Active and Reactive Power Control for DFIG Using PI, Fuzzy Logic and Self-Tuning PI Fuzzy Controllers

Wind turbine captures the kinetic energy of the wind and converts it into mechanical energy p_t then electricity p_e. For the wind turbine based on DFIG the electric power p_s delivery directly from DFIG stator to the electric grid. The active power transmitted to the grid is the algebraic sum of stator power P_s and rotor power P_r (Figure 1). In wind turbine the stator active and reactive power control of the DFIG is based on the theory of the induction machine vector control. The main duty of RSC is to control DFIG-produced active and reactive powers [27].

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Figure 1. Wind turbine control system with DFIG

2.1 Dynamic model of the DFIG

Dynamic model of DFIG can be described by the differential equations system for stator and rotor windings and by the equation of rotor motion equations (1, 2 and 3):

$\begin{align} & {{v}_{sd}}={{R}_{s}}.{{i}_{sd}}+\frac{d{{\varphi }_{sd}}}{dt}-{{\omega }_{s}}.{{\varphi }_{sq}} \\ & {{v}_{sq}}={{R}_{s}}.{{i}_{sq}}+\frac{d{{\varphi }_{sq}}}{dt}+\omega {}_{s}.{{\varphi }_{sd}} \\ & {{v}_{rd}}={{R}_{r}}.{{i}_{rd}}+\frac{d{{\varphi }_{rd}}}{dt}-{{\omega }_{r}}.{{\varphi }_{rq}} \\ & {{v}_{rq}}={{R}_{r}}.{{i}_{rq}}+\frac{d{{\varphi }_{rq}}}{dt}+{{\omega }_{r}}.{{\varphi }_{rd}} \\ \end{align}$                        (1)

$\begin{align}  & {{\varphi }_{sd}}={{L}_{s}}.{{i}_{sd}}+M.{{i}_{rd}} \\ & {{\varphi }_{sq}}={{L}_{s}}.{{i}_{sq}}+M.{{i}_{rq}} \\  & {{\varphi }_{rd}}={{L}_{r}}.{{i}_{rd}}+M.{{i}_{sd}} \\ & {{\varphi }_{rq}}={{L}_{r}}.{{i}_{rq}}+M.{{i}_{sq}} \\ \end{align}$                (2)

where, R_s, R_r, L_s and L_r are the resistances and leakage inductances of the stator and rotor windings;  M is the mutual  inductance;  v_sd,  v_sq,  v_rd,  v_rq ,  i_sd,  i_sq,  i_rd,  i_rq,  Φ_sd,  Φ_sq ,  Φ_rd and  Φ_rq are the d and q components of the space  vectors of stator and rotor voltages, currents, and fluxes; ω_s and ω_r are the angular frequencies of stator and rotor currents.

where: ωs = ωr + pΩ

Ω  is the rotational speed of the rotor and p is the pair pole number.

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Figure 2. Orientation of the (d, q) frame

The system dynamics, neglecting the friction loss, is given by:

$J\frac{d\Omega }{dt}={{T}_{g}}-{{T}_{em}}$       (3)

where, J is the turbine and the generator moment of inertia, Ω is the mechanical speed T_g is the gearbox  torque and T_em is the electromagnetic torque.

2.2 Control strategy of DFIG

Vector control of active and reactive power is decoupled power control in stator of DFIG. In order to simplify the control, for grid-connected DFIG system, the impact of stator resistance can be ignored [28-29]. The stator voltage vector is consequently in quadrature advance in comparison with the stator flux vector:

$\left(v_{s d}=0 \text { and } v_{s q}=v_{s}=\omega_{s} \varphi_{s d}\right)$

$\left(\emptyset_{s d}=\emptyset_{s} \text { and } \emptyset_{s q}=0\right)$

Hence, the active and reactive powers (Ps, Qs) between the stator and the grid can be written according to the rotor currents as:

$\begin{align} & {{P}_{s}}={{v}_{sq}}{{i}_{sq}}=-v\frac{M}{{{L}_{s}}}{{i}_{rq}} \\ & {{Q}_{s}}={{v}_{sq}}{{i}_{sd}}=\frac{v_{s}^{2}}{{{L}_{s}}{{\omega }_{s}}}-\frac{{{v}_{s}}M}{{{L}_{s}}}{{i}_{rd}} \\ \end{align}$          (4)

Rotor voltages can be expressed by:

$\begin{align} & {{v}_{dr}}={{R}_{r}}{{i}_{dr}}-g{{\omega }_{s}}({{L}_{r}}-\frac{{{M}^{2}}}{{{L}_{s}}}){{i}_{qr}}+({{L}_{r}}-\frac{{{M}^{2}}}{{{L}_{s}}})\frac{d{{i}_{dr}}}{dt} \\ & {{v}_{qr}}={{R}_{r}}{{i}_{qr}}-g{{\omega }_{s}}({{L}_{r}}-\frac{{{M}^{2}}}{{{L}_{s}}}){{i}_{dr}}+g\frac{{{v}_{s}}M}{{{L}_{s}}}+({{L}_{r}}-\frac{{{M}^{2}}}{{{L}_{s}}})\frac{d{{i}_{qr}}}{dt} \\ \end{align}$                     (5)

where, g is ω_r/ ω_s

In the decoupled system, P_s, Q_s, can be controlled by u_dr, u_qr respectively with R_p and R_q blocks, shown in Figure 3.

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Figure 3. Active and reactive power control system of DFIG

In this work, our objective is to evaluate system performance for the following types of controller: the conventional PI controller, fuzzy logic and self-tuning PI fuzzy controller.

2.3 Fuzzy logic controllers (FLC)

PID controller is a standard control structure for classical control theory. But the performance is greatly distorted and the efficiency is reduced due to nonlinearity in the process plant [30]. The fuzzy control is an intelligence control system to control complex system based on analogizing human ways of fuzzy thinking, which applies fuzzy sets, fuzzy linguistic variables and fuzzy logic inference knowledge. Fuzzy control with the advantage of good performances, robustness and so on need not require know mathematical model of object beforehand, On the other hand, fuzzy control provides a formal methodology for representing, manipulating, and implementing a human’s heuristic knowledge about how to control a system [31]. In the FLC, The reference value r(t) is compared with the actual value y(t) to obtain the error e(t) as shown in Figure 4. Also this error is compared with the previous error e(t-1) to get the change in error $\Delta e(t)$.

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Figure 4. Block diagram of FLC controller

There are two inputs for the fuzzy controller. The first input is the error E(t), the second input is the difference between successive errors ΔE(t) and is given by equation (6):

$\begin{align}  & E(t)={{G}_{e}}\,[r(t)-y(t)] \\ & \Delta E(t)={{G}_{\Delta e}}\,[e(t)-e(t-1)] \\ \end{align}$                  (6)

The output is the incremental change of the control signal U(t).U(t) is multiplied by the scale factor G_u to get the output signal u(t). In the system presented in this study, Mamdani type of FLC is used to produce the desired rotor reference voltages. Two independent FLCs are used to control active and reactive powers (Ps,Qs) in the stator of the DFIG. The fuzzy controller is composed of a fuzzifier, a rule inference, a rule base and a defuzzifier. The fuzzifier translates the error and change of error inputs from the real domain into a fuzzy domain. The rule base explains the knowledge to control the system. defuzzification is a mapping from a space of FLC actions defined over an output universe of discourse into a space of non fuzzy control actions. The 5 membership functions of error, variation of error and FLC output are shown in Figure 5.

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Figure 5. Membership functions for error, change of error and FLC output

The fuzzy sets have been defined as: NL, negative large; NS, negative small; ZE, zero; PS, positive small; and PL, positive large respectively.

The typical behavior of a controlled system is shown in Figure 5. The response process can be divided into several regions (r1, r2.....r10) by the peak value times. The typical behavior of a controlled system is shown in Figure 6.

For optimum control performance, control rules, membership functions and scaling factors are needed to be tuned properly. This is the fundamental problem in FLC design [32-33]. To overcome the difficulties in optimal control tuning, a standard phase plane technique has been used for designing robust rule base for PI-like fuzzy type control [34- 35].

The signs of ΔU in the regions described in Figure 6 are listed in Table 1, which can be summarized as follows [36]: IF E is zero then ΔU takes the sign of ΔE, else ΔU takes the sign of E.

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Figure 6. Operating regions of the time responses of error and change of error

Table 1. The signs of output (ΔU) in the FLC

Table 2. Fuzzy rules for 5 membership function

2.4 Self-tuning PI fuzzy controllers

With its advantages of simple structure, good stability, good robustness, reliability, convenient adjustment, PID control has become one of the main industrial control technologies. However, it has some limitations: when the control object is disturbed, the parameters of the controller cannot change automatically to adjust to the change of external environment [37]. The PI controller combined with fuzzy-logic is considered to be an effective solving technique to control the uncertain and nonlinear process.

In order to adjust the parameters of a PI controller, the conventional PI controller is combined with a FLC. Self-tuning PI fuzzy control provides a fuzzy logic supervised PI control scheme in which parameters of a PI controller are updated online as a function of the error and a variation of error, improving the behavior of a classical fixed PI controller. It combines the advantages of a FLC and a conventional PI controller, a self-tuning PI fuzzy control system structure is given in this paper, shown as Figure 7.

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Figure 7. Structure of a self-tuning PI fuzzy control

The objective of the FLC is to predict the value of k_p and k_i at any point of time from the systems response.

The membership function for E, ΔE, Δkp and Δk_i are expressed by triangle curve, as shown in Fig. 8. They are defined in the interval [-1, 1].

The fuzzy sets have been defined as: N, negative; ZE, zero; P, positive. The output of the FLC is the incremental change in the k_p and k_i parameters and the control signal is obtained by the following roles.

The role of proportional coefficient k_p is to accelerate the response speed of system, improve regulation accuracy of system. k_p bigger system response fester, the regulation accuracy higher, but easy to overshoot; k_p value is too small, will reduce accuracy and slow response speed. The role of integral coefficient k_i is to eliminate steady-state error. k_i large, the steady state error of the system eliminate faster, but k_i is too large, in the early stages of response process will produce integral saturation phenomenon, which led to a large overshoot in response process; if k_i is too small, will make the stable error of system difficult to eliminate[38].

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Figure 8. Membership function scheme of inputs and outputs of FLC

We can calculate output control u using self tuning PI parameters k_p, k_i, on line, according to discrete differential formula of PI control algorithm. The control signal is obtained by equations (7 and 8):

$\begin{align} & u(t)={{k}_{p}}\,E(t)+{{k}_{i\,}}\int_{0}^{t}{E(\tau )d\tau \,\,} \\ & \,\,\,\,\,\,\,\,\,=[{{k}_{p0}}E(t)\,+{{k}_{i0\,}}\int_{0}^{t}{E(\tau )d\tau ]} \\ & \,\,\,\,\,\,\,\,\,\,\,\,+[{{G}_{p}}\,\Delta {{k}_{p}}(E,\Delta E)\,E(t)+{{G}_{i}}\,\Delta {{k}_{i\,}}(E,\Delta E)\int_{0}^{t}{E(\tau )d\tau ]} \\ & \,\,\,\,\,\,\,\,\,=u(t)\,+\Delta u(t)(E,\Delta E) \\ \end{align}$              (7)

The PI parameters after adjusted are:

$\begin{align} & {{k}_{p}}={{k}_{p0}}\,+{{G}_{p\,}}\Delta {{k}_{p}}(E,\Delta E) \\ & {{k}_{i}}={{k}_{i0}}\,+{{G}_{i\,}}\Delta {{k}_{i}}(E,\Delta E) \\ \end{align}$            (8)

where, Δk_i and Δk_i are calibration proportional and integral PI controller gains, whose elements are fuzzy values, these parameters can be updated on-line, according to the process; kp₀ and ki₀ are initial parameter values of PI controller obtained by a Ziegler-Nichols method.

We can calculate the incremental change in the Δk_i and Δk_i parameters according to different error and variation of error (E, ΔE), the corresponding fuzzy rule is shown in Table 3.

Table 3. Δk_p and Δk_i fuzzy control rules