Quantitative analysis: comparative analysis

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5.4.3 Quantitative analysis: comparative analysis

The quantitative analysis is carried out using five different MPC strategies. They all use the same prediction model with a sampling time step of 10 seconds and a prediction horizon of 60 steps. Every controller is given a set of computation time budgets per network type. A simulation is run for every computation time budget. The optimization algorithms are allowed to start from various starting points until the CPU time budget is exceeded. The initial starting point is equal to the control signal obtained at the previous controller time step, if available. In the case that the CPU budget is not exceeded, the optimization is repeated from a new, randomly selected starting point.

The SLP-I algorithm was compared to the following optimization algorithms

SLP-II: The second optimization algorithm is similar to the SLP-I algorithm. How-ever, this algorithm does not consider the impact of past, upstream control sig-nals on the current turn fraction by removing the impact of upstream links in (5.46). In this way it can be studied whether the linearization procedure de-scribed in this paper leads to a better trade-off between computation time used and realized throughput.

SLP-I-FP: The third algorithm is similar to the first. However, the line-search step detailed in Section 5.3.5 is skipped by settings^∗ = 1 in (5.53). In this way, the added value of the line-search step can be studied.

SLP-II-FP: The fourth algorithm is similar to the second. However, the line-search step detailed in Section 5.3.5 is skipped by settings^∗ = 1. Compared to the SLP-I-FP algorithm, this algorithm does not consider the impact of past, upstream control signals on the current turn fraction.

SQP: The final algorithm tested is the optimization approach called SQP of the ‘fmin-con’ solver that is available in Matlab. This is a commonly used solver for non-linear optimization problems. This algorithm uses a numerical procedure to determine the gradient. A comparison with this algorithm can give insight into the computation time gain when using an analytic linearization. Also, when given sufficient computation time it can give an idea of the maximum achievable performance.

The comparison between the different algorithms enables to study the relative perfor-mance. By comparing between the SLP-I and SLP-II algorithm, the added value of taking into account more information when solving the linear optimization problem.

It is expected that the SLP-I algorithm may require more time but that it will lead to a better performance. By settings^∗ = 1 in the SLP-I-FP and SLP-II-FP algorithms, the added value of the line search in the optimization algorithm can be studied. The idea is that the line-search will increase the computation time, but on the other hand may also provide better convergence of the optimization algorithm. The reason for this is that settings^∗ = 1 may prevent these algorithms from reaching the optimum. Finally, the comparison with the SQP algorithm provides insight into the trade-off between computation time and performance.

Network 1: CPU time vs TTS

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Network 2: CPU time vs TTS

SLP-I

Figure 5.3: Impact of increasing the CPU time budget on the TTS for network 1 (left) and network 2 (right). The blue dashed horizontal line indicates the lowest TTS realized using the SQP algorithm.

The quantitative results are summarized in Figure 5.3 and Table 5.3 for the demand patterns reported in Table 5.2. The table and figure show the realized TTS for different CPU time budgets per iteration. It must be noted that both the CPU time budget and the average CPU time used over the iterations are reported here. The reason for this is that the algorithms cannot be stopped at an exact CPU time budget but only after the budget has been exceeded. The average CPU times are also used in Figure 5.3. It must also be pointed out that for network 2 with the SQP algorithm, only results with a very large CPU time budget are available. The reason for this is that the numerical linearization of the model takes a considerable amount of time so that it was only feasible to run the algorithm from a single starting point per iteration.

TRAILThesisseries

Table 5.2: O-D demands in veh/h from time 100 s to 1200 s for testing the quantitative behavior. The left table shows the demand pattern of network 1, and the right table shows the demand pattern of network 2.

Network 1 Network 2

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7 354 234 238 363 9 252 191 144 148 195 256

10 349 352 478 13 251 191 143 147 194 255

17 240 193 197 244 305

21 258 262 309 370

ter5.EfficientJointOptimizationofRoutingandIntersectionFlows141

caused by the convergence issue of the SLP-I-FP algorithm. **The SQP algorithm as run from 1 starting point without a CPU time budget, hence, this shows the best possible result.

Network 1

SLP-I SLP-II SLP-I-FP SLP-II-FP SQP

CPU budget

TTS CPU TTS CPU TTS CPU TTS CPU TTS CPU

30 166.23 31.66 177.50 31.45 168.98 30.94 182.70 30.90 391.54 244.10 60 167.08 61.58 176.59 61.16 167.30 61.12 182.72 60.88 391.54 234.72 90 166.26 91.45 177.23 91.19 168.45 91.07 182.76 90.89 177.53 161.66 120 166.26 121.63 175.52 121.71 171.88* 121.17 182.60 120.62 177.53 160.37 180 165.91 181.64 176.34 181.25 168.63 181.13 182.54 181.06 174.46 234.14 300 166.71 301.51 175.24 301.36 168.03 301.05 182.62 300.98 170.69 367.71 600 165.89 601.50 175.32 601.19 167.24 601.15 182.73 596.85 166.19 702.15

** - - - 164.54 3276.29

Network 2

SLP-I SLP-II SLP-I-FP SLP-II-FP SQP

30 222.00 36.17 256.03 34.35 221.93 34.53 255.42 33.11 -

-60 221.74 64.12 257.82 63.26 223.21 66.98 255.30 61.57 -

-90 220.62 94.58 254.74 95.34 223.56 94.15 255.29 92.21 -

-120 220.73 126.76 245.89 124.99 222.84 123.85 255.30 122.07 - -180 221.04 186.34 250.03 183.90 223.88 184.91 255.31 182.10 - -300 221.53 304.20 248.30 305.04 223.95 304.62 255.32 302.45 - -600 219.03 605.04 247.55 606.37 223.86 604.68 255.31 602.35 -

-** - - - 207.87 32585.85

The following observations can be made from the results:

• Both the SLP-I and SLP-I-FP algorithm show a lower TTS for different CPU time budgets compared to the SLP-II and SLP-II-FP algorithms. This indicates the added value of including the impact of the control signal on the turn fractions at downstream links in the future.

• The SLP-I algorithm outperforms the SLP-I-FP algorithm as does the SLP-II algorithm outperform the SLP-II-FP algorithm. This indicates that the inclusion of the line-search step when selecting a new point in the optimization algorithm does lead to better performance. An inspection of the algorithm indeed showed that the FP algorithms do not always converge.

• Extending the CPU time budget of the SLP-I algorithm does not lead to large TTS gains. This indicates that for the selected networks, the controller does not require many starting points to find the optimum, since, the increased CPU time budget mainly results in an increase of the number of starting points explored by the algorithm.

• The TTS of the SLP-I-FP algorithm increases when increasing the CPU time budget. This is probably related to the fact that it does not always converge so that adding more starting point leads to unexpected controller behavior.

• A clear decrease in TTS is visible when extending the CPU time budget of the SQP algorithm. When allowing the SQP algorithm as much time as needed to satisfy the stopping criteria from a single starting point, it is able to realize a better TTS compared to the SLP-I algorithm. This indicates that the SLP-I algorithm does not find the best possible solution. However, it does realize sub-optimal performance in much less CPU time, for instance, the SLP-I algorithm realizes a TTS of 166.23 veh·h in 31.7 seconds while the SQP algorithms re-quires 702.2 seconds to realize a comparable TTS of 166.19 veh·h for network 1. In the case of network 2, the SQP algorithm requires over 9 hours per iteration to find the optimum of 207.9 veh·h while the SLP-I algorithm is able to achieve a TTS of 220.6 veh·h in 90 seconds which is 6% higher but realized in much less time.

In conclusion, the quantitative results show the added value of the linearization proce-dure and the line-search step in the SLP optimization algorithm.