The results of the three systems are presented and discussed in the following section. All S-models lead within the numerical accuracy to the same results. Therefore the cost results are just shown for the one S-model (S). The time results are the mean of the two hardware configurations, whereby two runs on each configuration are performed. The reference model ((text {ref.})) optimizes the entire year with the original, hourly data without merging similar time series to typical days or typical periods. The modified model (mod.) optimizes the entire year with the hourly data aggregated by the typical days. This model allows the determination of the storage model usage error. The cost results are compared to both the reference and modified model. Costs are the combination of operating and component costs and the optimization objective. The relative error is calculated by the following equation:

$$begin{aligned} C_{rel} = left| 1 – frac{{C_{Operation}}_{{text {sce.}}} + {C_{investment}}_{{text {sce.}}}}{{C_{Operation}}_{{text {ref.} / text {mod.}}} + {C_{investment}}_{{text {ref.} / text {mod.}}}}right| . end{aligned}$$

(14)

A relative cost error of 2 % implies that the total cost difference between the reference ((text {ref.})) or modified (mod.) model to the considered storage scenario ((text {sce.})) (different model/ number of typical periods) is 2 %. This error will vary over the number of typical days since it compares the costs and not the energy system component sizes (like X kWh of battery). If the photovoltaic size is smaller than in the reference case, but the battery size is larger, this can lead to the same costs but with different component sizes.

For all cases, the ratio of the number of variables and constraints compared to the S-K-model is always lower than one. The ratio is less for the cyclic case since the storage is not linked. For the S-N-model, the ratio is logarithmically increasing over the number of typical days up to one since the number of the same typical periods in a row decreases with increasing typical days. For the S-G-model, it is slightly decreasing.

In General, the C-model is the fastest. The S-N-model is faster or nearly as fast as the S-K-model. Besides, the S-G-model is the slowest one.

CHP system

Figure 3 shows the calculation time for all storage models (a) and the total costs error (b). In most cases, the calculation time for a number of typical days below 27 is less than 35 % of the time of the complete year calculation. Especially the C-model stays below 5 % of the calculation time. This time reduction results from fewer variables and constraints used by the C-model. For the most number of typical days, the total cost error is less than 1 % compared to the reference model and less than 2.5 % compared to the modified model. The S and C-models lead to nearly the same error for a number of typical days less than 8. Afterwards, they differ. All S-models lead to an error of less than 0.2 %, which is slightly better than the C-model error of 0.6 %. This same error indicates that the primary storage usage is daily and not seasonal.

Figure 3c shows the number of variables (Var.) and constraints (Const.) as well as the calculation time relative to the S-K-model. In most cases, the S-N-model is more than 10 % faster and the C-model up to 80 %. The S-G-model is more than 40 % slower than the S-K-model in most cases.

Fig. 3
figure 3

Calculation time (a) and total costs error (b) for the different storage models and relative share of number of constraints and variables as well as time for C, S-G- and S-N- to S-K-model (c) for the CHP system

Air heat pump system

Figure 4 shows the calculation time for all storage models (a) and the total costs error (b). For a number of typical days below 28, the calculation time amounts to less than 40 % of the time for the complete year calculation. Especially the C-model stays below 5 % of the calculation time. The total costs error stays below 2 % for the most number of typical days compared to both the reference and modified model. The S- and C-models reduce this error below 1 % for 27 or more typical days. Both models lead to almost the same error. The concordance between the errors indicates that the storage usage is daily instead of seasonal.

Figure 4c shows the number of variables (Var.) and constraints (Const.) as well as the calculation time relative to the S-K-model. In most cases, for less than 10 typical days, the S-N-model is about 20 % faster than the S-K-model. For more than 10 typical days, the S-N-model is as fast as the S-K-model. The C-model is up to 80 % faster in most cases. The S-G-model is about 100 % slower than the S-K-model in most cases.

Fig. 4
figure 4

Calculation time (a) and total costs error (b) for the different storage models and relative share of number of constraints and variables as well as time for C, S-G- and S-N- to S-K-model (c) for the air heat pump system

Island system

Figure 5 shows the calculation time for all storage models (a) and the total cost error (b). The total cost error amounts to 45 % at its maximum. This error is significantly higher than the CHP and the Air heat pump system error. However, for the most numbers of typical days, the total cost error remains below 12 % compared to the reference model. Only in the case of more than 48 typical days does the cost error of the S-model differ from the C-model. The concordance between the errors in cases of up to 21 typical days indicates the storage usage as daily storage instead of seasonal storage. This statement is underlined by Fig. 6 since hydrogen storage is chosen for the number of typical days of 48 and more.

The cost error for a number of typical days less than 10 is below 2 % compared to the modified model. For a number of typical days of more than 48, the S and C-model errors differ. This trend indicates that the costs error for a number of typical days less than 10 is caused by the typical days and not by the storage models. For a number of typical days higher than 48, the error is caused by the storage model. So, the seasonal storage model has a lower error here.

For a number of typical days below 36, generally, the calculation time is lower than 14 % of the full-year calculation time. Especially the C-model stays below 5 % of the calculation time.

Figure 5c shows the number of variables (Var.) and constraints (Const.) as well as the calculation time relative to the S-K-model. Up to 12 typical days, the S-N-model often calculates at least 10 % faster than the S-K-model. It is approximately as fast as the seasonal model for more typical days. The calculation time reduction decreases by an increasing number of typical days because of nearly the same number of variables and constraints for both models. The C-model is up to 90 % faster in most cases, and the S-G-model is, on average, 230 % slower than the S-K-model.

Fig. 5
figure 5

Calculation time (a) and total costs error (b) for the different storage models and relative share of number of constraints and variables as well as time for C, S-G- and S-N- to S-K-model (c) for Island system

Figure 6 shows the total cost-share of the C-model, S-model, modified, and reference model. Until 21 typical days, the total costs increase for both storage models. Besides, the backup is used from a number of 6 typical days on. For less than 6 typical days, photovoltaic, wind turbine, and battery storage define the system. Hydrogen storage, electrolyser and fuel cell are the components allowing seasonal storage. As of 21 typical days, the seasonal storage components are included in the system optimized using a seasonal storage model. The seasonal storage model leads to nearly the same results as the reference case for a number of typical days of 48 and more. The C-model cannot get the same result because it cannot consider hydrogen storage as seasonal storage.

Fig. 6
figure 6

Comparison of total costs share for cyclic and seasonal storage model for Island system

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