Voltage source converter (VSC) based high-voltage direct-current (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 VSC-HVDC system for its utilization and exploitation, the improved power flow algorithms for VSC-HVDC system based on third-order and sixth-order Newton-type method are presented. The steady power model of VSC-HVDC system is introduced firstly. Then the derivation solving formats of multivariable matrix for third-order and sixth-order Newton-type power flow method of VSC-HVDC system are given. The formats have the feature of third-order and sixth-order convergence based on Newton method. Further, based on the automatic differentiation technology and third-order 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 two-terminal, multi-terminal, and multi-infeed DC with VSC-HVDC are carried out for the modified IEEE bus systems, which show the effectiveness and practicality of the presented algorithms for VSC-HVDC system.
Voltage source converter (VSC) based high-voltage direct-current (HVDC) is a new technology of HVDC transmission system. Based on pulse width modulation and VSC, the VSC-HVDC system has many merits and attracted wide publicity worldwide [
The main advantages of VSC-HVDC 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 multi-terminal and multi-infeed system, and so on. In the near future, a series of new VSC-HVDC 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 high-order 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 per-unit value system should be adopted both for AC system and DC system. In this paper the per-unit value system is introduced as follows [
The VSC-HVDC 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 two-terminal, multi-terminal, or multi-infeed DC system with VSC-HVDC, depending on the various different applications fields [
Schematic diagram of steady state physical model for multi-terminal VSC-HVDC.
Owning to having full controllable power electronic switch semiconductors such as insulated gate bipolar transistor and gate turn-off thyristor, VSC-HVDC 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 VSC-HVDC, 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 Newton-type 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 third-order convergence based on Newton-type 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 two-step 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 ADOL-C and ADC. The method of ADOL-C 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 ADOL-C method is used to realize the differential operation [
The steps of the improved AD algorithm based on third-order 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 high-order Newton methods, the modified IEEE 30-bus system with two-terminal and multi-infeed VSC-HVDC is analyzed in detail firstly. Then the simulation results of performance comparisons for the improved high-order Newton methods are presented among the modified IEEE 5-bus, IEEE 9-bus, IEEE 14-bus, IEEE 57-bus, and IEEE 118- bus text systems. At last, the AD based on third-order Newton method is evaluated for the modified IEEE 30-bus system with two-terminal of VSC-HVDC.
The proposed method has been applied to the modified IEEE 30-bus system [ In the system with two-terminal VSC-HVDC, the VSC1 and VSC2 are connected to AC line of bus 29 and bus 30, respectively. In the system with two-infeed VSC-HVDC, the VSC1, VSC2, VSC3, and VSC4 are connected to AC line of bus 12, bus 14, bus 29, and bus 30, respectively.
The modified IEEE-30 bus AC/DC system with VSC-HVDC.
The results of the power flow calculation of the AC system and DC system under different control modes for Newton, third-order and sixth-order 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 |
|
|
|
|
|
|
|
|
|
Sixth-order Newton |
|
|
|
|
|
|
|
|
|
| |||||||||
|
Newton |
|
|
|
|
|
|
|
|
Algorithm 1 |
|
|
|
|
|
|
|
|
|
Algorithm 2 |
|
|
|
|
|
|
|
|
|
Sixth-order Newton |
|
|
|
|
|
|
|
|
|
| |||||||||
|
Newton |
|
|
|
|
|
|
|
|
Algorithm 1 |
|
|
|
|
|
|
|
|
|
Algorithm 2 |
|
|
|
|
|
|
|
|
|
Sixth-order Newton |
|
|
|
|
|
|
|
|
|
| |||||||||
|
Newton |
|
|
|
|
|
|
|
|
Algorithm 1 |
|
|
|
|
|
|
|
|
|
Algorithm 2 |
|
|
|
|
|
|
|
|
|
Sixth-order Newton |
|
|
|
|
|
|
|
|
Results of the power flow calculation of DC system.
DC variable | Converter number | Control mode | ||||
---|---|---|---|---|---|---|
|
|
|
|
|||
|
Newton | VSC1 |
|
|
|
|
VSC2 |
|
|
|
|
||
Algorithm 1 | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
|
||
Algorithm 2 | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
|
||
Sixth-order Newton | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
|
||
| ||||||
|
Newton | VSC1 |
|
|
|
|
VSC2 |
|
|
|
| ||
Algorithm 1 | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
| ||
Algorithm 2 | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
| ||
Sixth-order Newton | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
| ||
| ||||||
|
Newton | VSC1 |
|
|
|
|
VSC2 |
|
|
|
|
||
Algorithm 1 | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
|
||
Algorithm 2 | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
|
||
Sixth-order Newton | VSC1 |
|
|
|
|
|
VSC2 |
|
|
|
|
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 | Sixth-order Newton | Newton | Algorithm 1 | Algorithm 2 | Sixth-order 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 VSC-HVDC is suitable for application in multi-infeed system [
Comparison of iteration times and computing time.
Control mode | Iteration times | Computing time (ms) | |||||||
---|---|---|---|---|---|---|---|---|---|
VSC1 + VSC2 | VSC3 + VSC4 | Newton | Algorithm 1 | Algorithm 2 | Sixth-order Newton | Newton | Algorithm 1 | Algorithm 2 | Sixth-order 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 118-bus systems are analyzed in this section [
The topology and parameter settings for different IEEE text systems.
Topology | Two-terminal | Two-infeed | Three-terminal | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Modified test system | IEEE-5 | IEEE-9 | IEEE-14 | IEEE-57 | IEEE-118 | IEEE-14 | IEEE-14 | IEEE-57 | ||||||||||||
Bus number | VSC1 | VSC2 | VSC1 | VSC2 | VSC1 | VSC2 | VSC1 | VSC2 | VSC1 | VSC2 | VSC1 | VSC2 | VSC3 | VSC4 | VSC1 | VSC2 | VSC3 | VSC1 | VSC2 | VSC3 |
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 | Sixth-order Newton | Newton | Algorithm 1 | Sixth-order Newton | ||
Two-terminal | IEEE-5 | 5 | 4 | 1.5 | 4.8287 | 4.4651 | 5.3692 |
Two-terminal | IEEE-9 | 4 | 3 | 1.5 | 3.1646 | 2.2094 | 2.7866 |
Two-terminal | IEEE-14 | 4 | 3 | 1.5 | 4.1375 | 2.6509 | 4.1890 |
Two-infeed | IEEE-14 | 4 | 3 | 1.5 | 4.3529 | 3.1544 | 3.9907 |
Three-terminal | IEEE-14 | 4 | 3 | 1.5 | 4.3546 | 2.9592 | 4.1260 |
Two-terminal | IEEE-57 | 4 | 4 | 2 | 22.0698 | 14.2854 | 23.6236 |
Three-terminal | IEEE-57 | 4 | 4 | 2 | 28.9858 | 24.7143 | 29.0658 |
Two-terminal | IEEE-118 | 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 third-order Newton method are presented. The IEEE-30 bus text system with two-terminal VSC-HVDC 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 | VSC1 |
|
VSC2 |
|
||
Algorithm 2 + AD | VSC1 |
|
|
VSC2 |
|
||
| |||
|
Algorithm 1 + AD | VSC1 |
|
VSC2 |
| ||
Algorithm 2 + AD | VSC1 |
|
|
VSC2 |
| ||
| |||
|
Algorithm 1 + AD | VSC1 |
|
VSC2 |
|
||
Algorithm 2 + AD | VSC1 |
|
|
VSC2 |
|
Comparison of iteration times and computing time
Control mode | Iteration times | Computing time (ms) | ||
---|---|---|---|---|
VSC1 + VSC2 | 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 third-order Newton method, as shown in Table Compared with the results of Table The result shows that AD technology is suitable for use in the third-order Newton method of VSC-HVDC 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 VSC-HVDC, the modified third-order Newton and sixth-order Newton methods have been presented to calculate the power flow of AC/DC systems with VSC-HVDC. The multivariate iteration matrix forms of the presented algorithms suitable for VSC-HVDC system are given. The proposed high-order Newton method has the third-order and sixth-order convergence, without solving the Hessian matrix. The task of the calculation is greatly reduced, and the efficiency is improved. Based on the third-order 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 two-terminal, multi-terminal, and multi-infeed VSC-HVDC have demonstrated the computational performance of the power flow algorithms with incorporation of VSC-HVDC 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).