## 3 Desired Power Sharing Calculation Method  
Because an energy conversion device is used as primary power source, we must take its traits for consideration. One is the efficiency characteristic. For example, an engine generator has a best efficiency point and a maximum efficiency curve, operating on which makes the engine under the best economy. The other is the lagging characteristic in response. For example, there is a time lag between opening the throttle and the increasing of engine's output power, or between increasing the hydrogen supply and power updating of fuel cell. Since the secondary power source is an energy storage device and it does not need additional controls, we set it as a power buffer, to compensate the lacking power or absorb the redundant power outputted by primary power source. Considering the two traits of the primary power source mentioned above, a desired power sharing calculation method based on a first order inertial element is proposed, as shown in Figure 4.  
This calculator has two inputs, which are \(P_{mot}\) and \(P_{inv1\_opt}\). Where \(P_{mot}\) is the input power of OW-PMSM, identical to the summation of two inverters' output power, and can be determined as follow:  
\[P_{mot}=\overset{\rightarrow}{u_{s}}\cdot\overset{\rightarrow}{i_{s}} \tag{8}\]  
\(P_{inv1\_opt}\) is the optimal output power of inverter1, under which the primary power source obtains maximum efficiency. \(P_{inv1\_opt}\) can be set as a constant, corresponding to the best efficiency point; or set as a variable, corresponding to the maximum efficiency curve, following the changes of its output power. Then the power deviation of inverter1 \(\Delta P_{inv1}\) can be obtained by subtracting \(P_{inv1\_opt}\) from \(P_{mot}\):  
\[\Delta P_{inv1}=P_{mot}-P_{inv1\_opt} \tag{9}\]  
\(\Delta P_{inv1}\) indicates the difference between the total desired power and the ideal power output of inverter1. Thus, we use \(\Delta P_{inv1}\) as the input of the first order inertial element, to generate the compensating power of inverter1. In the frequency domain, we have:  
\[\Delta P_{inv1}^{*}(s)=\frac{K}{Ts+1}\Delta P_{inv1}(s) \tag{10}\]  
where \(\Delta P_{inv1}^{*}\) is the compensating power of inverter1, \(K\) and \(T\) are the gain and the time constant of the inertial element, respectively. In the time domain, \(\Delta P_{inv1}^{*}\) and \(\Delta P_{inv1}\) satisfy the following differential equation:  
\[T\frac{d}{dt}\Delta P_{inv1}^{*}(t)+\Delta P_{inv1}^{*}(t)=K\Delta P_{inv1}(t) \tag{11}\]  
In a discrete system, the current \(\Delta P_{inv1}^{*}\) can be obtained by Equation (12), where \(\Delta P_{inv1}^{*}(t)\) is the valve of \(\Delta P_{inv1}^{*}\) in current sample step and \(\Delta T\) is the step size of the system. \[\Delta P_{inv1}^{*}(t)=\frac{K\Delta P_{inv1}(t)-\Delta P_{inv1}^{*}(t-\Delta T )}{T}\Delta T+\Delta P_{inv1}^{*}(t-\Delta T) \tag{12}\]  
After \(\Delta P_{inv1}^{*}\) is obtained, the desired output power of inverter1 \(P_{inv1}^{*}\) can be determined:  
\[P_{inv1}^{*}=P_{inv1\_opt}+\Delta P_{inv1}^{*} \tag{13}\]  
Figure 4: Calculation diagram of desired power of inverter1. By adjusting \(K\) and \(T\), different power sharing characteristics can be achieved. We have \(K\in[0,1]\), and it affects the response amplitude of \(\Delta P^{*}_{inv1}\). The larger \(K\) is, the greater amplitude \(\Delta P^{*}_{inv1}\) responds to \(\Delta P_{inv1}\). For example, when \(K=0\), we have \(P^{*}_{inv1}=P_{inv1\_opt}\) at all time, totally regardless of the influence of \(\Delta P_{inv1}\). In this circumstance, the efficiency of the primary power source is maximized, but the secondary power source will also take the largest power fluctuation. When \(K=1\), \(P^{*}_{inv1}\) is completely following \(P_{mot}\) except the lag caused by the inertial element.