Abstract— Increased penetration of wind energy into the electrical power grid over the last two decades has led to renewed research emphasis on stability and the dynamic behavior of power systems. In this paper, PID controller synthesis for a DFIG model is considered. The design is based on analytical characterization of the roots of closed-loop characteristic polynomial via the Hermite-Biehler theorem. The PID design provides improved transient response compared to the traditional PI control of the current loop. Computer simulations of the PID controller show a well-damped response to changes in power demand with no oscillations.

I. INTRODUCTION

The increased penetration of wind energy into the power systems over the last two decades has led to concerns about its influence on the dynamic behavior of the power systems. The transients in power systems must be simulated and analyzed for getting insight into what impacts these new power generators have on the power systems [1]. Methods and tools for simulation (fast or real-time) of wind turbines in large power systems are therefore needed. Real-time simulation is also required for field and in-plant testing in order to evaluate performance of control and protection systems. In response to these concerns power system operators have developed grid codes for connecting equipment to the electrical grid [2, 3]. These regulations are extended to wind farms connected to the transmission grid, including both onshore and offshore farms. The attention in these requirements is drawn to both the wind turbine fault ride-through capability and the wind turbine grid support capability, i.e. their capability to assist the power system by supplying ancillary services. The ancillary services represent a number of services required by the power system operators, such as voltage and frequency control, in order to secure safe and reliable grid operation.

The effect of wind farm integration into the power grid depends on both the design and implementation of power system to which the wind farms are connected and the ability of the wind farms to fulfil the power grid requirements. This ability depends of course on the technology used for manufacturing wind turbine/wind farms. This fact has challenged wind turbine manufactures and spurred research activity concerning the ability of wind turbines to comply with high-power system requirements. There is presently much research activity world-wide involving model

K. Iqbal is with University of Arkansas at Little Rock, 2801 S.

University Ave, Little Rock, Arkansas 72204 (phone: 501-371-7617; fax: 501-569-8698; e-mail: kxiqbal@ualr.edu).

K.S. Altmayer is with University of Arkansas at Little Rock, 2801 S. University Ave, Little Rock, Arkansas 72204 (e-mail: ksaltmayer@ualr.edu).

Anindo Roy is with University of Maryland, College Park, Maryland (e-mail: ARoy@som.umaryland.edu).

simulation studies to understand the impact of system disturbances on wind turbines and consequently on the power system itself [4]-[8].

Wound rotor induction machines of the type Doubly Fed Induction Generators (DFIG) are commonly used with wind turbines in wind power generation. The DFIG connected with back to back converter at the rotor terminals provides a very economic solution for variable speed applications [9]. In the DFIG, three-phase alternating supply is fed directly to the stator. The rotor is connected to the supply via ac-dc-ac converter-inverter that uses power electronics devices. The network side converter is controlled using Field Oriented Control (FOC) that involves transformation of the currents into a synchronously rotating d-q reference frame aligned with the stator flux. The Direct Torque Control (DTC) is used for the rotor side converter in order to achieve good dynamic performance. The rotor-side converter is used to control the wind turbine output power and the voltage (or reactive power) measured at the grid terminals. The grid-side converter is used to regulate the voltage of the DC bus capacitor, also used to generate or absorb reactive power.

In this paper we address the problem of PID stabilization of a DFIG model. We present a framework for PID control of doubly-fed induction generator connected to wind turbine. The PID control results in improved dynamic stability in the face of voltage transients on the electrical grid.

II. MATHEMATICAL MODEL OF THE DFIG

This section starts with a description of the functional aspects of DFIG. Then, a mathematical model of the DFIG is developed that is used in later simulations.

A. DFIG Operation

The wind turbine connected to the DFIG is assumed to operate in the variable speed mode, whereby the rotor speed is controlled to operate within a variable speed range ( ) centered on the generator synchronous speed (Fig. 1). The variable speed mode requires a controller to vary the pitch of the wind turbine based on the wind speed. The pitch controller aims for maximum power efficiency by optimizing the tip-speed ratio defined as: . In the variable speed configuration, stator windings of the induction generator are connected to the grid via a step-up transformer.

The rotor windings are connected via slip rings to back-to-back voltage source converters, and onward to the transformer and the grid (Figure 1). The converter makes it possible to supply or obtain power from the grid through the rotor terminals. This prevents the generator from switching to motor operation while driving at sub synchronous speed. The rotor operates in the super-synchronous operation mode (negative slip) at higher wind speeds whereby the power flows to the grid from both the stator and the rotor. At lower

PID Controller Synthesis for Improved DFIG Transient Response Kamran Iqbal, Senior Member, IEEE, Kumud S. Altmayer, and Anindo Roy, Senior Member, IEEE

wind speed, the sub-synchronous mode is activated and the rotor absorbs active power from the grid for rotor winding excitation. The power output from the DFIG to the grid is the sum of both stator power and the rotor power ( ) as realized by the stator and the grid side converter power ( ). Neglecting losses, the total output power equals the mechanical power ( ) extracted from wind by the wind turbine.

Fig.1: The DFIG with wind turbine (from [4])

B. DFIG Model

The model of the DFIG is based on the governing electrical equations for the stator and rotor windings that can be found in any power systems text book. The governing equations are described in the synchronous (d-q) coordinates with the d-axis aligned with the stator field (air gap flux) and q-axis aligned with the stator voltage (Figure 2, [10]). Orienting machine equations to stator field has two advantages [7]: 1) the alternating current, voltage, and flux become stationary and can be treated as dc variables; and, 2) it is possible to decouple the control of active and reactive power.

Fig. 2: Rotor and stator flux orientation (from [10])

The basic stator and rotor equations are described as follows, where the variables are represented by complex vectors and include direct (d) and quadrature (q) components: 岫 岻

where are stator and rotor flux linkages, are stator and rotor voltages, are the synchronous frequency and the rotor electrical frequency, are stator and rotor resistances, and are the stator, rotor, and magnetizing inductances. The transformer ratios are assumed to be 1:1. The above equations include the stator and rotor voltages as inputs.

Let be a transformation variable for the rotor circuit, and define: . Further,

let then, a -equivalent model is obtained as described by the following equations [6]:

where .

For an induction machine model, the choice of state

variables includes [7]: [ ] [ ] , where represent the magnetizing current and flux, respectively. As there are four governing electrical equations, two variables can serve as state variables and can be used to define four state equations in both d- and q-coordinates. A common choice of the state variables for control of the DFIG is the stator flux and the rotor current: . Then, from the -equivalent model, we obtain the governing equations in terms of these variables as [6]: ( ) 岫 岻 ( )

where 岫 岻 The last term on the LHS of the second equation is defined as the back EMF of the machine ( 岻. The state equations for the machine model are obtained by separating the above equations into their respective d and q components and are given in the appendix. A simplification of the model is obtained by assuming that the air gap flux remains constant as the stator voltages are constant in amplitude, frequency, and phase. Further, the d-axis of the rotating frame is aligned with so that:

Then,

C. Power Calculations

The mechanical power extracted by the turbine from the wind is given as: , where is the mechanical power, 岫 岻 is the available wind power (a function of wind speed and blade diameter) and is the turbine efficiency. Assuming gear ratio 1:1, The mechanical power and the stator electric power output are defined by:

where are the mechanical and electromechanical torques and are the turbine and the synchronous speeds. The turbine speed is related to the rotor frequency by where denotes the electrical poles on the rotor.

The ideal stationary power distribution through the generator depends on the slip of the generator defined as: 岫 岻 . In the steady state , where the stator power is given as: 岫 岻 and the rotor power is: 岫 岻 . Generally, is only a fraction of ( ).

The expressions for the active and reactive powers are given below: { } ( ) 岫 岻 { } ( ) 岫 岻 岫 岻

Power calculations can be simplified under reasonable

assumptions 岾 峇 so that

the active power is controlled by the q-component of the stator current and the reactive power is controlled by the d-component of the stator current. Thus, 岫 岻

Finally, the rotor speed dynamics are described as: 岫 岻

where are the combined rotor-turbine inertia and damping, is the number of machine poles, and are the electrodynamic (motor) and load (wind turbine) torques, respectively. The former is given as: { }

D. Control of the DFIG

Traditionally, proportional integral (PI) control of electrical machines and DFIG is employed [11]-[13]. The integral term is needed because of the approximations used in obtaining a simplified model of the machine. The basic control loop involves the control the rotor current based on the machine side converter (MSC) voltage. The rotor side voltage and current are related as: ( )

The last term in the above equation may be treated as feedforward compensation, so that the following first-order transfer function from to is obtained: 岫 岻

This transfer function is used as the basis for internal model control (IMC) of rotor current as suggested by [6] and other researchers. The current reference for the control of RSC is based on the power demand and is obtained as follows:

( )

Two additional control loops are used to regulate the active and reactive power: based on and where is obtained from the speed-mechanical power curve of the turbine. The is normally zero, but can be imposed in the case of voltage sag following a grid fault.

E. DFIG Transient Response

The fourth order state-space model of the induction machine presented in the appendix can be used to analyze the transient response of the DFIG. The synchronous model (at zero slip) includes a pair of eigenvalues with low damping ratio ( ). This lightly damped mode results in current fluctuations in the event of power exchanges. Use of active damping to overcome the oscillations is proposed [6]. As an alternative, in the following we propose a PID controller for the model. The analytical design of the proposed PID controller is based on the Hermite-Biehler framework.

III. HERMITE-BIEHLER FRAMEWORK FOR PID CONTROLLER

SYNTHESIS

The Hermite-Biehler (HB) theorem [14] and its generalizations [15]-[16] have been used in mathematical analysis to study stability of polynomials. The HB theorem divides the polynomial into even and odd parts, and provides stability characterization in terms of interlacing property of the real, non-negative, and distinct zeros of the even and odd parts of a polynomial. The generalized Hermite-Biehler theorem additionally provides information on the signature of the polynomial, defined in terms of the difference between the number of LHP and RHP zeros of the polynomial. These theorems are given below:

Definition: Let 岫 岻 , be a real polynomial of degree . Let 岫 岻 岫 岻 岫 岻 where 岫 岻 岫 岻 are the components of 岫 岻 made up of even and odd powers of s. Let 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 Let denote the nonnegative real zeros of 岫 岻 and let denote the nonnegative real zeros of 岫 岻, both arranged in ascending order of magnitude. Then,

Theorem 1 (The Hermite-Biehler theorem) [15]: 岫 岻 is Hurwitz stable if and only if all the zeros of 岫 岻 岫 岻 are real and distinct, and are of the same sign, and the non-negative real zeros satisfy the following interlacing property:

Theorem 2 (The Generalized Hermite-Biehler Theorem) [15]: Let 岫 岻 ∑ with a root at the origin of multiplicity . Let 岫 岻 岫 岻 岫 岻. Let be the zeros of 岫 岻 that are real, distinct and nonnegative and define 岫 岻岫 岻 [ 岫 岻] . Then, [ 岫 岻] 岫 岻 [ [ 岫 岻岫 岻] ∑ 岫 岻 [ 岫 岻] 岫 岻 [ 岫 岻]] [ 岫 岻] [ 岫 岻] 岫 岻 [ [ 岫 岻岫 岻] ∑ 岫 岻 [ 岫 岻] ] [ 岫 岻]

where [ ] 岫 岻 岫 岻 is the signature of the polynomial.

The generalized Hermite-Biehler theorem is used to synthesize PID controllers for stable and unstable plants [17]-[18]: the problem involves PID controller design for a unity gain feedback system, where the plant and controller transfer functions are given as: 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻

The closed-loop characteristic polynomial of the unity-gain feedback system is formed as: ( ) 岫 岻 岫 岻 岾 岫 岻 岫 岻峇

Let define a set of PID gains; then, the design problem can be stated as follows: given the process model, find PID controller parameters such that 岫 岻 is Hurwitz stable, i.e., has its roots in the open lhp. To proceed further, let 岫 岻 岫 岻 and define: 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻

Then [ 岫 岻 岫 岻] [ 岫 岻] [ 岫 岻], i.e., 岫 岻 is Hurwitz if and only if 岫 岻 and 岫 岻 岫 岻 have the same number of rhp roots [18]. Further, 岫 岻 岫 岻 entertains a separation of controller parameters, where the even part of 岫 岻 岫 岻 contains , the odd part contains . This separation can be exploited for PID controller design as follows, where denote the degrees of 岫 岻 and 岫 岻, respectively.

Let 岫 岻 岫 岻 岫 岻 ( ) where 岫 岻 岫 岻 岫 岻岫 岻 and ( ) 岫 岻 岫 岻. Then, a necessary condition for stability is that ( ) has at least the number of real, nonnegative distinct roots of odd multiplicity given as:

{ 岫 [ 岫 岻] 岻 even 岫 [ 岫 岻] 岻 odd

The ranges of satisfying this condition are called allowable and denoted as ( ). Next, for some ( ), let where are the real nonnegative distinct roots of ( ) of odd multiplicity, and define [ 岫 岻] and [ ( )] Let 岫 岻 [ 岫 岻], then the stability condition reduces to [19]: [ 岫 岻] {

( ∑ 岫 岻 岫 岻 ) even ( ∑ 岫 岻 ) odd

The string of integers that satisfy the above conditions are called admissible and denoted at . For each ( ) the set of stabilizing PID controllers: 岫 岻 { [ ] [ ]} is found from a set of inequalities developed as [Ho97]: [ ( )]

These inequalities are combined into matrix form as [20]: [ ] [ ][ ] where [ ] [ 岫 岻 岫 岻 岫 岻]

[ ] [ 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻]

[ ] [ ]

A. PID Controller with Guaranteed Stability Margins:

The Hermite-Biehler framework allows the design of a PID controller to achieve pre-specified gain and phase margins for a given plant 岫 岻 岫 岻 岫 岻 [17]. Let and denote the desired gain and phase margins respectively. Then the PID controller gains that allow the closed-loop system to achieve those margins must satisfy the following conditions:

1. 岫 岻 ( ) 岫 岻 is Hurwitz stable for all [ ]

2. 岫 岻 ( ) 岫 岻 is Hurwitz stable for all [ ]

IV. PID CONTROLLER SYNTHESIS FOR THE DFIG MODEL

The Hermite-Biehler framework presented above is applied to the DFIG model to synthesize a PID controller. By taking the Laplace transform of the DFIG equations we get ( ):

[ ] [ ] [ ] [ ] [ ] The above equation is solved for the rotor currents: [ ] 岫 岻 [ ] ([ ] [ ] [ ])

where 岫 岻 岫 岻 岫 岻 . Following [13], an approximation to the terms involving the stator currents is assumed: [ ] 岫 岻 [ ] [ ] [ ] [ ] For PID controller design, the following transfer function from to is assumed: 岫 岻 岫 岻

The design involves the following closed-loop (CL) characteristic polynomial: ( ) 岫 岻 岫 岻 岫 岻 岫 岻 岫 岻

In order to achieve controller gain separation, we form the product 岫 岻 岫 岻 and substitute to get: 岫 岻 岫 岻 ( )岫 岻 岫 岻岫 岻 岫 岻

The even and odd parts of this polynomial are given as: 岫 岻 岫 岻岫 岻 岫 岻 岫 岻 岫 岻 岫 岻 ( )岫 岻 岫 岻 岫 岻 Next, we would like to examine the roots of 岫 岻 to find the allowable ranges of stabilizing gain . To do that, we write 岫 岻 in the rootlocus form: 岫 岻 岫 岻 (岫 岻 )

and plot the rootlocus for both positive and negative values of . By examining the rootlocus properties, 岫 岻 has a single nonnegative distinct root at for the following range of : {岾 峇 岫 岻} and it has two

nonnegative distinct roots for the range of given as: 岾 峇. Since (even), and ( 岫 岻) , hence, for a given fixed , a necessary condition for the existence of and is that 岫 岻 has at least one real distinct nonnegative root.

Following the procedure for HB synthesis outlined above [18][19], we compute the stabilizing controller as follows: for the first interval for , an admissible string of integers is given as: , which results in the following linear inequalities:

For the second interval for , an admissible string of integers is given as: , which results in the following linear inequalities: 岫 岻

To illustrate the design aspects, we take two values of stabilizing gains. First, let [ ] [ ] Then, for the DFIG model parameters (Appendix), the resulting characteristic polynomial has roots at: , with a very low phase margin. Next, let [ ] [ ] Then, the resulting characteristic polynomial has roots at: , with a nice 50 phase margin.

The above PID controller design was based on the second order model of the DFIG. In order to test the controller, we simulated the fourth order machine model given in the appendix with the PID controller. The DFIG was simulated at rated power ( ). The PID controller gains used for the simulation were [ ] [ ] The response for the rotor currents is shown in Figure 2. As seen from the figure, the PID controller damps out the oscillations seen when using only the PI controller.

(a)

(b)

Fig. 2: The step response of 4th order model to rated power ( ): 岻 岻 .

V. CONCLUSION

Traditional PI controllers for DFIG result in lightly damped modes that produce oscillations during power transients. We presented a scheme for PID control of DFIG based on Hermite-Biehler framework from systems theory. The design was shown to provide good dynamic stability.

APPENDIX

The state equations of the induction machine are given as: [ ] 岫 岻 ( ) 岫 岻

The parameters for the DFIG model are taken from the

DFIG simulation model in Matlab SimPowerSystems toolbox [Mat], and are given in p.u. quantities:

rs=.00706;

rr=.005;

ls=.171;

lr=.156;

lm=2.9;

ws=1;

ACKNOWLEDGMENT

K. Iqbal would like to thank University of Arkansas at Little Rock for their support.

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