

models, we vary the length of the original time series from only 2 time steps up to 365 time

steps.

As benchmark, we directly solve these synthesis problems. The benchmark and the RiSES4

calculations are performed using 4 Intel-Xeon CPUs with 3.0 GHz and 64 GB RAM. All LP problems

are solved using CPLEX 12.6.3.0. The optimality gap for RiSES4 εRiSES is set to 2 %. As the synthesis

problem is modeled as LP, all optimizations are solved to global optimality, thus the benchmark

solutions have an optimality gap of 0 %. Additionally, the B&C branch is not active for LP problems,

as no binary variables exist.

For very small time series (< 12 time steps), the benchmark solves the synthesis problem faster

than RiSES4, Fig. 6. However, with increasing size of the employed time series, RiSES4 solves the

LP problems significantly faster. For 30 time steps, RiSES4 is 3 times as fast, and for 40 time steps

RiSES4 is already 200 times faster than the benchmark. For 60 time steps, the benchmark does not

find a feasible solution within the time limit of 2*106 seconds (23 days), whereas RiSES4 provides an

optimal solution within the optimality gap εRiSES in less than 2000 seconds which is at least 1000

times faster. For the time-series length of 365 time steps (=365), RiSES4 still provides a feasible

solution with an optimality gap of 4 % within the calculation time of 1 day (86400 s).

Figure 6. Necessary computational time of RiSES4 to reach the optimality gap εRiSES of 2 % and of

the benchmark CPLEX (gap ε of 0 %) as function of the length of the original time series .

Figure 7 shows the total annualized cost TAC of the lower bound of the R&A branch and the upper

bound of the A&O branch of RiSES4 for the investigated time series length .

Figure 7. Total annualized cost TAC of the final solutions of the A&O branch (upper bound) and

the R&A branch (lower bound) of RiSES4 and of the benchmark CPLEX (gap ε of 0 %) as function

of the length of the original time series .

Additionally, Figure 7 shows the total annualized cost TAC of the benchmark solutions. For time

series length of maximal 30 time steps (≤30), the objective TAC of the benchmark lies in between

the upper and the lower bounds of RiSES4. For the last found solution of the benchmark with 40 time

steps (=40), the benchmark provides a wrong optimal solution due to numerical errors, while

RiSES4 still provides optimal solutions within the optimality gap εRiSES fast, Fig. 6.

The design candidates within the optimality gap εRiSES of RiSES4 employ a wide mix of conventional

power plants, but gas-based power plants take the largest share in conventional power production.

The renewable power generation is mainly based on onshore wind turbines and a smaller portion of

photovoltaics. For the residual heating sector, thermal isolation, gas boilers and heat pumps provide

the heat demand, whereas oil boilers are not in use anymore. In the industrial heating sector, only a

small portion of conventional heat technologies is replaced by heat pumps and electric boilers. The

private vehicle sector is only based on conventional gasoline and diesel cars and no power-to-fuel

technologies are built.

As storage systems, few large battery systems (8000 MWh) are added to the existing infrastructure.

These battery systems are mainly operated for short storage cycles. However, the public gas grid is

operated in seasonal cycles, Fig. 8. Figure 8 shows the storage level of the public gas grid of optimal

design candidate of the largest investigated time series (=120). For this time series, the optimality

gap εRiSES can still be satisfied. As comparison, the storage level of the public gas grid of the

operational optimization employing the full time series is shown. The storage level in both

optimizations is very similar, showing that RiSES4 captures seasonal storage in the aggregated

synthesis optimizations leading to designs with excellent solution quality.

Figure 8. Storage level of the public gas grid of the aggregated solution of RiSES4 satisfying the

optimality gap εRiSES of 2 %. For comparison, the storage level of the operational optimization with

the full time series is shown. Both storage levels are shown for an original time series length of

120 time steps.

4. Conclusions

The synthesis of energy systems typically results in large-scale (MI)LP optimization problems which

are computationally challenging and often not solvable within reasonable computational time or

memory limits. Long-term time-coupling constraints, e.g., due to storage systems, further increase

the complexity.

To obtain a feasible solution with known quality, we propose the rigorous synthesis method RiSES4

(Rigorous Synthesis of Energy Supply Systems with Seasonal Storage). RiSES4 provides feasible

solutions via time-series aggregation. To model seasonal storage coupled typical periods are used.

RiSES4 includes a general under- and overestimation of input parameters to rigorously solve synthesis

problems including time-dependent input parameters, such as energy demands and renewable

resources.

RiSES4 is applied to 2 complex synthesis problems and the results are further validated in

computational studies. RiSES4 provides fast convergence, outperforming the state-of-the-art solver

CPLEX. The RiSES4 method is generally applicable to two-stage time-dependent synthesis problems

with coupling variables and constraints.

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