AN INTEGRATED POWER FLOW SOLUTION OF FLEXIBLE AC TRANSMISSION SYSTEMS CONTAINING WIND ENERGY CONVERSION SYSTEMS

This paper presents a general steady-state modelling approach of power networks having wind farms and FACTS devices. A mathematical formulation is then proposed where the state variables associated with wind farms and FACTS devices are combined with the nodal voltage magnitudes and angles of the transmission network for a unified iterative solution of the power flow problem. Additionally, a suitable wind generator model based on a permanent magnet synchronous generator for power flow studies is introduced, which allows for a direct voltage magnitude’s control at the high-voltage side of the wind generator’s transformer. Lastly, two numerical examples are presented which show how the suggested approach performs.


Introduction
Even though the use of wind generators for converting wind energy into electricity is beneficial from the environmental standpoint, their consideration in the active power dispatch makes the already complex task of achieving power system controllability even more demanding. Consequently, the quantification of the effects that large-scale integration of wind generation will cause on the network is a very important matter that requires special attention when planning and operating an electrical power system. Arguably, power flow analysis is the most popular computational calculation performed in a power system's planning and operation, and this study has been selected to quantify the electrical response of wind generators when they are integrated in Flexible AC Transmission Systems.
Mathematical models of several types of wind generators have been developed in which their active and reactive power outputs are obtained based on the steady-state equivalent representation of the induction machine. The power injection method is then used to include these models into the power flow formulation, which is solved by using a sequential approach to obtain an operating point of the power system. In this approach, only the network's state variables are calculated through a conventional power flow algorithm, while a subproblem is formulated for updating the state variables of wind generators as well as their power injections at the end of each power flow's iteration.
Instead of using the power injection concept, another way of representing a wind generator is by means of an equivalent variable impedance expressed in terms of the slip of the generator and its rotor and stator winding parameters. This impedance is included in the system's admittance matrix, and the network nodal voltages are computed through the power flow analysis. Based on these voltages, the air-gap power of the wind generator is calculated and used to iteratively compute the value of the slip of the induction generator that produces the match between the air-gap power and the mechanical power extracted from the wind.
In general terms, all the methods discussed above share the characteristic of using a sequential approach to calculate the state variables of the wind generators, and none of them considers the integration of FACTS controllers in the network where the WECSs are embedded.
A fundamentally different approach for the modelling of WECS, within the context of the power flow problem, is a method that simultaneously combines the state variables associated with the wind generators, the FACTS controllers and the transmission network in a single frame-of-reference for a unified iterative solution through a Newton-Raphson (NR) technique. From the convergence standpoint, the unified method is superior to the sequential one because the interaction between the network, FACTS controllers, and wind generators is better represented during the iterative solution. Furthermore, it arrives at the solution with a quadratic convergence regardless of the network size. Hence, the key contribution of this work is to provide a comprehensive and general approach for the analysis of power flows in Flexible AC Transmission systems containing wind generators in a unified single-frame of reference.

Power flow including FACTS controllers and wind generators
In this paper, the unified approach suggested in [1] is extended to compute the power flow solution of a power system containing FACTS controllers and WECS and represented by a single set of nonlinear power flow mismatch equations f(X AC , X F , X WG )=0, where X AC is a vector of all nodal voltage magnitudes and angles, X F stands for the state variables of the FACTS controllers, and X WG is a vector of all state variables associated with the wind generators. The linearized power flow mismatch equations corresponding to the wind farms are then combined with those associated with the FACTS controllers and the rest of the network, as given by (1), which are solved iteratively by the NR method: where ΔR F and ΔR WG represent the mismatch equations of the FACTS controllers and wind generators, respectively. The NR method starts from an initial guess for all the state variables and updates the solution at each iteration i until a pre-defined tolerance is fulfilled. In this unified solution, all the state variables are adjusted simultaneously in order to compute the steady-state operating condition of the power system. Hence, this method has strong convergence characteristics.

Modelling of FACTS devices
Among all FACTS devices used to improve the steady-state performance of power systems [1], the Static Var Compensator (SVC) and Thyristor-Controlled Series Capacitor (TCSC) are the controllers considered in this paper.

Static VAR compensator
An integrated SVC and step-down transformer model is obtained by combining the admittances of both components Y T-SVC = Y T + Y SVC as proposed in [1]. The linearised power flow equations are given by (2) considering the firing angle α SVC of the SVC as the state variable within the NR method where G T-SVC and B T-SVC are functions dependent on α SVC :

Thyristor-controlled series compensator
The TCSC firing angle power flow model is represented as an equivalent series reactance which is associated with the firing angle α TCSC. This reactance can be expressed as [1] (see Appendix): Assuming the TCSC controls the active power flowing from bus k to bus m to a specified value of sp km P , the set of linearised power flow equations is where the power flow mismatch for the TCSC module is defined as

Modelling of wind generators
Wind generators are categorized according to how they operate when they are connected to the grid. The Fixed-Speed Wind Generators (FSWG) are called so because their speed is mainly set according to the system's frequency [2]. In this category are the Stall-Regulated Fixed-Speed Wind Generators (SR-FSWG) and the Pitch-Regulated Fixed-Speed Wind Generators (PR-FSWG). A variant of the previous models is the semi-variable speed wind generator (SSWG), which uses a wound-rotor induction generator with an external resistor added to the rotor circuit in order to achieve a power regulation when wind speeds are above the rated one [3]. Also, variable-speed wind generators are being employed worldwide with the doubly-fed induction generator being the most used. However, another emergent topology that is being widely accepted is the wind generator based on a Permanent Magnet Synchronous Generator (PMSG) with a full-scale converter in which the gearbox can be omitted [4].
In this paper, the mathematical modelling of FSWG, SSWG and PMSG-based wind generators for power flow studies is addressed. In reference [5], the models of fixed-and semivariable speed wind generators are suitably derived for power flow analysis and can be readily integrated in the formulation presented herein. For this reason, only a brief description of these models is given next.

Fixed-speed wind generators
This generator is directly connected to the network through a step-up transformer, and its final operating point depends upon the electrical frequency as well as the nodal voltage at the generator's terminals. The generated reactive and active powers are determined by Equations (5) and (6), respectively, and the stator and rotor currents of the induction generator can be expressed according to Equations (7) and (8) [5] : where s is the machine's slip, V is the terminal voltage, and the constants from A to W are as given in the Appendix. Furthermore, the power converted from mechanical to electrical form (P conv ) can be computed by using Equation (9), where R 2 represents the rotor resistance

Stall-regulated fixed-speed wind generator
The mechanical power P m [W] extracted from the wind by this generator is given by [6]  , ω T is the angular speed of the turbine [rad/s], and the constants c 1 to c 9 are the parameters of the wind turbine's design. Thus, assuming that the SR-FSWG is connected at bus k, the power mismatches equations are (11)-(13), and the set of linearised equations that has to be assembled and combined with the Jacobian matrix and the power mismatch vector of the entire network is shown in Equation (14) [5]: where P g (V,s) and Q g (V,s) are given by (6) and (5), respectively, P LK and Q LK represent the active and reactive powers drawn by the load at bus k, respectively, and P k cal and Q k cal are active and reactive power injections given by

Pitch-regulated fixed-speed wind generator
Since this wind generator has a blade pitch angle mechanism which actuates to limit the power extracted from the wind [7], the generated active power P g,pr can be obtained from its power curve and is considered constant at a value , sp g pr P through the iterative process; however, the reactive power Q g,pr =Q g needs to be calculated [5]. Therefore, the internal power equilibrium point in the wind generator has to be computed by Equation (17): where P losses,s and P losses,r are the three-phase stator and rotor power losses, respectively, and the core losses in the induction machine are neglected. As mentioned previously, the set of linearised power flow mismatch equations regarding the PR-FSWG is given by Equations (18)-(21) when the generator is connected at bus k:

Semi-variable speed wind generator
In this type of generator, the slip of the induction machine cannot be regarded as the state variable because of the external resistance R ext added in the rotor circuit [5]. Hence, a total rotor circuit resistance, R x = (R 2 +R ext ) /s, is considered as a single-state variable associated to the rotor circuit, which is adjusted to satisfy the power mismatch equations during the NR power flow calculation [5,8]. Hence, the reactive and active powers, the stator and rotor currents as well as the power converted from mechanical to electrical form will be dependent functions on R x and can be expressed as where the internal energy balance in the induction machine ΔP WT3 = -(P m,ss -P conv ) is derived by using Equations (26)-(27).

PMSG-based wind generator
This type of wind generator possesses a PMSG and a full-rated converter to connect the generator to the network, resulting in complete speed and reactive power control [9]. Hence, all the generated power is supplied to the power system through a machine-side converter and grid-side converter. The schematic diagram of this topology is shown in Fig. 1(a). Reactive power support is one of the characteristics that make this machine attractive for wind power production. In this case, the inclusion of the explicit representation of the wind generator step-up transformer is considered, which allows for direct voltage magnitude control at the high-voltage side of the transformer. The proposed model for power flow studies is shown in the Fig. 1(b) in which P g,pmsg represents the output power set by the wind generator power curve for a given wind speed, V msc and V gsc are the voltage at the machine-side converter and grid-side converter terminal, respectively, and Z st is the step-up transformer impedance.
For the active and reactive powers at bus k, the subscripts gsc and k are exchanged in Equations (32) and (33). Therefore, the NR-based power flow formulation is given by Note that Equation (36) represents the power constraint in the AC/DC/AC converter in which active power losses are neglected.

Machine-side converter
Grid-side converter

DC-link
Network connection Step-up transformer

Case studies with wind farms and FACTS devices
This section shows how the proposed approach performs when considering a power system having FACTS devices and wind generators.

5-bus test system with FSWGs, SSWGs and a SVC
The typical 5-bus test system is used to provide an example with the inclusion of a wind farm consisting of ten SR-FSWGs, ten PR-FSWGs and ten SSWGs operating at a wind speed of 16 m/s with which all wind generators are injecting their maximum power. Additionally, a SVC is placed at bus 5 in order maintain its terminal voltage magnitude at 1 pu. The conventional generators are set to control voltage magnitudes at 1 pu. Parameters of the wind farm are given in the Appendix.  Table I.
The simulated wind farm is a reactive power consumer since it lacks of a reactive power control as seen from Table 1. When no SVC is considered, its reactive power absorption exceeds 10 MVAr, resulting in a low voltage magnitude at node 6. On the other hand, when the SVC is set in operation, not only the low-voltage side of the wind farm transformer is boosted, but also the system voltage profile. Since all wind generators are operating at the same wind speed, clearly the state variables calculated for each group of wind generators corresponding to each model will result in the same value. Furthermore, if the system voltage profile changes, as occurs with the inclusion of the SVC, another operating point is found at each wind generator as shown in Table 2.

5-bus test system with PMSG-based wind generators and a TCSC
In this case, a PMSG-based wind farm is located at bus 4 with 30 wind generators operating at rated wind speed, i.e. 15 m/s. Be aware that each PMSG-based wind generator provides reactive power support by controlling its terminal voltage magnitude. Also, a TCSC is placed for controlling the active power flowing through the transmission line connected between nodes 4 and 5 at The results are reported in Table 3. The system voltage profile is improved when the PMSG-based wind farm is integrated to the grid as seen from Table 3. This is mainly due to two reasons; one is the fact that there is a redistribution of power flows in the network, e.g. the load connected at node 4 is being supplied locally by the wind farm. On the other hand the PMSG-based wind farm is providing voltage support, resulting in a voltage magnitude of 1 pu at the low-voltage side of the wind farm transformer even when the TCSC is set in operation to increase the power transfer to 20 MW in the line connecting nodes 4 and 5.   The data for each wind farm is (on a base power of 100 MVA): wind farm step-up transformer impedance is 0.2 pu and the impedance of each wind generator transformer is 4.1667 pu. Also, the data of each wind generator model are given in