Low variance case

In this section, we are going to analyze the CPU time, the percent error and the “Relative Efficiency” for different methods over a number of seasons when the demand and wind output are drawn from the lognormal distribution with the standard deviation of one. All of the numerical results for this section are given in Table 6.2.

Table 6.2: Numerical results when the demand and fuel cost are drawn from the lognormal distribution with σ = 1. The running time is rounded to minutes.

CPU time (minutes) % error of objective value Relative Efficiency Number of time periods (seasons) 2 4 6 8 2 4 6 8 2 4 6 8 Uncluster 38 172 401 670 2.23 3.06 4.27 5.46 85.63 525.10 1713.76 3656.02 Uncluster + MCMC 34 149 348 559 1.97 2.83 3.46 4.58 66.55 420.54 1203.22 2558.92 Tech 7 44 119 175 3.08 4.58 6.21 7.03 22.54 203.35 737.75 1231.66 Tech + MCMC 5 42 112 166 3.04 4.06 5.74 6.89 15.91 169.57 644.03 1141.79 Tech + Size 14 73 175 290 2.77 3.83 4.97 6.52 38.23 280.36 867.76 1893.41 Tech + Size + MCMC 10 55 147 219 2.64 3.74 4.58 6.07 26.58 204.64 672.65 1331.56

Firstly, Figure 6.3 shows the CPU time of different methods for a number of seasons.

According to Figure 6.3, it can be seen that the order of the CPU time for different methods does not change over the number of season and it is shown in Table 6.3. Figure 6.3 and Table 6.3 show that the clustered formulations take significantly less amount of CPU time than the clustered formulation.

2 3 4 5 6 7 8 0 100 200 300 400 500 600 700

Number of time periods (seasons)

Time (minutes)

Running time Unclustered

Unclustered with MCMC−IS Clustered−by−Tech

Clustered−by−Tech with MCMC−IS Clustered−by−Tech−and−Size

Clustered−by−Tech−and−Size with MCMC−IS

Figure 6.3: The CPU time of different methods for a number of seasons when the demand and wind output are drawn from the lognormal distribution with the standard deviation of one

1. Clustered-by-Tech with MCMC-IS 2. Clustered-by-Tech

3. Clustered-by-Tech-and-Size with MCMC-IS 4. Clustered-by-Tech-and-Size

5. Unclustered with MCMC-IS 6. Unclustered

Table 6.3: The order of the CPU time of different methods over a number of seasons (increasing order)

1. Unclustered with MCMC-IS 2. Unclustered

3. Clustered-by-Tech-and-Size with MCMC-IS 4. Clustered-by-Tech-and-Size

5. Clustered-by-Tech with MCMC-IS 6. Clustered-by-Tech

Table 6.4: The order of the % error of the objective cost for different methods over a number of seasons (increasing order)

The level of speed up is proportional to the aggregation level of the problem. For example: Since the size of the unclustered problem is about five times bigger than of the “Clustered-by-Tech” problem, it takes about four to five times greater than of the latter. The similar result can be seen when the CPU time of the “Unclustered” formulation is compared with the “Clustered-by- Tech-and-Size” one. More specifically, as the problem size of the former is about 2.5 times bigger than the size of the latter, it takes two to nearly three times longer for the “Unclustered” method to converge. Furthermore, the results show that the MCMC-IS algorithm takes less CPU time to solve a particular formulation than the SDDP algorithm. As the number of time periods increases, the advantage of the MCMC-IS algorithm becomes clearer (Figure 6.3). This implies that the cutting-planes construction for approxi- mating the expected cost-to-go function in the MCMC-IS algorithm are more effective at every time period, and then accumulative over the number of time periods. The only case when the MCMC-IS algorithm does not show its clear advantage over the SDDP algorithm is in the “Clustered-by-Tech” formula- tion. This is because the size of the “Clustered-by-Tech” problem is relatively small, so the Monte Carlo method for generating samples in the SDDP algo- rithm is efficient enough for producing accurate results. On the other hand, the MCMC-IS algorithm takes some time for generating the MCMC samples and constructing the IS distribution to correct the bias. Therefore, the ad- vantage of the MCMC-IS is easier to be realized in the large-scale problems such as the “Clustered-by-Tech-and-Size” and “Unclustered” formulation. The % error of the objective cost for different methods for a number of sea- sons is shown in Figure 6.4.

According to Figure 6.4, the % error increases as the number of seasons increases. The order of the % error for different methods is shown in Table 6.4.

2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

Number of time periods (seasons)

Percent error

Percent error of the objective value Unclustered

Unclustered with MCMC−IS Clustered−by−Tech

Clustered−by−Tech with MCMC−IS Clustered−by−Tech−and−Size

Clustered−by−Tech−and−Size with MCMC−IS

Figure 6.4: The % error of the objective cost for different methods for a number of seasons when the demand and wind output are drawn from the lognormal distribution with the standard deviation of one

aggregation reduces the CPU time to solve a problem, it increases the per- centage error. This is because it assumes that all of the generators in the cluster are identical and hence, the clustered formulations will lose some de- tails of every generator. Nevertheless, the % error increases very small, as shown in 6.2: All of the clustered methods have less than 2% increase in the % error relatively to the unclustered methods. For example: The % error increases from the “Unclustered” to “Clustered-by-Tech” method is 0.85%, 1.52%, 1.94% and 1.57% for the 2, 4, 6, 8−seasons respectively, whereas the CPU time of the “Clustered-by-Tech” method is 5.2, 3.9, 3.4, 3.9 times faster than of the former.

The “Relative Efficiency” that investigates the trade-off between the CPU time and the % error is shown in Figure 6.5.

According to Figure 6.5, the order of the “Relative Efficiency” of different methods over a number of seasons is shown in Table 6.5. According to Table 6.5, the top four methods use the clustered formulation while the bottom two positions belong to the unclustered formulations. This shows the advantage of the aggregation. For a given clustered formulation, the MCMC- IS algorithm always performs better than the SDDP algorithm. Therefore, the results indicate the advantage of MCMC-IS and clustering for solving this large scale problem.

2 3 4 5 6 7 8 0 500 1000 1500 2000 2500 3000 3500 4000

Number of time periods (seasons)

Efficiency (minutes * percent error)

Efficiency Unclustered

Unclustered with MCMC−IS Clustered−by−Tech

Clustered−by−Tech with MCMC−IS Clustered−by−Tech−and−Size

Clustered−by−Tech−and−Size with MCMC−IS

Figure 6.5: The “Relative Efficiency” of different methods over a number of seasons when the demand and wind output are drawn from the lognormal distribution with the standard deviation of one

1. Clustered-by-Tech with MCMC-IS 2. Clustered-by-Tech

3. Clustered-by-Tech-and-Size with MCMC-IS 4. Clustered-by-Tech-and-Size

5. Unclustered with MCMC-IS 6. Unclustered

Table 6.5: The order of the “Relative Efficiency” of different methods over a number of seasons. 1 = the best method. 6 = the worst method

2 3 4 5 6 7 8 0 200 400 600 800 1000 1200 1400

Number of time periods (seasons)

Time (minutes)

Running time when the variance increases Unclustered

Unclustered with MCMC−IS Clustered−by−Tech

Clustered−by−Tech with MCMC−IS Clustered−by−Tech−and−Size

Clustered−by−Tech−and−Size with MCMC−IS

Figure 6.6: The CPU time of different methods for a number of seasons when the demand and wind output are drawn from the lognormal distribution with the standard deviation of two