## 4 Experiments and Evaluation
### Adaptive Scheduling: Performance Analysis  
This section details our evaluations and analysis to optimize the scheduling process, specifically, how the chromosome design and the weights of power and speed in the fitness function of our GA algorithm influence the performance of the scheduling process. First of all, in this experiment, we use the digital twin (the simulated model) in order to remove the effect of random and unknown factors that could be introduced on the real conveyor system, such as, unbalanced frictions or inaccurate sensor readings. The simulation also allows us to simulate a wide range of abnormal conditions, which is hard and dangerous to realize on the real system. A speed range of 6 to 32 \(\tau/\min\) is used during this experiment as this is the recommended speed range of the real conveyor system according to its motor specifications.  
To effectively evaluate the performance of the algorithm under different conditions, we propose a customized efficiency calculation having both power (energy) and speed (distance) as the evaluation metrics with the selected weights of importance (i.e., \(\alpha\) and \(\beta\)). The evaluation metrics are selected to reflect our optimization goals of safety (minimum power) and performance (maximum speed). It is defined as follows:  
\[\eta=(1-\frac{E_{\Delta}}{E_{nom}})\cdot\alpha+\frac{S_{real}}{S_{nom}}\cdot\beta, \tag{7}\]  
where \(\alpha\) and \(\beta\) are the same weights of the fitness function in Equation (2), that is, the weights of speed and power components. \(E_{\Delta}\) is the absolute deviance between the energy consumption under a constant requested speed (i.e., the nominal energy consumed, \(E_{nom}\)) and the real energy consumption during an operation. \(S_{real}\) is the actual amount of travelling distance of the conveyor belt during an operation. \(E_{nom}\) is calculated using the total expected traveling distance under a requested speed, \(S_{nom}\). Since the real-time power consumption is changing all the time, \(E_{\Delta}\) can be calculated by the integral of absolute power difference over time:  
\[E_{\Delta}=\int_{t_{0}}^{t_{0}+\Delta t}\int|P_{real}-P_{nom}|dt, \tag{8}\]  
where \(P_{real}\) is the real power value measured during an operation and \(P_{nom}\) is the nominal power value expected under the given conditions. \(S_{real}\) can be calculated by the integral of real-time speed over time:  
\[S_{real}=\int_{t_{0}}^{t_{0}+\Delta t}v_{real}dt, \tag{9}\]  
where \(v_{real}\) is the measured speed values. \(E_{nom}\) and \(S_{nom}\) can be calculated as:  
\[E_{nom}=P_{nom}\cdot\Delta t \tag{10}\]  
\[S_{nom}=v_{req}\cdot\Delta t. \tag{11}\]  
In our case, a good adaptive control algorithm should provide a reliable response to a load torque increase and decrease. In other words, when a disturbance is introduced and/or withdrawn, the goal is to keep the speed as close as possible to the originally requested one while keeping the motor in a safe operation (lesser power consumption when there is an abnormal behavior predicted). The control mechanism should also satisfy a smooth behavior during the operation by not allowing sudden increase or decrease on the speed when the load drastically changes.