High-demand realization

Naturally, it is of interest to investigate how the deterministic and the stochastic model adjust the production to meet the realized demand. Figure 6.14(a) and 6.14(b) shows the final production plan from the stochastic and the deterministic model, respectively, to meet the heat demand in the realization of one scenario. For the stochastic model this corresponds

to the addition of the second stage decision. The forecast for the heat demand and spot price are included in the figure together with the realization in the specific scenario. The scenario constitutes a high-demand scenario where the realization for the demand is higher than the expected value.

Figure 6.14 – Heat production plan, resulting from the stochastic programming model (a) and deterministic in-sample (b), to meet the heat demand in one realization of heat demand and spot price.

A significant difference between the production plans resulting from the deterministic and the stochastic model, is the occurrence of uncovered demand in the deterministic model results, Figure 6.14(b). This is generally very undesirable, as it might lead to the start-up of expensive and non-sustainable gas and oil boilers. The reasoning behind this behavior is found when reviewing the initial heat production schedule presented in figure 6.12 as well as the corresponding sold and consumed power, displayed in Figure 6.15. The latter compares the scheduled accumulated power production and consumption resulting from the deterministic (Det) and stochastic (Stoch) models.

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Figure 6.15 – Comparison of power production (to be sold) and power consumption in the deter-ministic and stochastic optimization results.

The limiting factor between the first and second stage is the requirement of a constant net power production. This means that any additional power production in the second stage should be matched by an increase in power consumption of equal size. In Figure

6.3 Chapter summary 73

6.15 at hour 12, a difference between the stochastic and deterministic power production to be sold is observed. According to the deterministic optimization results, 310 MWh power (maximum total power production from CHP and CHP2) is sold at this time. On the contrary, the results from the stochastic model show a smaller amount, approximately 300 MWh. Furthermore, a planned power consumption around hour 12 is only apparent in the stochastic schedule.

Full power capacity is already sold from CHP2, which means it is not able to produce any heat. Furthermore, the HP and EB are limited as they require an additional power consumption, not available in this system. With the heat demand being realized at a higher value the CHP2 is not able to produce heat, as this would require a decrease in net power production. This explains the uncovered demand in the deterministic solution for the realization of a high demand scenario. Furthermore, this example illustrates the benefits of stochastic compared to deterministic optimization.

6.3 Chapter summary

This chapter analyzed and discussed results from the deterministic and the stochastic op-timization model. Analytical calculations of marginal heat production costs as a function of the spot price were presented, and compared to a simplified version of the deterministic model. The comparison showed an agreement between the simplified model and analytical calculations.

Extending the model to its full form, i.e. taking into account start-up, ramping etc. allowed for comparisons and further analyses to be made. Based on the presented results it is con-cluded that the deterministic model provides adequate results, where ramping constraints and start-up costs are important features. This allows for the subsequent use of the model to provide numerical result presented in Chapter 7.

Additionally, the stochastic model was investigated for two days in November, using 100 scenarios for the heat demand and spot price as input. The heat production schedule was compared to the schedule obtained with the deterministic model. Especially, an increased use of storage was observed, consistent with the increased need for flexibility due to the presence of uncertainty. A high-demand scenario, representing one realization of the heat demand and electricity price, was analyzed. This resulted in uncovered heat demand in the deterministic optimization result, while the stochastic model was still able to meet the heat demand. Analyzing the corresponding power production, allowed for an explanation of the uncovered demand. This emphasized the difference between the stochastic and the deterministic model.

Both the deterministic and the stochastic optimization model were found to provide rea-sonable results consistent with intuitive, analytical and numerical expectations. The results from the models, thus provide an operational strategy for the introduction of a HP and EB in the Nordic power market and Copenhagen district heating system.

This allows for an analysis of the monetary benefits of stochastic optimization as opposed to deterministic which is presented in the following chapter together with numerical results for the impact of HPs and EBs.

Chapter 7

Numerical results

This chapter presents numerical results from solving the deterministic and the stochastic model developed in Chapter 4. The difference in heat production costs obtained from the stochastic and the deterministic model are analyzed and the influence of different production capacities for the HP and EB is investigated. Numerical results from three case studies, based on the stochastic model, are presented, analyzed and mutually compared to each other.

7.1 Computational performance

Optimally, the comparison between the stochastic and the deterministic model should be made considering a full year. However, the stochastic model is cumbersome to solve with just 100 scenarios. Consequently, instead of reducing the number of scenarios, the model is only solved for four different weeks, chosen to represent the yearly variation. This means that the first week of February, May, August and November is used. The computation time for solving a single day using the stochastic model vary from a few minutes to more than an hour. An optimality gap of maximum 0.5% is therefore accepted as well as a time limit of one hour for each day is used in the solver. This might lead to a small deviation from the optimal solution for the stochastic model, and a slightly suboptimal solution. This will favor the deterministic model slightly and reduce the benefits from the stochastic optimization model.

The solver, CPLEX, was used together with the interface program GAMS to solve the de-veloped optimization models that in their nature are mixed integer linear programs (MILP).

The computer used has an Intel^r Core^TM i5 processor at a clock-speed of 1.7 GHz, 4GB ram and Windows 8 64 bit.