## 3 The Algebraic Sign of Virtual Power Flows
Powers \(P_{x}\) and \(P_{x}^{y}\) have the same algebraic sign only if the following is true:  
\[\frac{P_{x}^{y}}{P_{x}}=\frac{\omega_{x}-\omega_{y}}{\omega_{x}}>0 \tag{5}\]  
or  
\[\frac{\omega_{y}}{\omega_{  
**Case 2:**\(\omega_{x}\) and \(\omega_{y}\) have the same algebraic sign, or \(\text{sign}\bigg{(}\dfrac{\omega_{y}}{\omega_{x}}\bigg{)}=+1\).  
Under this hypothesis, Equations (6) and (8) cannot hold simultaneously.  
**Subcase 2a:** When \(\dfrac{\omega_{y}}{\omega_{x}}<1\), then \(P_{x}^{y}\) and \(P_{x}\) have the same algebraic sign. The described situation is depicted in Figure 3, 1 st working mode.  
**Subcase 2b:** When \(\dfrac{\omega_{x}}{\omega_{y}}<1\), then \(P_{y}^{x}\) and \(P_{y}\) will have the same algebraic sign. The described situation is depicted in Figure 4, 1 st working mode.  
The previous analysis can be summarized as follows:  
1. When \(\omega_{x}\) and \(\omega_{y}\) do not have the same algebraic sign, the power ratios \(\dfrac{P_{x}^{y}}{P_{x}}\) and \(\dfrac{P_{y}^{x}}{P_{y}}\) will have simultaneously the same algebraic sign. For all the links, the direction of the virtual power-flow is the same as the actual power flow (see Figure 2).
2. When \(\omega_{x}\) and \(\omega_{y}\) have the same algebraic sign, power ratios \(\dfrac{P_{x}^{y}}{P_{x}}\) and \(\dfrac{P_{y}^{x}}{P_{y}}\) cannot have simultaneously the same algebraic sign. In particular, the condition \(\dfrac{P_{x}^{y}}{P_{x}}>0\) is fulfilled when \(\dfrac{\omega_{y}}{\omega_{x}}>1\) (See Figure 3). Conversely, the condition \(\dfrac{P_{y}^{x}}{P_{y}}>0\) is fulfilled when \(\dfrac{\omega_{x}}{\omega_{y}}>1\).  
Under kinematic inversions that fix links \(x\) and \(y\), respectively, assuming without loss of generality \(P_{x}^{y}>0\), the following relations hold (see 1 st working mode of Figure 2):  
\[T_{x}\omega_{xy}\eta_{y(x-z)}+T_{z}\omega_{zy} =0 \tag{9a}\] \[T_{y}\omega_{yx}\eta_{x(y-z)}+T_{z}\omega_{zx} =0 \tag{9b}\]  
The overall power balance condition is as follows:  
\[\eta_{(x,y-z)}=\frac{|T_{z}\omega_{z}|}{T_{x}\omega_{x}+T_{y}\omega_{y}} \tag{9c}\]  
Solving the system composed of the Equations (9), one obtains the following:  
\[\eta_{(x,y-z)}=\left|\frac{\omega_{z}\omega_{yx}\eta_{y(x-z)}}{\omega_{x} \omega_{zy}-\frac{\eta_{y(x-z)}}{\eta_{x(y-z)}}\omega_{y}\omega_{zx}}\right| \tag{10}\]  
**Subcase 2a:**\(\dfrac{\omega_{y}}{\omega_{x}}<1\) or \(\text{sgn}\left(\dfrac{\omega_{x}}{\omega_{y}}\right)=+1\) (see Figure 3, 1 st working mode)  
Under kinematic inversions that fix links \(x\) and \(y\), respectively, assuming without loss of generality \(P_{x}^{y}>0\), the following relations hold:  
\[T_{x}\omega_{xy}\eta_{y(x-z)}+T_{z}\omega_{zy} =0 \tag{11a}\] \[T_{y}\omega_{yx}+T_{z}\omega_{zx}\eta_{x(z-y)} =0 \tag{11b}\]  
Solving the system composed of the Equations (9c) and (11), one obtains the following:  
\[\eta_{(x,y-z)}=\left|\frac{\omega_{z}\omega_{yx}\eta_{y(x-z)}}{\omega_{x} \omega_{zy}-\eta_{x(z-y)}\eta_{y(x-z)}\omega_{y}\omega_{zx}}\right| \tag{12}\]  
The analytical Expression (10) can be combined into the following formula:  
\[\boxed{\eta_{(x,y-z)}=\left|\frac{\omega_{z}\omega_{yx}\eta_{y(x-z)}}{\omega_{ x}\omega_{zy}-A\omega_{y}\omega_{zx}}\right|} \tag{13}\]  
with  
\[A=\left\{ \begin{aligned} &\dfrac{\eta_{y(x-z)}}{\eta_{x(y-z)}}& \text{when sign}\left(\dfrac{\omega_{x}}{\omega_{y}}\right)=-1\\ &\eta_{x(z-y)}\eta_{y(x-z)}&\text{when }\left(\dfrac{ \omega_{y}}{\omega_{x}}\right)<1\text{ and sign}\left(\dfrac{\omega_{x}}{\omega_{y}} \right)=+1\end{aligned} \right. \tag{14}\]  
The proposed formula is more general than the one proposed by [PERSON] (see Equation (11) of [26]). When \(\omega_{x}=\omega_{y}=\omega_{z}\), the Equations (9b) and (11) are identically satisfied and Formula (13) is not valid. However, under these kinematic conditions, there is no meshing power loss. Under our hypotheses, the mechanical efficiency is equal to one.