Newton Raphson Method Example
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INTEGRATED AC/DC POWER SYSTEM USING MODIFIED NEWTON-RAPHSON LOAD FLOW ANALYSIS
The Newton-Raphson method or the other name called Newton Method, is a powerful technique for solving equations numerically. Like so much of the differential calculus, it is based on the simple idea of linear approximation. The Newton Method, properly used, usually homes in on a root with devastating efficiency. In this research, the Newton-Raphson method load flow analysis will be modified. It is modified to attain compatibility for the AC/DC systems with unified DC links in the ac network. The modified Jacobian equation includes the DC real and reactive power at the AC/DC buses and their dependency on the AC system variables. The AC/DC test system with a load flow computation in MATLAB will be evaluated by using modified Newton-Raphson method.
Electric power transmission was primitively developed with direct current. The availability of transformers and the development and of induction motors at the beginning of the 20th Century, led to greater appeal and use of AC transmission. The research and development on multi-electrode grid controlled mercury arc valve for high powers and voltages was carried out in 1929. There use in conversion processes for transmission and frequency changing was carried out in 1930’s. DC transmission now became practical when long distances were to be covered or where cables were required. After the Second World War, the research on HVDC got stimulated, particularly in Sweden and in Russia. In 1950, a 116 km experimental transmission line was commissioned from Moscow to Kasira at 200 kV .
HVDC transmission is now an integral part of the delivery of electricity in many countries throughout the world. In view of rapid growth in demand and supply of electricity, electric power system is becoming increasingly large and more complex. Moreover, regular electric supply is the sheer necessity for growing industry and other fields of life. The power industry planners are demanding stronger trend towards supplying electric power of higher quality by improving the system security and its impact on environment in parallel with pursuit of economy. In real life situation, the criterion of perfection is never met, because there are deviations between the model and reality. Load flow and state estimation analysis are important tools for deciding the stable operation and control of power system as well as future planning of power systems [2, 3].
The first commercial HVDC line built in 1954 was a 98 km submarine cable with ground return between the island of Gotland and the Swedish mainland . Thyristors were applied to DC. Transmission in the late 1960’s and solid state valves became a reality. In 1969, a contract for the Eel River DC. Link in Canada was awarded as the first application of sold state valves for HVDC transmission .
Why use DC transmission in the transmission system? This question often asked by the people. Some of people are response that the losses will are lower. Unfortunately, the answer given not correct. Therefore, the level of losses is planed into a transmission system and modulate by the selected conductor size. Either as overhead transmission lines or submarine cables can have lower losses but at higher expense since the larger cross-sectional area will generally result in lower losses but cost more for the DC and AC conductors. It is mostly by economic choice, when converters are used for DC transmission in preference to AC transmission driven by one of the following reasons:
- An overhead DC transmission line with its towers can be designed to be less costly per unit of length than an equivalent AC line designed to transmit the same level of electric power. However it is more costly at each end of the DC converter stations than the terminating stations of an AC line. Therefore, there is a breakeven distance above which the total cost of DC transmission is less than its AC transmission alternative. Lower visual profile can have by the DC transmission line than an equivalent AC line. Indeed, it lead to a lower environmental impact. There are other environmental advantages to a DC transmission line through the electric and magnetic fields being DC instead of ac.
- The breakeven distance is lesser than overhead transmission, if transmission is by submarine or underground cable. It is not practicable to consider AC cable systems exceeding 50 km but DC cable transmission systems are in service with length which is in hundreds of kilometers and even distances greater than or equal 600 km have been considered executable.
- Neighboring networks are not synchronized with some AC electric power systems even though their physical distances between them are quite small. This situation occurs in Japan. It is 60 Hz network for half the country and the 50 Hz system for the other side. It is physically impossible to connect the both of them together by direct AC methods in order to exchange electric power between them. Nevertheless, it is likely to transfer the needed power flow even though the AC systems, if a DC converter station is located in each system with an interconnecting DC link between them indeed connected remain asynchronous .
The main objectives of this research are:
- To study the Newton-Raphson method for the load flow analysis.
- To modified the Newton-Raphson method to attain compatibility for AC/DC systems with integrated DC links in the AC network.
SIGNIFICANT OF STUDY
Load flow study is a crucial tool that involve with the numerical analysis applied to a power system. A power flow study usually uses simplified notation such as a one-line diagram and per-unit system and focuses on various forms of AC power (i.e. voltages, voltage angles, real power and reactive power) . Normal steady-state operation is how the power system is analyze. Exist a few number of software implementations of power flow studies. Load flow analysis is probably the most important of all network calculations since it concerns the network performance in its normal operating conditions. It is performed to enquire the magnitude and phase angle of the voltage at each bus and the real and reactive power flows in the system components. The important of this study is to attain compatibility for AC/DC systems with integrated DC links in the AC network by using the modification of the Newton-Raphson power flow analysis method.
SCOPE OF WORK
The scope of this research is to analyze IEEE test system with the modified Newton-Raphson algorithm that will be develop and to solve the HVDC link power system load flow. This algorithm will be develop from the Newton-Raphson equation and some other equation from the DC link equations. The Jacobian equation for the AC power flow is altered to attain compatibility for systems with integrated DC link(s) in the AC network. Treated as voltage dependent PQ-buses are the converters. This grant the DC variables to be included in the power flow equations. Both the AC network and the DC link will be comprise by the new residual vector and Jacobian matrix .
The studies of integrated ac/dc power system, their advantages and analysis of load flow in integrated ac/dc power system and their implementation is much developed here the literature review on these aspects is briefly summarized as:
K. Ayan, U. Arifoglu, U. Kilic represent the Load Flow (LF) analysis of pure AC power systems is solved saperately. It is by numerical analysis methods and the heuristic methods. The load flow of integrated ac/dc power systems only has been implementing by numerical methods so far. A lot of methods to implement load flow analysis of integrated ac/dc power systems in literature. Examples of these methods are Newton-Raphson, Fast Decoupled and Broyden. By using Genetic Algorithm the consecutive load flow analysis of AC/DC system is implemented. The suggest method is examine on IEEE 9-bus test system. Heuristic method is apply for load flow analysis of the integrated AC/DC power systems for the first time .
Panosyan and Oswald presented the model of a two-terminal HVDC link and its integration into the Newton-Raphson method for the load flow analysis taking into consideration the control strategies of the HVDC converter stations. It is suggested that the simplest way of integrating a dc link into the ac load flow is by representing it by constant active and reactive power injections at the two terminal buses in the ac systems. Thus the two terminal ac/dc buses are delineate as a PQ-bus with a constant voltage independent active and reactive power .
Sanghavi and Banerjee suggested a sequential approach for performing the load flow analysis of an integrated ac/dc power system. In this approach a simplified model of the dc link has been developed and the means of integrating the link equation with a standard ac load flow program has been explained. in this model minimum amount of modifications have been made with a standard ac load flow program .
Mobarak presented the well-known Newton-Raphson method for the load flow analysis of system consisting of HVDC link. The original NR method is modified to attain compatibility for AC/DC systems with integrated DC links in the AC network. The altered Jacobian equation includes the DC real and reactive power at the AC/DC buses and the dependency on the AC system variables .
Marsafawy and Mathur proposed a new method for the load-flow calculations of, integrated AC/DC systems. They proposed use of fast decoupled load flow method, which handles all AC/DC equations simultaneously, and fully exploits the sparsity techniques. They have shown application of the technique on two systems. The systems are one with a point to-point DC transmission and the other with a 3-terminal mesh HVDC sub-system. It was a fast technique for load-flow studies of an integrated AC/DC system .
Definition of Terms/Concepts
MATLAB: MATLAB®is a high-level language and interactive environment for numerical computation, visualization, and programming. Using MATLAB, data can be analyze, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable us to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages.
Power flow equations formulated in polar form.
Expressing in polar form;
Separating the real and imaginary parts;
Expanding equation above in Taylor's series
The Jacobian matrix;
DC Link Equation:
Suppose the rectifier maintains constant current (CC), the inverter operates with constant extinction angle (CEA), maintaining adequate commutation margin under normal operation. That is:
If we assume the rectifier maintains constant current (CC), the inverter operates with constant voltage (CV), maintaining adequate commutation margin under normal operation. That is:
In a nutshell, the objective of this study will be achieved after all the research have been done. The Newton-Raphson method will be study and will be applied in the load flow analysis. Then, the Newton-Raphson method will be modified and applied in the AC/DC systems with the integrated DC links in the AC network to achieve it compatibility. The research will able to make the calculation for the AC/DC systems with integrated DC links in the AC network become easy. A new method to include dc systems in power flow calculations will be proposed. The method is more efficient than previous methods and more importantly, the method is easy to implement and developments of ac power flow solution techniques can be combined with the other method.
O. Osaloni and G. Radman, "Integrated AC/DC systems power flow solution using Newton-Raphson and Broyden approaches," in System Theory, 2005. SSST'05. Proceedings of the Thirty-Seventh Southeastern Symposium on, 2005, pp. 225-229.
K. Ayan, U. Arifoglu, and U. Kilic, "Integrated AC/DC systems Load Flow using Genetic Algorithm," in Power Engineering and Optimization Conference (PEOCO), 2011 5th International, 2011, pp. 404-409.
A. Panosyan and B. Oswald, "Modified Newton-Raphson load flow analysis for integrated AC/DC power systems," in Universities Power Engineering Conference, 2004. UPEC 2004. 39th International, 2004, pp. 1223-1227.
Y. A. Mobarak, "Notice of Violation of IEEE Publication Principles<BR>Modified load flow analysis for integrated AC/DC power systems," in Power System Conference, 2008. MEPCON 2008. 12th International Middle-East, 2008, pp. 402-405.
M. M. El-Marsafawy and R. M. Mathur, "A New, Fast Technique for Load-Flow Solution of Integrated Multi-Terminal DC/AC Systems," Power Apparatus and Systems, IEEE Transactions on, vol. PAS-99, pp. 246-255, 1980.
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