Vector Analysis and Optimal Control for the Voltage Regulation of a Weak Power System with Wind Energy and Power Electronics

This paper deals with the voltage regulation in a weak system which contains large inductive loads and wind turbines using Doubly Fed Induction Generators (FDIGs). The DFIGs demand large amounts of reactive power from the grid and as a result, there is a voltage drop in the system which may be extra deteriorated if large inductive loads and motors are also present in the same line. The problem of the voltage regulation in these cases is treated with the installation of a Static Var Compensator (SVC) besides the capability of the DFIGs to partially regulate the voltage themselves. In this paper, new modeling procedures based on optimal control are developed for the design of the SVC controller and a novel strategy for the grid side converter of the DFIG is presented. The nonlinear system is simulated in the SIMULINK software so that the performance of the new controllers is validated.


Introduction
The increased need for wind energy development makes the installation of wind turbines in 'weak' ac grids necessary.On the other hand, many voltage instability incidents have taken place around the world the last years (Custem & Vournas, 1998;Berizzi, 2004).The distributed generation with wind power stations installed in weak distribution systems may enlarge this problem especially when large inductive loads are connected to the same line.So, voltage regulation has become a major research area in the field of power systems (Chondrogiannis, 2007;Ledesma, 2002;Kesraoui, 2016).This paper deals with the design of the necessary control loops so that good performance of the grid voltage can be attained in a very weak system which contains a wind park (WP) and large inductive loads.The WP consist of wind turbines with Doubly Fed Induction Generators (DFIGs).The DFIGs demand reactive power from the grid.These amounts of reactive power make the grid voltage very sensitive to load variations.The voltage performance can be improved by means of FACTS devices and better voltage controllers inside the DFIG.
The system under study is shown in Figure 1.A medium voltage line is connected to the main grid at bus 1 with short circuit capability of 150 MVA.There is a steam power generation system (SPGS) at bus 2 with rated power of 50 MVA and a wind park at bus 3 connected to this line.This system can be a part of a local grid in an island to which wind parks are to be installed.
The WP includes 11 wind turbines each with rated real power of 1.5 MW.At bus 4 there are inductive loads with rated power of 2 MVA and power factor 0.9 lagging.These loads also include three asynchronous motors each rated 300 kW.The nominal line voltage of the system is 25 kV.The variation of the reactive power demanded from the WP causes the load voltage at all buses to deviate from the rated values despite the presence of the SPGS in the system.At t = 50 sec there is an increase in the wind speed from 8 m/s to 14 m/s and at t = 75 sec the large induction motors start to operate. Figure 2 shows the rms value of the load voltage at bus 4. The real power produced from the WP is shown in Figure 3 and the reactive power from the WP is shown in Figure 4.The real power production of the SPGS is kept constant at 15 MW.

Mathem
The mathe where: ovement of the ator (SVC) at t purpose of this ontroller and t stem to stabiliz nes to be conn ystems if good ption and mod matical model ematical model e the d,q comp he d,q compone he d,q compon e the d,q comp X md ) is the d ax -X 2 md /X fd ) is X' q ( = X q -X 2 mq /X 1q ) is the q axis transient reactance, ω is the rotor electrical angular speed,  is the electrical speed of the magnetic flux, δ is the power angle, H is the inertia constant, T m is the mechanical torque, T D is the damping torque (being neglected from now on).
Finally, T' do , T' qo are time constants on field and damper winding respectively (we consider the machine to have one damping winding on q axis) and R s is the stator resistance.
By neglecting the equations regarding the stator magnetic fluxes (equations (a) and (b) above) and replacing the relevant values from the Appendix we have: It also is: As we have already seen, the SPGS is connected to the bus 2 and there is a small line up to the main bus 1.The Figure 7 depicts the vectors of the voltages at the buses.The voltage v 2 at the bus 2 will be a little ahead of the voltage v 1 at the bus 1 (approximately 3 degrees).We consider the main axes D, Q and the axes d, q internally in the synchronous generator to which the various quantities of the SG have been analyzed in the equations ( 1b)-(4b).We arbitrarily consider that the voltage v 2 is lying on the D axis.
The stator current i of the SG with its components i d , i q onto the axes d, q is also the current I of the line between the buses 2 and 1 with the components I From the line analysis on the axis Q we have: By taking into account that v 2Q = 0 and by replacing the values we have: The equations ( 1b)-( 4b) can be rewritten if we replace the quantities i d , i q by I D , I Q using the equation (5a).Then we can reach in: (4) The nonlinear system that consists of the voltage v 1 at the bus 1 and the internal quantities of the SG is actually described by the equations ( 1)-(6).By linearizing the above equations around the operating point given at the Appendix we finally conclude to the linear system given by the following equations: The equations ( 7)-( 12) form the linearized model of the SG and the line between the buses 1 and 2.

Model of the grid -side converter of the WT
A schematic configuration of the grid side converter, is shown in Figure 8 in which the grid phase voltages are denoted as e a , e b , e c and the converter phase voltages as v a , v b , v c are respectively.The d,q components (i d , i q ) of the line currents i a , i b ,i c concerning the d,q components (v d , v q ) of the converter voltages v a , v b , v c can be given by the following equations (Vittal & Ayyanar, 2013): The model choice of t replacing t As there i frequency the parame system and  a signal that wi rrent i q of the s  =  Δ = 0.866 ations ( 7)-( 12) equation ( 14

SVC C In order fo
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