Online adaptive type-2 fuzzy logic control for load frequency of multi-area power system

Abstract

Load frequency control (LFC) is one of the important control problems in design and operation of power systems as permanent deviation of frequency from nominal value affects power system operation and reliability. This paper presents a control method based on neural network for LFC of a two-area power system containing re-heat thermal plants. System parameters are assumed to be unknown and the proposed type-2 fuzzy controller is designed online, is adaptive and does not require initial adjustment by the operator. The training method of the type-2 fuzzy controller includes error back-propagation and gradient descent. In this paper, since the dynamics of the system is unknown, it is modelled using multilayer perceptron (MLP) structure, and Jacobian of the system is extracted to determine system model. In order to evaluate the robustness of proposed online adaptive fuzzy type-2 controller (OADF) against parameter changes, a time-variant parameter is added to the system. The performance of the controller is compared with the PI, PID, N-PID, fuzzy-PI and neural network controllers. Simulation results illustrate the improved performance of LFC and its capability to overcome uncertain and time-variant parameters.

Keywords

LFC adaptive type-2 fuzzy control multi-area power system MLP back-propagation and gradient descent

1 Introduction

Frequency control is one of the most important control problems in design and operation of power networks. Maintaining frequency and power exchange in adjacent areas, is one of the main objectives of load frequency control (LFC) in multi-area power systems. Frequency deviation, changes the power of frequency-sensitive loads. Small changes of loads in different areas changes frequency and also the exchanged power between areas. Since active power should be generated as required and since load consumption changes at different hours of the day, controlling generated power of generators is of great importance [1]. Frequency deviation from nominal value is selected as a signal for excitation control system of generators, where active power balance indicates system frequency is constant. LFC only responds to small and slow changes of load and frequency while larger frequency deviations are controlled by automatic generation control (AGC). Also, due to changing nature of the load during the day, classic constant-gain controllers will not perform well for controlling load changes. In order to overcome this problem, variable-structure controller is suggested [2 –4] and adaptive control methods are used for automatic control [5 –9] which require a complete model for designing controller. LFC has become a challenging problem due to increasing size, changing structure and complexities of interconnected power systems. As size and complexity of deregulated power systems increase and considering different uncertainties and disturbances in control system, using novel control mechanisms for frequency control gains more importance. Many researchers have tried to improve the performance of classic controllers. Recently, fuzzy logic has widely been used for identification, modelling and control of systems with nonlinear dynamics [10, 11]. Various combinations of control methods have been proposed for improving performance of fuzzy PI and PID controllers [12 –16]. Process of adjusting PID coefficients can be costly and time-consuming [17, 18]. Many efforts have been made for adjusting coefficients of controller for reducing error and changing its rate. In [19], a smart method has been proposed for online adjustment of parameters of PI controller. During the last decade, various methods have been proposed for adaptive fuzzy logic frequency control [20 –25]. An adaptive method has been proposed in [20] for conventional PI control and optimal LFC which has employed a Sugeno fuzzy system for finding gains of fuzzy controller under different operating conditions. An intelligent fuzzy control structure has been proposed in [21] for synchronization and optimization of control gains under load changes. In [24], a fuzzy-PI controller has been designed for LFC based on genetic algorithm and particle swarm optimization (PSO).

These conducted studies are mainly performed in two ways: a) either system dynamics are known: the system is then modeled linearly and the controller is designed online. b) system dynamics are assumed to be unknown and the controller is designed offline. In this case, evolutionary algorithms are used to optimize the controller and optimized parameters are applied to the system; in fact, system parameters are indirectly assumed to be known. Shortcomings of these methods are their being time-consuming, the optimal operating point might not be obtained which increases computation work, and it may result in control instability.

In this paper, system parameters are assumed to be unknown but the controller is designed online so that the output of the system is taken online. In addition, parameters of fuzzy system are adjusted such that frequency variation error converges to zero. The cost function is based on gradient descent which requires system Jacobian. Since system dynamics are not known, system Jacobin is not known and it will be extracted by modelling the system using MLP neural network, online.

This remainder of this paper is organized as follows. In Section 2, the system is modelled. Section 3 introduces the proposed control method describing MLP neural network structure, system Jacobian and structure of type-2 neuro-fuzzy controller. In Section 4, training fuzzy controller based on error back-propagation and gradient descent are presented. Simulation results are provided in Section 5. Finally, the paper is concluded in Section 6.

2 System modelling

Figure 1 Shows a schematic representation of a two-area power system where its parameters are adopted from [26]. Each area includes a governor, re-heating steam turbine and generator.

Fig. 1

General scheme of two-area interconnected power system.

Dynamic relationship between power changes ΔP_gi - ΔP_di and frequency deviation Δf_i is represented in Equation (1): $Δ {\dot{f}}_{i} = - \frac{1}{T_{pi}} Δ f_{i} - \frac{K_{pi}}{T_{pi}} Δ P_{tie, i} + \frac{K_{pi}}{T_{pi}} Δ P_{gi} - \frac{K_{pi}}{T_{pi}} Δ P_{di} + \frac{1}{T_{gi}} Δ P_{ci}, i = 1, 2$ (1)

where

Δf_i: frequency deviation

ΔP_tie,i: tie-line active power deviation

ΔP_gi: generator output power deviation

ΔP_di: load disturbance

ΔP_ci: controller output

T_pi: system time constant

K_pi: system gain constant

Dynamics of the thermal turbines is described in Equation (2): $Δ {\dot{P}}_{gi} = - \frac{1}{T_{ti}} Δ P_{gi} - \frac{1}{T_{ti}} Δ P_{ri}$ (2) where

ΔP_ri: Re-heat turbine output power deviation

Dynamic equations of the governor are as in Equation (3). $Δ {\dot{X}}_{gi} = - \frac{1}{T_{gi} R_{i}} Δ f_{i} - \frac{1}{T_{gi}} Δ X_{gi}$ (3) where

T_gi: thermal governor time constant

R_i: speed drop due to governor action

ΔX_gi: governor valve position deviation

Re-heating part of the turbine can be described as in Equation (4).

$\begin{matrix} Δ \dot{P_{ri}} = - \frac{K_{ri}}{T_{gi} R_{i}} Δ f_{i} + (\frac{1}{T_{ri}} - \frac{K_{ri}}{T_{gi}}) \\ Δ X_{gi} - \frac{1}{T_{ri}} Δ P_{ri} \end{matrix}$ (4) where

K_ri: re-heat turbine gain

Power exchanged between area i and j is described as follows: $Δ {\dot{P}}_{tie}^{ij} = T (Δ f_{i} - Δ f_{j}), Δ P_{tie}^{ij} = - Δ P_{tie}^{ji}$ (5) where

T: interconnection gain between control area i and j

Total power change between area i and j can be calculated as in Equation (6). $Δ {\dot{P}}_{tie, i} = \sum_{j = 1, j \neq i}^{2} Δ {\dot{P}}_{tie}^{ij} = \sum_{j = 1, j \neq i}^{2} T (Δ f_{i} - Δ f_{j})$ (6)

3 Structure of the proposed controller

Power system has nonlinear dynamics and uncertainties; under such conditions constant-gain controllers do not perform well. Thus, methods such as artificial neural network (ANN), fuzzy logic and fuzzy neural networks are utilized in LFC [27 –32]. However, these controllers are not robust against uncertainties and noise, and do not perform well against parameter variations. This paper employs type-2 neuro-fuzzy controller which can model uncertainty unlike type-I fuzzy systems assuming that system dynamics are unknown. The proposed controller reduces sensitivity to system parameters, increases speed of output response and reduces computations thanks to type reduction, and adapts itself to new conditions.

In this section, the proposed controller based on neural network is introduced. Block diagram of the proposed method is shown in Fig. 2 where the output of the neuro-fuzzy network constitutes the control signal. Parameters of this neuro-fuzzy network are trained such that frequency changes Δf tend to zero. When Δf tends to zero, the control objective is satisfied. In the subsection, structure of the neuro-fuzzy network and adjustment of its weights are described.

Fig. 2

Block diagram of proposed structure.

The Main advantages of our proposed controller are:

Unlike current studies, the system dynamics are completely uncertain.

The controller is adaptive and does not require initial adjustments by the operator.

Controller is able to overcome uncertain and time-variant parameters of the system.

3.1 Structure of the MLP neural network

The neural network, with its structure shown in Fig. 3, is used to model the system for adaptive calculation of the system Jacobian.

Fig. 3

Neural network structure for system modeling.

The description of the used notation in Fig. 3 is as follows:

u (t - z₁), u (t - z₂), . . . , u (t - z_n): inputs of the neural network

z₁, . . . , z_n: constant delays

u (t): sum of system output and control signal at instant t

$w_{11}^{1}, w_{12}^{2}, . . ., w_{1 n}^{1}$ : weights connected to first neuron in the middle layer

$w_{21}^{1}, w_{22}^{1}, . . ., w_{2 n}^{1}$ : weights connected to second neuron in the middle layer

$w_{h 1}^{1}, w_{h 2}^{1}, . . ., w_{hn}^{1}$ : weights connected to neuron h, h being the number of neurons in the middle layer

w₂₁, w₂₂, . . . , w_2h: weights connected to output and neurons of the layer.

Output of this neural network is obtained step-by-step as follows:

Input of the neural network is the control signal and system output at previous time samples.

Output of neurons of the middle layer are obtained as follows: $\begin{array}{l} n e t_{i} = w_{i}^{1} U \\ o_{i} = g (n e t_{i}), i = 1, ..., h \end{array}$ (7) where

$\begin{array}{l} U = {[u (t - z_{1}), u (t - z_{2}), ..., u (t - z_{n})]}^{T} \\ w_{i}^{1} = [w_{i 1}^{1}, w_{i 2}^{1}, ..., w_{i n}^{1}] \end{array}$ (8) $g ({net}_{i}) = \frac{1 - exp (- {net}_{i})}{1 + exp (- {net}_{i})}$

Output of the neural network is obtained as follows:

y = w_{2} o

(9) where

o = [o_{1}, o_{2}, . . ., o_{h}]^{T}

w = [w_{21}, w_{22}, . . ., w_{2 h}]

(10) Weights of this neural network are trained such that cost function E is minimized:

E = \frac{1}{2} e_{est}^{2} = \frac{1}{2} (y_{d} - y)^{2}

(11) Where y_d is the desired output and y is output of the neural network. Weights at t + 1 are

w (t + 1) = w (t) - η \frac{\partial E}{\partial w}

. Gradient descent and back-propagation error are used for training. In order to obtain

\frac{\partial E}{\partial w}

, the chain differentiation of

\frac{\partial E}{\partial w} = \frac{\partial E}{\partial e} \frac{\partial e}{\partial y} \frac{\partial y}{\partial w}

applies. By substituting

\frac{\partial E}{\partial e} = e, \frac{\partial e}{\partial y} = - 1, \frac{\partial y}{\partial w} = 0

, rule of training weights is obtained as in Equation (12):

w_{2} (t + 1) = w_{2} (t) + η e_{est} o

(12) Adaptive law for weights of the first layer is obtained as in Equation (13)

w_{i}^{1} (t + 1) = w_{i}^{1} (t) + η e_{est} \dot{g} ({net}_{i}) w_{2 i} U

(13) Where

w_{i}^{1}

is vector of weights connected to ith neuron in the middle layer, w_2i is weight connected to output and ith neuron in the middle layer.

\dot{g} ({net}_{i})

is the derivative of g (net_i) with respect to net_i which is constant adaptive rate.

Equation (13) is an adaptive equation based on gradient descent and differentiation with respect to weights of hidden layers.

3.2 System Jacobian

Finally, the obtained model is used to calculate system Jacobian:

$\begin{matrix} \frac{\partial Δ f}{\partial u_{c}} \\ = ([w_{11}^{1}, w_{21}^{1}, . . ., w_{h 1}^{1}] diag [\dot{g} ({net}_{1}), . . ., \dot{g} ({net}_{h})] w_{2}) \end{matrix}$ (14) where

$\frac{\partial Δ f}{\partial u_{c}}$ : the derivative of the output with respect to control input

$[w_{11}^{1}, w_{21}^{1}, . . ., w_{h 1}^{1}]$ : vector of weights connected to the first input and neurons of the middle layer

$[\dot{g} ({net}_{1}), . . ., \dot{g} ({net}_{h})]$ : vector of derivative of output of neurons of the middle layer with respect to their input

w₂: vector of weights connected to output and neurons of the middle layer.

3.3 Structure of Type-II Neuro-Fuzzy Controller

Structure of the neural network is represented in Fig. 4.

Fig. 4

Structure of Type-2 Neuro-Fuzzy controller.

Number of neurons in the middle layer is M and number of inputs of the network is 3. Feedforward output of the controller is obtained as follows:

Firing fuzzy rules are obtained as in Equation (15): ${\bar{o}}_{k} = exp (- \frac{| | X - C_{k} | |^{2}}{{\bar{σ}}_{k}^{2}})$ ${\underline{o}}_{k} = exp (- \frac{| | X - C_{k} | |^{2}}{{\underline{σ}}_{k}^{2}})$ (15) $k = 1, . . ., M$

Where C_i and σ_i, i = 1, . . . , n are center and width of the Gaussian function, and the input vector is: $X (t) = [Δ f (t), \frac{d}{dt} Δ f (t), \int_{0}^{t} Δ f (τ) d τ]^{T}$

Output of the fuzzy system is obtained based on order reduction of Nie-Tan: $u_{c} = w^{T} ξ$ (16)

Where w is vector of weights of the output layer and ξ is defined as follows: $ξ = [z_{1}, . . ., z_{m}]^{T}$ $z_{i} = ({\bar{o}}_{i} + {\underline{o}}_{i}) / \sum_{i = 1}^{M} ({\bar{o}}_{i} + {\underline{o}}_{i})$ (17)

Where M is number of rules or number of neurons in the middle layer.

4 Fuzzy controller training based on error back-propagation and gradient descent

In this section, training of weights based on error back-propagation procedure and gradient descent is presented. To begin, instantaneous square error be-tween desired response and network output at instant t is considered as cost function: $E = \frac{1}{2} e^{2} = \frac{1}{2} (Δ f)^{2}$ (18) Where Δf is the system output. According to learning rules of error back-propagation, this cost function should be differentiated with respect to parameters of the neural network. In other words: $w (t + 1) = w (t) - η \frac{\partial E}{\partial w}$ (19)

Where w is vector of weights of the subsequent sections defined in the previous section, and η is the training rate of gradient descent.

In order to find $\frac{\partial E}{\partial w}$ , chain differentiation gives: $\frac{\partial E}{\partial w} = \frac{\partial E}{\partial Δ f} \frac{\partial Δ f}{\partial u_{c}} \frac{\partial u_{c}}{\partial y} \frac{\partial y}{\partial w}$ (20) Where the system Jacobian, $\frac{\partial Δ f}{\partial u_{c}}$ , is calculated from Equation (14), and the other three terms can be calculated as follows: $E = \frac{1}{2} e^{2} = \frac{1}{2} (Δ f)^{2} \Rightarrow \frac{\partial E}{\partial Δ f} = Δ f$ $u_{c} = y \Rightarrow \frac{\partial u_{c}}{\partial y} = 1$ (21) $y = w^{T} ξ \Rightarrow \frac{\partial y}{\partial w} = ξ$

5 Simulation results

In order to evaluate the performance of our proposed controller, time-domain simulations are performed in Matlab environment. Block diagram of two-area LFC is shown in Fig. 5 with the system parameters as listed in Table 1 [27]:

Fig. 5

Two-area load frequency control.

Table 1

Nominal parameters of two-area power system

Parameter	Value
T _gi	0.08 s
T _ti	0.3 s
T _pi	20 s
T _ri	10 s
R _i	2.4 Hz/pu.MW
β _i	0.425 pu.MW/Hz
K _pi	120 Hz/pu
T	0.545 pu/MW

For comparison, the values of parameters of PI = ${\overset{´}{K}}_{p} + {\overset{´}{K}}_{i} / s$ and PID = ${\overset{´}{K}}_{p} + {\overset{´}{K}}_{i} / s + {\overset{´}{K}}_{d} s$ controllers are adjusted using Ziegler-Nichols (ZN).

For area 1 and 2, the coefficients of the PID controller are:

Area 1: ${\overset{´}{K}}_{p} = 0, {\overset{´}{K}}_{i} = 0.0018, {\overset{´}{K}}_{d} = 0$

Area 2: ${\overset{´}{K}}_{p} = - 0.59, {\overset{´}{K}}_{i} = - 0.28, {\overset{´}{K}}_{d} = - 0.116$ .

For PI controller, coefficients are:

${\overset{´}{K}}_{p} = 0.05, {\overset{´}{K}}_{i} = 1$ .

Frequency deviations in area 1 using OADF, PID and PI are shown in Fig. 6. According to this figure, using our proposed OADF method, overshoot of frequency changes in area 1 is 0.03 and it settles down in 2s, while using PID controller, overshoot is 0.3 and it settles down in 10s. In addition, using PI controller, the system becomes unstable and oscillates. Frequency deviations in area 2 are also shown in Fig. 7. An overshoot of 0.04 and a settling time of 4s in our proposed OADF method illustrates its better performance compared to the other two controllers.

Fig. 6

Frequency deviation in area 1 with OADF, PID and PI controllers.

Fig. 7

Frequency deviation in area 2 with OADF, PID and PI controllers.

Control signal of area 1, and the deviation of the tie-line power are shown in Figs. 8 and 9, respectively. As can be seen in these figures, PID controller oscillates considerably at the beginning and becomes stable after a few seconds while in our proposed method, there is no oscillation and it becomes stable in a shorter time. In addition, OADF performs better for control signal of area 1 and deviation of the tie-line power in terms of stability, settling time and damping.

Fig. 8

Control signal in area 1 with OADF, PID and PI controllers.

Fig. 9

Deviation of the tie-line (T) power with OADF, PID and PI controllers.

In order to evaluate the performance of the OADF controller, a time-variant parameter is added to the system, and the results are given in Fig. 10. As can be seen, OADF controller is robust against uncertain dynamics of time-variant parameter. Frequency changes in area 1 and 2 are not oscillatory becoming stable after 3s. Control signal of area 1 and 2, shown in Fig. 11, indicate superiority of our proposed method. For comparison, two-area system is also simulated using PID controller with time-variant input. As can be seen in Fig. 12, oscillation amplitude is larger and settling time is about 15s.

Fig. 10

Frequency deviation in area 1 and 2 with OADF controller with time-variant parameter.

Fig. 11

Control signals in area 1 and 2 with OADF controller with time-variant parameter.

Fig. 12

Frequency deviation in area 1 and 2 with PID controller.

The proposed OADF controller is additionally compared with fuzzy-PI and N-PID controllers of [33] to study transient response. Controller gain is optimized using differential evolution particle swarm optimization (DEPSO) and N-PID structure described in [33].

As shown in Fig. 13, our proposed controller has smaller overshoot and better transient response compared to fuzzy-PI controller. Figure 14 also illustrates that the overshoot of frequency deviation with N-PID controller is large compared to proposed controller. Thus, the OADF controller leads to a small overshoot and also the system settles very early irrespective of the disturbance.

Fig. 13

Frequency deviation in area 1 and 2 with fuzzy-PI controller.

Fig. 14

Frequency deviation in area 1 and 2 with N-PID controller.

Additionally, the proposed OADF is compared with adaptive fuzzy controller of [34]. As a shown in Fig. 15, our proposed method has better performance.

Fig. 15

Frequency deviation in area 1 and 2 with adaptive fuzzy controller [34].

In this paper, integral squared error (ISE) is used to validate the results. Table 2 presents values of ISE for area 1 and 2: with adding a time-variant parameter to system output (R1-R5) and without adding a time-variant parameter to system output (L1-L5). It can be noted that OADF controller has lowest and PI controller has maximum ISE values for both area 1 and 2.

Table 2

ISE for OADF, PID, PI, N-PID, fuzzy-PI and NN-PID controllers without time-variant parameter (L1-L5) and with time-variant parameter (R1-R5) in five iteration (N = 1,2,...,5)

Controller	ISE_i	L1 N = 1 R1	L2 N = 2 R2	L3 N = 3 R3	L4 N = 4 R4	L5 N = 5 R5
OADF	ISE₁	0.0779 \| 0.0027	0.0009104 \| 0.0026	0.00616 \| 0.0038	0.00209 \| 0.0038	0.00218 \| 0.0055
	ISE₂	0.1388 \| 0.0030	0.0008203 \| 0.0030	0.00490 \| 0.0023	0.00211 \| 0.0036	0.00235 \| 0.0296
PID	ISE₁			0.45730 \| 0.8242
	ISE₂			0.09175 \| 0.1332
PI	ISE₁			4.4110 \| 4.566
	ISE₂			1.621 \| 1.20
N-PID	ISE₁			0.005210 \| 0.00612
	ISE₂			0.00835 \| 0.0083
fuzzy-PI	ISE₁			0.00308 \| 0.0328
	ISE₂			0.03850 \| 0.0385
NN-PID	ISE₁			0.03112 \| 0.0041
	ISE₂			0.05127 \| 0.0061

The simulation results and various performance indices illustrate that the proposed OADF controller has better control performance than the PID, PI, N-PID, fuzzy-PI and NN-PID controllers.

6 Conclusion

In this paper, a control method is proposed for adjusting frequency of a two-area system with re-heat thermal plants. The system dynamics are unknown and parameters of the fuzzy system are adjusted by minimization of a cost function using gradient descent. The system is modelled online using MLP neural network to obtain system Jacobian. Type-2 neuro-fuzzy control method is used to adjust frequency deviations online. In addition, in order to investigate the robustsness of the proposed controller, a time-variant parameter is added to the system output and then output responses of the proposed method are compared with PI and PID controllers adjusted using ZN method and N-PID, fuzzy-PI and NN-PID controllers. In order to validate the proposed method, integral squared error (ISE) criterion is used. Simulation results indicate superiority of the proposed method.

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