## 3 Proposed Energy Management Strategy
### Proposed State Machine Control  
The SMCS is a deterministic rule-based method that can be used for vehicle movement condition detection, battery SOC division, and UC VDR classification. This paper defines the SMC-based proposed mathematical relations for different states. However, the alone use of this strategy cannot guarantee the operation of HESSs at optimal points. Generally, the previous studies have used three conditions of acceleration/ braking/stop as different vehicle movement states [26]. However, in the presented operation states of the vehicle, five states are defined based on the demand power (\(P_{Demand}(t)\)), speed (\(v(t)\)), and acceleration (\(a(t)\)), as shown in Figure 2. These five vehicle movement states are idling, acceleration, fixed speeding cruise, braking, and regenerative braking. The vehicle movement states is formulated by (9) [27]:  
\[VSM(t)=sign(a(t))+sign(v(t))+sign(P_{Demand}(t)) \tag{9}\]Furthermore, the optimal operation of the battery and UC depend on the charge/discharge efficiency. In this study, battery combined efficiency (\(BCE\)) and UC combined efficiency (\(UCE\)) are formulated to divide the different states of SOC and VDR. The battery combined efficiency of lead-acid battery (\(\eta_{BCE}\)) is formulated in Table 1 and shown in Figure 3.  
Figure 1: Multi-level EMS flowchart.  
In this study, \(BCE\) is used to determine the different states of the battery SOC. The \(BCE\) expectation (\(\mu_{BCE}\)) is defined as Table 1. However, more indices are required for dispersion analysis of BCE. The standard deviation is a proper index to determine the efficiency's dispersion from the median point. This index is calculated using by (10), in which uniform probability distribution function (\(P_{BCE}(\eta_{i})\)) is in the range of \([0.1,0.9]\).  
\[\sigma_{BCE}=\sqrt{E\{(BCE-\mu_{BCE})^{2}\}}=\sqrt{\sum_{i=1}^{n}(\eta_{i}-\mu _{BCE})^{2}P_{BCE}(\eta_{i})} \tag{10}\]  
The battery combined efficiency, mean, and standard deviation for different SOCs of a lead-acid battery at rated power are shown in Figure 4. It can be seen that maximum BCE is 92.35% occurring when SOC is 0.5; minimum BCE is 88.31% occurring when SOC is 0.1; average BCE is 90.89%; and the standard deviation of BCE is 1.42%, which are used to divide states of the battery.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element** & **Combined Efficiency** & **Combined Efficiency Expectation** \\ \hline Lead-acid battery & \(\eta_{BCE}=(\frac{\eta_{dis}+\eta_{delay}}{2})\) & \(\mu_{BCE}=\sum_{i=1}^{n}\eta_{i}P_{BCE}(\eta_{i})\) \\ MAXWELL ultracapacitor & \(\eta_{UCE}=(\frac{\eta_{UCE}+\eta_{UCE}}{2})\) & \(\mu_{UCE}=\sum_{i=1}^{n}\eta_{ui}P_{UCE}(\eta_{ui})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The combined efficiency.  
Figure 3: Typical Lead-acid battery combined efficiency.  
Figure 2: Vehicle state movement function values.  
The high power density of UC discriminates it from other conventional ESSs and makes fast reactions in a short time possible. The charge/discharge efficiency of the UC depends on the voltage, current, and VDR. The UC combined efficiency can be used for analyzing the division of UC states. The mathematical calculations of UC combined efficiency are given in Table 1, and its curve is shown in Figure 5.  
The expectation of the UC combined efficiency (\(\mu_{UCE}\)) is defined as in Table 1, where uniform PDF (\(P_{UCE}\) (\(\eta_{ui}\))) is in the range of \([40,100]\). The UC combined efficiency at the rated power is divided into three operational regions as shown in Figure 6. It can be seen that as VDR is increased, UCE increases. The median of the combined efficiency is 95.2%, and its average is 92.96% at the maximum power of UC, which are used to separate the operational states of UC.