## 2 Modelling and Theoretical Analysis for Power Flow Gradients
### Theoretical Analysis for Regulation Principles of the PFG to CIVs  
Foremost, it is essential to illuminate the regulation meanings or targets for PFG to CIVs by UPFC. When the CIVs are regulated to certain levels or regions, the APFG and RPFG to CIVs at different SCP will also change to new levels or regions, therefore the power flows will transmit to the requisite values by higher or lower efficiencies.  
To be specific, if the power flow gradients are regulated to a higher level, the power flows in the system are changing much more intensely. In this case, the regulation efficiencies or changing rate of power flows are higher and the power flows can transmit to the requisite values more rapidly. However, the resulting impulsive shocks to the system are more serious at the same time.  
In contrast, if the power flow gradients are regulated to a lower level, the power flows in the system are changing much more mildly, the regulation efficiencies or changing rate of power flows are lower and the power flows may transmit to the required values more slowly. Moreover, the resulting impulsive shocks to the system are weaker at the same time.  
Next, according to the mathematical model above, the regulation principles of all the PFG to each CIV at each SCP can be carried out theoretically.  
First of all, according to Equations (1)-(3), the PFG to CIV1 are only dependent on CIV2 of phase angle \(\theta\), and similarly, the PFG to CIV3 are only dependent on CIV3. However, the PFG to CIV2 is dependent on CIV1 and CIV3 simultaneously. Therefore, the PFG to CIV1 and CIV3 at SCP1 can both be regulated by one CIV, namely that they have both only one degree of freedom for regulations. In contrast, the PFG to CIV2 has one more degrees of freedom for regulations by UPFC, therefore, the regulation modes or manners for PFG to CIV2 are more complicated and sufficient.  
Based on Equations (4)-(6) for the PFG at SCP2, the PFG to CIV1 are similar to that of SCP1, yet the PFG to CIV2 and CIV3 both depend on all the three CIVs; therefore, they both have three degrees of freedom for regulations by UPFC, indicating that the PFG at SCP2 can be regulated more complicatedly and sufficiently than that of SCP1.  
According to Equations (7)-(9), the PFG at SCP3 are all dependent on three CIVs and they all have three degrees of freedom for regulation by UPFC, revealing that the PFG at SCP3 have become the most complicated and sufficient along the whole power system after the insertion of OSIV and the regulations by UPFC.  
Similarly, based on Equations (10)-(12), the PFG to CIV1 at SCP4 is dependent on CIV2 and CIV3 has one degree of freedom, and the PFG to CIV2 and CIV3 at SCP4 both depend on three CIVs and have three degrees of freedom. Therefore, the PFG at SCP4 have similar complexity for regulations to that of SCP2.  
In conclusion, the PFG at SCP1 have the fewest degrees of freedom and the simplest regulation principles for regulations by UPFC. The PFG at SCP2 and SCP4 have the similar degree of freedom for regulations by UPFC, and the PFG at SCP3 owns the most degrees of freedom and the most complicated regulation principles for regulations by UPFC, indicating that the PFG of the power system can be regulated by UPFC efficiently after the insertion of OSIV at SCP2.