## 4 Experiments and Evaluation
### Adaptive Scheduling: Performance Analysis
#### 4.5.2 The Effect of Different Operational Limits on the Performance  
This experiment is intended to evaluate the performance of different chromosome designs, that is, different operational limits that are defined as the allowed speed interval to operate between \(v_{min}\) and \(v_{max}\) values. We create populations with the speed intervals of sizes \(0.25,0.5,1,2\) and \(8\) r/min, that is, each operation mode (chromosome) selected only allows for a speed change defined by the interval value and it is the same for all chromosomes. For the experiments, initially requested speed is \(25\) r/min and the same sinusoidal disturbance in the parameter tests is applied on the system. For this experiment, we pick the parameters of \(\alpha=0.55\) and \(\beta=0.45\) as discussed in Section 4.5.1. The experiment results (speed, power, and efficiency) of different sizes of intervals are shown in Figure 10.  
Figure 9: System behaviors for \(\alpha\) and \(\beta\) combinations under an abnormal load condition. In Figure 9(a) we show that systems with bigger interval sizes (allowed speed intervals) react slower than the ones with smaller sizes. This is mainly due to their larger allowed speed and power changes, causing longer reaction times to changing environmental conditions. Since the threshold to trigger for a new scheduling is also set proportional to the allowed interval, a larger interval makes the system more resistive to mode changes (see in Figure 5). Another finding is that the decreased rate of speed is almost the same for different cases. This is because when the speed is decreasing, the calculated PID error is usually much larger than the threshold causing faster genetic algorithm calls. Finally, the cases with smaller intervals exhibit better system recoveries, that is, when the torque is completely withdrawn they show better approximation to the original speed request. As a comparison, when \(interval=8\) the speed is not able to increase after the disturbance becomes zero. This is also reflected in power consumption (in Figure 9(b)) and in the efficiency analysis (in Figure 9(c)). Figure 10: Performance comparisons among various intervals. The average efficiencies calculated by Equation (7) are shown in Table 3. From the table, we conclude that the smaller the interval size, the higher the efficiency. This is expected, as a small interval size indicates a more fine-grained control. The system schedules for another chromosome faster and more frequently since suggested interval sizes are smaller and so exceeded faster. For a better comparison, we run ANOVA (anaylsis of variance) on each two neighboring intervals. The results suggest that, starting from the interval of 0.5, there are significant differences between the efficiencies (\(F(1,3671)=26.99,p=2.16E-7\)). However, the significance disappears between 0.25 and 0.5 intervals (\(F(1,3671)=0.5084,p=0.4759\)). The similarity on the efficiency of interval groups 0.25 and 0.5 can be explained by the computational bottleneck. Normally, with the interval of 0.25 and smaller we should see a more fine-grained control as discussed. However, in practice, since the scheduling by the genetic algorithm takes time, that is, in this case it is particularly not fast enough to respond to such frequent calls, the advantage of smaller interval size disappears due to the slower response times. This also causes fluctuations on the curves. Since the torque is very dynamic, it constantly changes causing the system to regularly trigger for a new mode selection. In an allowed interval, the system increases the speed and the power consumption; however, a continuous change in the torque triggers the adaptive control switching to a different operation mode (different limitations on the speed). These fluctuations should be theoretically compensated by smaller interval sizes but the computation time of the genetic algorithm is a limiting factor.