Voltage source converter (VSC) based highvoltage directcurrent (HVDC) system is a new transmission technique, which has the most promising applications in the fields of power systems and power electronics. Considering the importance of power flow analysis of the VSCHVDC system for its utilization and exploitation, the improved power flow algorithms for VSCHVDC system based on thirdorder and sixthorder Newtontype method are presented. The steady power model of VSCHVDC system is introduced firstly. Then the derivation solving formats of multivariable matrix for thirdorder and sixthorder Newtontype power flow method of VSCHVDC system are given. The formats have the feature of thirdorder and sixthorder convergence based on Newton method. Further, based on the automatic differentiation technology and thirdorder Newton method, a new improved algorithm is given, which will help in improving the program development, computation efficiency, maintainability, and flexibility of the power flow. Simulations of AC/DC power systems in twoterminal, multiterminal, and multiinfeed DC with VSCHVDC are carried out for the modified IEEE bus systems, which show the effectiveness and practicality of the presented algorithms for VSCHVDC system.
Voltage source converter (VSC) based highvoltage directcurrent (HVDC) is a new technology of HVDC transmission system. Based on pulse width modulation and VSC, the VSCHVDC system has many merits and attracted wide publicity worldwide [
The main advantages of VSCHVDC system are as follows: no synchronization problem of AC system, the feature of supplying power to passive network, the simultaneous and independent control for active power and reactive power, the easy achievement of inversion for power flow, the more flexible control modes, the suitable application for multiterminal and multiinfeed system, and so on. In the near future, a series of new VSCHVDC transmission system will be built and put into operation worldwide [
The Newton method is a fundamental and important technology to solve the power flow of power system [
In recent years, the solution of nonlinear equation has made great progress, especially the modified Newton method with highorder convergence performance [
The remaining of this paper is arranged as follows. In Section
For the simulation and calculation of AC/DC hybrid power systems, the unified perunit value system should be adopted both for AC system and DC system. In this paper the perunit value system is introduced as follows [
The VSCHVDC system consists of at least two VSC stations, one operating as a rectifier station and the other as an inverter station. The VSC stations can be connected as twoterminal, multiterminal, or multiinfeed DC system with VSCHVDC, depending on the various different applications fields [
Schematic diagram of steady state physical model for multiterminal VSCHVDC.
Owning to having full controllable power electronic switch semiconductors such as insulated gate bipolar transistor and gate turnoff thyristor, VSCHVDC has the ability to independent control active and reactive power at its terminal. So for each VSC, a couple of regular used control goals can be set [
AC active power control: determines the active power exchanged with the AC system.
DC voltage control: is used to keep the DC voltage control constant.
AC reactive power control: determines the reactive power exchanged with the AC system.
AC voltage control: instead of controlling reactive power, AC voltage can be directly controlled, determining the voltage of the system bus.
The general used control means of VSC include the following four categories:
For the AC/DC systems with VSCHVDC, the power flow equations are given as follows [
Pure AC bus equation
DC bus equation
VSC converter equation
DC network equation
The mathematical description of multivariable iterative form for Newton method is given by
The equivalence form of linear equation solution for (
The single variable iterative algorithm format based on modified Newtontype method is given by
The iterative format of (
The multivariable matrix equivalent form of (
The gotten Jacobian matrix and its triangular factorization are being utilized fully in the algorithm iterative process of (
Another single variable iterative algorithm format with thirdorder convergence based on Newtontype method is given by:
The iterative format of (
The multivariable matrix equivalent form of (
For the above presented Algorithm 1 and Algorithm 2, the two iterative formats have the advantages for fast convergence speed of Newton method and less computations of simplified Newton method. The application of Algorithm 1 and Algorithm 2 is a twostep process.
The prediction based on the Newton method [
The correction for the obtained predicted value of
The simplified realization of the iterative procedure for (
An effective implement of the iterative process of (
The approximate value of
The (
The AD technique could always be decomposed to complex computations of basic functions and basic mathematical operations, such as the four arithmetic operations of add, subtract, multiply, and divide, the basic functions of trigonometric function, exponential function, and logarithmic function. Here an instance is given to illustrate the application of AD. The function expression of a certain model is given by
The independent variables and intermediate variables of (
Independent variables and intermediate variables of (
Independent variables  Intermediate variables 





 
 

Forward and backward mode.
Forward mode  Backward mode 











Now there are two kinds of implementation method for AD, the source code transform method and operator overloading method. The typical representative softwares for the former is ADIFOR and ADIC. The typical representative software for the latter is ADOLC and ADC. The method of ADOLC realizes the differentiation of C++ program automatically by using operator overloading and can calculate any order derivative by forward and backward mode. In this paper, the ADOLC method is used to realize the differential operation [
The steps of the improved AD algorithm based on thirdorder Newton method are listed below.
Return to Step 3.
In this part, in order to validate the correctness and suitability of the proposed algorithms, three sections are presented.
For the modified highorder Newton methods, the modified IEEE 30bus system with twoterminal and multiinfeed VSCHVDC is analyzed in detail firstly.
Then the simulation results of performance comparisons for the improved highorder Newton methods are presented among the modified IEEE 5bus, IEEE 9bus, IEEE 14bus, IEEE 57bus, and IEEE 118 bus text systems.
At last, the AD based on thirdorder Newton method is evaluated for the modified IEEE 30bus system with twoterminal of VSCHVDC.
The proposed method has been applied to the modified IEEE 30bus system [
In the system with twoterminal VSCHVDC, the VSC1 and VSC2 are connected to AC line of bus 29 and bus 30, respectively.
In the system with twoinfeed VSCHVDC, the VSC1, VSC2, VSC3, and VSC4 are connected to AC line of bus 12, bus 14, bus 29, and bus 30, respectively.
The modified IEEE30 bus AC/DC system with VSCHVDC.
The results of the power flow calculation of the AC system and DC system under different control modes for Newton, thirdorder and sixthorder Newton methods are shown in Tables
Results of the power flow calculation of AC system.
Control mode  Method  Bus Number  

1  2  3  4  










Newton 








Algorithm 1 









Algorithm 2 









Sixthorder Newton 









 

Newton 








Algorithm 1 









Algorithm 2 









Sixthorder Newton 









 

Newton 








Algorithm 1 









Algorithm 2 









Sixthorder Newton 









 

Newton 








Algorithm 1 









Algorithm 2 









Sixthorder Newton 








Results of the power flow calculation of DC system.
DC variable  Converter number  Control mode  







Newton  VSC_{1} 




VSC_{2} 





Algorithm 1  VSC_{1} 





VSC_{2} 





Algorithm 2  VSC_{1} 





VSC_{2} 





Sixthorder Newton  VSC_{1} 





VSC_{2} 





 

Newton  VSC_{1} 




VSC_{2 } 



 
Algorithm 1  VSC_{1} 





VSC_{2} 



 
Algorithm 2  VSC_{1} 





VSC_{2} 



 
Sixthorder Newton  VSC_{1} 





VSC_{2} 



 
 

Newton  VSC_{1} 




VSC_{2} 





Algorithm 1  VSC_{1} 





VSC_{2} 





Algorithm 2  VSC_{1} 





VSC_{2} 





Sixthorder Newton  VSC_{1} 





VSC_{2} 




It can be seen from Table
In Table
Both in Tables
The comparisons of iteration times and computing time for the four proposed Newton methods are shown in Table
Comparisons of iteration times and computing time.
Control mode  Iteration times  Computing time (ms)  

Newton  Algorithm 1  Algorithm 2  Sixthorder Newton  Newton  Algorithm 1  Algorithm 2  Sixthorder Newton  

4  3  3  2  8.6885  8.8367  0.1040  12.6763 

4  3  3  2  9.8464  5.6506  0.1023  9.5408 

4  3  3  1.5*  9.3356  5.6121  0.1045  8.4520 

4  3  3  1.5*  9.4767  5.6403  0.0990  8.2271 
For the flexible control performance and particular technical advantages, the VSCHVDC is suitable for application in multiinfeed system [
Comparison of iteration times and computing time.
Control mode  Iteration times  Computing time (ms)  

VSC_{1} + VSC_{2}  VSC_{3} + VSC_{4}  Newton  Algorithm 1  Algorithm 2  Sixthorder Newton  Newton  Algorithm 1  Algorithm 2  Sixthorder Newton 


4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 






4  3  3  1.5 




It can be seen from Table
The modified IEEE 5, 9, 14, 57, and 118bus systems are analyzed in this section [
The topology and parameter settings for different IEEE text systems.
Topology  Twoterminal  Twoinfeed  Threeterminal  

Modified test system  IEEE5  IEEE9  IEEE14  IEEE57  IEEE118  IEEE14  IEEE14  IEEE57  
Bus number  VSC_{1}  VSC_{2}  VSC_{1}  VSC_{2}  VSC_{1}  VSC_{2}  VSC_{1}  VSC_{2}  VSC_{1}  VSC_{2}  VSC_{1}  VSC_{2}  VSC_{3 }  VSC_{4}  VSC_{1}  VSC_{2}  VSC_{3}  VSC_{1}  VSC_{2}  VSC_{3} 
4  5  8  9  13  14  56  57  75  118  12  14  29  30  12  13  14  55  56  41 
Performance comparison for different text systems.
Topology  Modified test system  Iteration times  Computing time (ms)  

Newton  Algorithm 1  Sixthorder Newton  Newton  Algorithm 1  Sixthorder Newton  
Twoterminal  IEEE5  5  4  1.5  4.8287  4.4651  5.3692 
Twoterminal  IEEE9  4  3  1.5  3.1646  2.2094  2.7866 
Twoterminal  IEEE14  4  3  1.5  4.1375  2.6509  4.1890 
Twoinfeed  IEEE14  4  3  1.5  4.3529  3.1544  3.9907 
Threeterminal  IEEE14  4  3  1.5  4.3546  2.9592  4.1260 
Twoterminal  IEEE57  4  4  2  22.0698  14.2854  23.6236 
Threeterminal  IEEE57  4  4  2  28.9858  24.7143  29.0658 
Twoterminal  IEEE118  4  3  1.5  120.8938  70.8048  115.1420 
It can be seen in Table
In this part, the simulation results for AD algorithm based on Algorithm 1 and Algorithm 2 of thirdorder Newton method are presented. The IEEE30 bus text system with twoterminal VSCHVDC is employed to demonstrate the validity of the proposed AD algorithm. The results of the power flow calculation of DC system of control mode
Results of the power flow calculation of DC system.
DC variable  Converter number  Control modes of 



Algorithm 1 + AD  VSC_{1} 

VSC_{2} 


Algorithm 2 + AD  VSC_{1} 


VSC_{2} 


 

Algorithm 1 + AD  VSC_{1} 

VSC_{2} 
 
Algorithm 2 + AD  VSC_{1} 


VSC_{2} 
 
 

Algorithm 1 + AD  VSC_{1} 

VSC_{2} 


Algorithm 2 + AD  VSC_{1} 


VSC_{2} 

Comparison of iteration times and computing time
Control mode  Iteration times  Computing time (ms)  

VSC_{1} + VSC_{2}  Algorithm 1 + AD  Algorithm 2 + AD  Algorithm 1 + AD  Algorithm 2 + AD 

2  2  0.15  0.31 

2  2  0.31  0.31 

2  2  0.31  0.32 

2  2  0.31  0.32 
From the results of Tables
Compared with the results of Algorithm 1 and Algorithm 2 of thirdorder Newton method, as shown in Table
Compared with the results of Table
The result shows that AD technology is suitable for use in the thirdorder Newton method of VSCHVDC system. And the application of AD technology reduces the work of hand code greatly. The efficiency of code programming is improved.
In this paper, based on the steady mathematical model of VSCHVDC, the modified thirdorder Newton and sixthorder Newton methods have been presented to calculate the power flow of AC/DC systems with VSCHVDC. The multivariate iteration matrix forms of the presented algorithms suitable for VSCHVDC system are given. The proposed highorder Newton method has the thirdorder and sixthorder convergence, without solving the Hessian matrix. The task of the calculation is greatly reduced, and the efficiency is improved. Based on the thirdorder Newton method, the automatic differentiation technology is used to increase the efficiency of hand code. Some numerical examples on the modified IEEE bus systems with twoterminal, multiterminal, and multiinfeed VSCHVDC have demonstrated the computational performance of the power flow algorithms with incorporation of VSCHVDC models.
This work is supported in part by the National Natural Science Foundation of China (Grant no. 61104045 and U1204506), in part by the Youth Project of National Social Science Fund (Grant no. 09CJY007), and in part by the Fundamental Research Funds for the Central Universities of China (Grant no. 2012B03514).