In order to compare the stochastic and the deterministic model, an in-sample approach is used. This means that the deterministic solution is used as a fixed first stage solution in the stochastic model. Subsequently, this model is solved resulting in the optimal second stage
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behavior for the deterministic model, given the initial solution based on the expected value for the heat demand and power price. This allows for a comparison of the numerical results from the deterministic and the stochastic model.
7.2.1 Model comparison
The first case study concerns the deterministic and the stochastic model described in Chap-ter 4. The in-sample approach, described above, is here used to compare the two types.
February May August November
0 0.5 1 1.5 2 2.5x 106
Time [Day]
Total cost [DKK]
Deterministic Stochastic
Figure 7.1 – Comparison of the stochastic model results and the deterministic in-sample results for the first week of February, May, August and November 2013. A small difference between the stochastic and deterministic model results are observed, most prominant during the days in August.
Figure 7.1 shows the total daily heat costs, for the first week of February, May, August and November for the deterministic in-sample and the stochastic model. The differences between the stochastic and the deterministic model costs are numerically small during all weeks. However, the relative differences are found higher in August compared to other weeks. Calculating the difference between the stochastic and the deterministic model costs for each of the weeks, as a percentage, results in:
wF eb= 0.5% (7.1)
wM ay = 1.4% (7.2)
wAug = 17.4% (7.3)
wN ov= 1.5% (7.4)
The reason for these very small differences in February, May and November might be due to the high degree of flexibility present in the system. As long as the net power production is not changed, the production units can adjust production according to the demand and spot price realization. In Chapter 5 the standard deviation of the heat demand was found in (5.5), toσ = 26.66 MWh. As this standard deviation is small relative to the capacity of the HP and EB, the system might be flexible enough for the deterministic model to handle
7.2 Deterministic and stochastic comparison 77
the deviations almost good as the stochastic optimization model. In May, the demand is very low, and the CHP is generally the only unit used for heat production, as was seen in Figure 6.10. In this situation the flexibility is limited and thus a higher benefit from using the stochastic model is experienced.
7.2.2 Capacity impact
This section carries out an analysis to investigate the impact of the HP and EB capacity on the appropriability of stochastic as opposed to deterministic optimization.
A number of relevant scenarios are selected and both the stochastic and the deterministic in-sample model are solved for each scenario. The three scenarios chosen are:
1. 100% capacity for both the HP and EB (reference case 0) 2. 50% capacity for both the HP and EB
3. 0% capacity for both the HP and EB
A comparison of the three scenarios is displayed in Figure 7.2. For each scenario and week, the difference between the deterministic and the stochastic model results are found as a percentage of the stochastic result. Thus a positive difference implies that the stochastic model provides lower heat costs compared to the deterministic in-sample.
February May August November
0 5 10 15 20 25 30 35 40
Time [Day]
Relative difference [%]
100% capacity (case 0) 50% capacity
0% capacity
Figure 7.2– Comparison of the stochastic results and the deterministic in-sample for the first week of February, May, August and November 2013. A large positive value signifies that the stochastic model results in lowered heat costs.
For January and November the same pattern appears. Production at full capacity on the HP and EB results in a small relative difference. However, these months experience the highest costs due to a high demand, see Figure 6.10. The high demand forces many production units to operate simultaneously, which increase the flexibility.
Decreasing the capacity of the HP and EB, the impact of the stochastic approach increases.
Especially when the capacity for the EB and HP is fixed to zero, i.e. removing the HP and EB from the system, the stochastic solution is significantly better than the deterministic, which again reflects the influence of system flexibility.
In May and August the benefits from the stochastic model are generally higher and almost constant for August. This is likely due to the low heat demand during these months and the corresponding choice of production unit. Section 6.1 and 2.2 illustrated how the back-pressure CHP was most economical at almost any electricity price. During periods with low heat demand, such as May and August, the back-pressure CHP preferably satisfies the demand solely and the HP and EB do not get operating hours, see Figure 6.10. Therefore, a change the in capacity does not influence the results significantly compared to periods where the HP and EB are used frequently.
This analysis, thus, indicates a negative correlation between the capacity of the HP and EB, and the significance of stochastic optimization, where capacity reduction results in increased benefits from the stochastic model compared to the deterministic model. This should be taken into consideration when deciding specific HP and EB capacities for a system.