## 4 Conclusions  
This paper has researched and compared the regulation principles and efficiencies of the power flow gradients to multiple characteristic independent variables by multiple regulation modes and scenarios at several critical points of the UPFC embedded power system. The mathematical model of APFG and RPFG at each SCP are built based on detailed real circuit, and it was tested and analyzed in different typical operation conditions and in planes and spaces. The main conclusions are summarized as below. Figure 12: Spatial surfaces of PFG to CIV2 at SCP4 in spaces: (**a–d**) Spatial surfaces of PFG to CIV2 by Reg.1 mode of OSIV in four operation conditions; (**e–h**) Spatial surfaces of PFG to CIV2 by Reg.2 mode of OSIV in four operation conditions. Figure 13: Spatial surfaces of PFG to CIV2 at SCP4 in spaces: (**a–d**) Spatial surfaces of PFG to CIV2 in four regulation scenarios. 1. The PFG to each CIV at different SCP have different degrees of freedom for regulations. 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 degrees 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 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. 2. In the two-dimensional planes, the APFG and RPFG to CIV1 at SCP3 and SCP4 in the normal operation conditions have the similar curves and focus within their individual certain regions, while the curves of APFG and RPFG in the severe operation condition stay at different regions with the curves in the normal operation conditions. The PFG to CIV2 at each SCP can be regulated to larger or smaller required amplitudes in different regions by Reg.1 and Reg.2 modes, the curves of APFG to CIV2 against RPFG to CIV2 by Reg.1 and Reg.2 modes can form special axial or rotational symmetrical closed shapes, the regulation ranges and areas by Reg.2 mode in the third and fourth operation condition are much larger than others, which are beneficial to be regulated throughout an overall closed region and adaptive to different operation conditions. Both the curves of PFG to CIV3 at SCP1, SCP3 and SCP4 show nearly the same shapes with the similar magnitudes and phases in different regulation scenarios, while those at SCP2 are with different magnitudes and phases, indicating the regulations of PFG by UPFC at SCP2 can be efficiently adaptive to different operation conditions. 3. In the three-dimensional spaces, the spatial surfaces of PFG to CIV1 at SCP3 have non-uniform dome shapes humping upwards, and nearly the same regulation efficiencies by Reg.1 and Reg.2 modes which are larger than SCP1 and SCP2. The spatial surfaces of PFG to CIV2 at each SCP present similar funnel shapes in all operation conditions, and the surface curvatures and areas by Reg.1 mode at each SCP are similar, but those by Reg.2 mode at SCP1, SCP3 and SCP4 are larger than SCP2, revealing the OSIV of UPFC at SCP2 has enough large efficiencies to regulate the power flows at other SCP. The spatial surfaces of PFG to CIV3 at SCP2 present similar funnel shapes, while those at SCP3 and SCP4 have two funnel peaks and inverse varying tendencies within different regulation regions, which is more beneficial to occupy larger surface areas and adaptive to different regulation requirements. 4. When the PFG to CIVs have larger regulation efficiencies, the resulting impulses or shocks to the power system during the regulation processes may also be more intense, and the requirements for suppressing the resulting shocks need to be more rigorous at the same time, so as to maintain the stability and normal operations of the power system. [PERSON] conceived and wrote the original manuscript, [PERSON] is the supervision, [PERSON] reviewed and edited the paper critically, [PERSON] prepared the resources and [PERSON] searched literatures.