CHAPTER 4 AN EFFICIENT ANFIS BASED SELF TUNING OF PI CONTROLLER FOR CURRENT HARMONIC MITIGATION

Transcription

1 92 CHAPTER 4 AN EFFICIENT ANFIS BASED SELF TUNING OF PI CONTROLLER FOR CURRENT HARMONIC MITIGATION 4.1 OVERVIEW OF PI CONTROLLER Proportional Integral (PI) controllers have been developed due to the unique characteristic features that the systems with open loop transfer functions of type 1 have zero steady state error with respect to a step input. The realization of PI controller is shown in Figure 4.1. (4.1) Where E and U represents input error signal and controllers output, s is the Laplace variable and are proportional (P) and integral (I) gain of PI controller. (a) Figure 4.1 (Continued)

2 93 (b) Figure 4.1 Proportional-Integral (4.2) (4.3) Where denotes the resistors and denotes the capacitor. Figure 4.1(b) represents proportional integral is represented in the form of resistor and capacitor Working of Proportional-Integral Controller PI controller works by summing the current controller error and the integral of all previous errors. The main aspect is to continuously adjust the value at no error so as to eliminate the offset. The integral reset time variable has been introduced in the controller to evaluate the amount of integral action. If this number is smaller, the sensitivity of adjustment would be lesser which in turn increases the overall accuracy.

3 Disadvantages The disadvantages of PI controller are that it results in higher maximum deviation, a longer response time and a longer period of oscillation than with proportional action alone. Therefore, this control technique has been deployed in scenarios the above limitation can be compensated and offset is in desirable. 4.2 AN OVERVIEW OF PI CONTROLLER TUNING Due to the limitations of the PI controllers, the overall performance of the system gets affected considerably. Hence, tuning of PI controllers has become an active research area and it has been used in various applications PI Controllers Tuning Rules The process model and controller transfer functions are given by: (4.4) Where G(s) is first order plus dead time transfer function,, K are integral time, proportional gain, process static gain. is represents the time, is a dead time. In literature, a number of works have been carried out to tune the PI controllers. This section discusses some of the most important tuning methods carried out in the past years. Tavakoli & Fleming (2003) proposed an optimal method based on a dimensional analysis and numerical optimization techniques, for the tuning of the PI controllers for first order plus dead time systems (FOPDT). This dimensional analysis leads to relations:

4 95 (4.5) Functions and in (4.5) are determined for a step change in the set point so that the integral of the absolute error is minimized. Then genetic algorithms are used to find the best values for each. Eventually functions and are determined using curve-fitting techniques: (4.6) (4.7) Frequency-response method has been designed by Ziegler and Nichols (ZN). This formulation is based on the two parameters such as the ultimate gain and ultimate period that characterize the process dynamics. It can be determined by a relay feedback. Ziegler and Nichols then investigated the effect of disturbance and the effect of load change with a proportional controller. Their conclusion pointed out that, a good negotiation between large offset and large amplitude decay ratio has been chosen for tuning. Then, load change is carried out to find the best response with a PI controller where the gain controller is. The best response has been attained with an integral time This approach gives good results when the dead-time is short. When there is a large dead-time, the closed loop keeps robust but parameters of the controllers are de-tuned, the response is then vague. Cohen & Coon (1953) presented a technique to determine the adjustable parameters for a preferred degree of stability. The tuning is attained under a FOPDT system with a dimensionless equation. Harmonics in response after a heavy side step has been ignored and the amplitude ratio of

5 96 the fundamental is set to The integral time is determined with the objective of a 0.25 amplitude ratio and a compromise between a minimum control area and a maximum stability. Cohen-Coon method has small gain margin and phase margin when the process dead-time is short. This issue considerably decreases when the dead-time of the process increases. Thus, CC tuning design has been frequently used with processes that present a large dead-time. Hang et al (1991) presented the refinements of the Ziegler-Nichols (RZN) tuning formula. It has been derived from a dimensional analysis where the dimensionless variables used are the scaled process gain and the scaled dead time. A step response with 10% overshoot and 3% undershoot is required which defines the tuning rule. Smith presented a control approach for single-input single-output systems, which has the ability to improve the control of loops with dead-time. It has been observed that this approach provides good results when the model is correctly identified. The Smith predictor has been designed as a four block unit which includes the internal controller, the process, the process model and the process model without delay. The internal controller can be a PI controller. An open loop control is initially obtained, based upon an undelayed prediction, the controller being tuned from the model without delay. Feedback action is provided through the (possibly filtered) difference between the prediction (including the delay) and the real measurement that is added to the setpoint. The above discussed methods are the motivating factors for tuning the PI controllers in order to improve its overall performance. The following sections discusses about the model based PI tuning methods available in the literature.

6 Model-Based PI Tuning Methods This section present a model based tuning approach in which the tuning formulas obtained when methods are applied to first-order-plus-dead time plants. It has been assumed that the process steady state gain K > 0, which would imply that the correct controller action is increase-decrease or reverse. The tuning rules are easily adapted for negative-gain plants by replacing K with its absolute value and configuring a direct-acting controller. Rivera et al (1986) presented Internal Model Control (IMC) tuning method for PI controller tuning. Instead of approximating the time delay as a first-order Taylor series, Rivera et al (1986) attempted to account for its effect on the process dynamics by inflating the time constant to. (4.8) (4.9) The controller settings have been shown to produce a FOPDT closed-loop response to step changes in set point. The user-specified parameter can be interpreted as the desired closed loop time constant. Rivera et al (1986) suggested that the parameter be chosen greater than max. It is to be observed that, for this algorithm, the approximate IMC filter constant Decreasing improves the performance at the cost of greater control effort and reduced robustness. In the Skogestad Internal Model Control (SIMC) approach, the proportional gain is computed as

7 98 (4.10) and the integral time as the lesser of and, where (4.11) (4.12) denotes a positive constant selected by the user. When, SIMC coincides with the direct synthesis technique in which plays the role of the approximate closed-loop time constant. PI controller configured in this way can be tuned for excellent performance, but its regulatory response may be poor for lag-dominant processes, i.e., when the dead time-to time constant ratio. When high-performance regulatory control of such a process is required, SIMC offers the flexibility of choosing in order to avoid the pole-zero cancellation. In DS-d approach, Chen & Seborg (2002) reformulated the direct synthesis equations to generate a critically damped regulatory closed-loop transfer function with time constant. The DS-d (direct synthesis for disturbance rejection) settings are (4.13) (4.14) As is the case with in the IMC and SIMC methods, an increase in yields a slower closed-loop response (lower performance).

8 99 However, there is a limit to the extent to which the DS-d controller can be detuned. One must restrict (4.15) in order that KC and remain positive. The Wang & Shao (2000) (WS) approach has been based upon a frequency-domain model of the process dynamics Tuning PI Controllers Parameters The general approach for tuning is: 1. Initially have no integral gain ( large) 2. Increase until get satisfactory response 3. Start to add in integral (decreasing ) until the steady state error is removed in satisfactory time (may need to reduce if the combination becomes oscillatory). By manually tuning the parameter to remove the steady state becomes more difficult and more time would be needed to process. In order to overcome this major drawback and to reduce the time complexity of tuning the parameters, optimization approaches have been widely used. Optimization based techniques or supervised learning methods produce best outcomes that automatically tune the parameters. In the present research work, Neuro fuzzy network which uses the input layers to tune the number of parameters and hidden layers activation

9 100 function to tune the parameters. Neuro fuzzy network removes the steady state error than the manual tuning parameter. 4.3 A THOROUGH ANALYSIS OF CONTROLLER DESIGNS The extensive use of proportional integral controller in various applications is mainly due to its effectiveness in the control of steady-state error of a control system and also its easy implementation. However, one major limitation of this conventional controller which has been discussed earlier is its inability to improve the transient response of the system. The conventional PI controller (Figure 4.2) has the form of Equation (4.16), where Y is the control output which is fed to the PWM signal generator. and are the proportional and integral gains respectively, these gains depending on the system parameters. represents the error signal, which is the difference of the injected current to the reference current. (4.16) Equation (4.16) shows that the PI controller introduces a pole in the entire feedback system, consequently, making a change in its original root response. The effect is the reduction of steady-state error. PWM Figure 4.2 Conventional PI controllers

10 101 On the other hand, the constants and determine the stability and transient response of the system, in which, these constants depend on their universe of discourses (4.17) The minimum and maximum proportional and integral gains are obtained through proper experimental evaluations. The design of the conventional PI controller is completed based on the knowledge of the expert. When the compensator constants go beyond the permissible values, the control system would result in an unstable state. After the determination of the domain of the proportional and integral constants, the tuning of the instantaneous values of the constants takes place. Based on the value of the control system. The constants and are adjusted to guarantee that the steady-state error of the system is minimized to almost zero Fuzzy Logic Controller Fuzzy Logic Controller (FLC) is appropriate for systems that are hard to manipulate mainly due to the existing non linear complexities. This is because, unlike a conventional PI controller, rigorous mathematical formulation is not needed to design a good fuzzy controller. The database which comprises of membership functions lies between 0 and 1. The main processes in FLC are fuzzification, interference mechanism and defuzzification. The interference method uses a set of linguistic rules to convert the input conditions into a fuzzified output. Finally, defuzzification is used to convert the fuzzy outputs into required data. The fuzzy controller configuration is shown in the Figure 4.3.

11 102 Knowledge base Rule base Input Scaling factors normalization Fuzzification Inference Defuzzification denormalizeation Plant Output Output scaling factors normalization Error Measurement Figure 4.3 Fuzzy controller block diagram Fuzzification Fuzzification is an essential aspect in fuzzy logic theory. Fuzzification is the process in which the crisp values are converted to fuzzy. By identifying certain uncertainties present in the crisp values, the fuzzy values have been formulated. The conversion of fuzzy values is denoted by the membership functions. Defuzzification Defuzzification results in the fuzzy to crisp conversions. The fuzzy results generated cannot be used as such to the applications, hence it is necessary to convert the fuzzy quantities into crisp quantities for further processing. FLC Design Methodology The design of the fuzzy logic controller comprises of the following steps. Identifying the input signals to FLC.

12 103 Determining the number of membership function, and Deciding upon the type of membership function. Membership function The number of membership function determines the quality of control which can be achieved using FLC. As the number of membership function increase, the quality of the controller improves at the cost of increased computational time and memory. Investigations are carried out considering seven membership function for each input and output signal. The rule base table for fuzzy controller diagram is shown in the Figure 4.4. e de NL NM NS ZE PS PM PL NL NL NL NL NL NM NS ZE NM NL NL NL NM NS ZE PS NS NL NL NM NS ZE PS PM ZE NL NM NS ZE PS PM PL PS NM NS ZE PS PM PL PL PM NS ZE PS PM PL PL PL PL NL NM NS ZE PS PM PL Figure 4.4 Example Fuzzy rule base Fuzzy control scheme for APF In the fuzzy logic control algorithm for APF, error and change in error are the two input values considered. The two inputs are related by the

13 104 member functions. The membership functions are expressed as Negative Large (NL), Negative Middle (NM), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Middle (PM) and Positive Large (PL). Actual voltage is compared with the reference voltage, based on that error will be produced. It can be compensated by using fuzzy logic controller. Actual current is compared with the reference current, and error is compensated by fuzzy controller. Fuzzy sets support a flexible sense of membership functions. The structure of fuzzy APF controller is shown in the Figure 4.5. Shunt Active Filter with Non- Linear load - + PWM + + Defuzzification Membership function Fuzzification Error Change in Error Figure 4.5 Structure of the fuzzy for APF controller A triangular membership function has the advantage of simplicity and easy implementation and is adopted in the application. Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. The centroid method of defuzzification is generally used, but the

14 105 disadvantage of this method is, it is computationally difficult for complex membership functions Adaptive Fuzzy PI Controller Fuzzy controllers have been widely used in various industrial processes. Especially, fuzzy controllers are effective techniques when either the mathematical model of the system is nonlinear. The fuzzy control system adjusts the parameter of the PI controller by the fuzzy rule. In most of these studies, the Fuzzy controller used to drive the PI is defined by the authors from a series of experiments. The expression of the PI is given in the Equation 4.18: (4.18) Where represents the output of the control, represents the input of the control. The error of the reference current is denoted by and the injected speed is denoted by ; represents the parameter of the scale; represents the parameter of the integrator. The correspondent discrete equation is: (4.19) Where represents the output on the time of k th sampling; represents the error on the time of sampling; denotes the cycle of the sampling; Simple transformations applied to Equation (4.19) leads to:

15 106 (4.20) Tuning The tuning equation for and are shown above: (4.21) (4.22) Neural Network Controller One of the essential issues that assess the quality of the delivered power is the estimation of fundamental components from distorted current or voltage waveforms. In order to provide high-quality power supply electricity, it is necessary to accurately estimate or extract the time varying fundamental components such as the magnitude and the phase angle, to mitigate harmonic components using active power filters. Artificial Neural Networks (ANN) controller has been used in the mitigation of current harmonics. ANN is observed to be the most useful technique in solving the complex optimization problem. Generally, a low pass filter is utilized for separating the basic component from voltage which is inefficient in actual conditions introduced earlier. Moreover, it has problems related to accurate phase and gain tuning and three additional current sensors are required for sensing load currents. It is clearly observed that the particular order and cut-off frequency plays a major role in designing a filter. Hence, in the proposed approach, a neural network is employed for extracting fundamental components from

16 107 each phase of source currents instead of load currents for non-ideal mains supply and resulting real power due to fundamental components of currents. The architecture of proposed ADALINE neural network has two layer (input and output) network having n-inputs and a single output shown in Figure 4.6 and Figure 4.7. Linear T.F Output Bais Figure 4.6 Internal blocks of proposed Neural Network Figure 4.7 Input /Output relationship of purlin transfer function The basic blocks of this network are input signal delay vector, a purelin transfer function, weight matrix.

17 108 (4.23) where denotes the represents the denotes the input of the neural network. The input to the network is a time delayed series of the signal whose fundamental component is to be extracted. The length of this delay series is 61, which has been decided considering expected maximum distortion and unbalance in 3-phase input signal. The input to the ANN system is supply voltage and current and the output of the system is APF reference current. The weight adjustment is performed during the training process of the ADALINE using Widrow-Hoff delta rule. 4.4 PROPOSED HARMONIC MITIGATION APPROACH WITH PI BASED ANFIS Several number of research works have been devoted to improve the ability of fuzzy systems as discussed in Wilamowski & Jaeger (1996), such as evolutionary strategy and neural networks as discussed in Cupal &Wilamowski (1994). The combination of fuzzy logic and neural networks results in a hybrid intelligent system by integrating human-like reasoning style of neural networks. The architecture of the neuro fuzzy systems is shown in the Figure 4.8.

18 109 X Fuzzification Multiplication Sum Division All weights equal expected values Fuzzifier Out Y Fuzzifier All weights equal 1 Z Fuzzifier Figure 4.8 Neuro fuzzy system architecture Adaptive Neuro-fuzzy Principle A typical architecture of an ANFIS is shown in Figure 4.9, in which a circle indicates a fixed node, whereas square indicates an adaptive node. considered in the Figure 4.9. Among several FIS models, Sugeno fuzzy model has been extensively used in various applications due to its high interpretability and computational efficiency as discussed in Lin & Lee (1996). For a first order Sugeno fuzzy model, a common rule set with two fuzzy if then rules can be expressed as: Rule 1: if x is and y is, then (4.24)

19 110 Rule 2: if x is and y is, then (4.25) where and are the fuzzy sets in the antecedent, and, and are the design parameters that are determined during the training process. Back propagation algorithm N Input Output AND N Normalizer Output reference value Figure 4.9 Adaptive neuro fuzzy structure As shown in Figure 4.9, the ANFIS comprises of five layers as discussed in Lin & Lee (1996) and Miloudi et al (2007). The tasks carried out in each layer are given below. Layer 1: It composed of a number of computing nodes whose activation functions are fuzzy logic membership functions (triangular functions). Layer 2: This layer selects the minimum value of the inputs.

20 111 Layer 3: Normalizes each input with respect to the others (The i th node output is the i th input divided by the sum of all the other inputs). Layer 4: i th th node output and the ANFIS input signals. Layer 5: Sums all the incoming signals. ANFIS structure can be tuned automatically by a least-square estimation (for output membership functions) and a back propagation algorithm (for output and input membership functions) as discussed in Miloudi et al (2007). The proposed architecture is shown in the Figure PI using neuro fuzzy + Park transformation abc-dq Butterworth lowpass filter Butterworth lowpass filter Park Transformati on dq0-abc - Figure 4.10 Park Transformations and Harmonic Current Injection Circuit

21 Offline tuning of PI Controller Offline tuning of PI controller using ANFIS is proposed for the tuning of input SAPFs by developing the adjustment rules defined in terms of and for updating the SAPFs, in dependence on the performance of the closed loop system. Offline tuning mechanism simply means that the tuning of input gains based on error and change in error. Based on this mechanism, the incremental change in and is obtained by Equation (4.26). (4.26) where and are the updating factors for incremental change in and which are computed online based on neuro fuzzy logic reasoning using the error and change in an error at each sampling time. are the scaling factors. is an error signal. Thus the input gains of the auto tuning PI controller does not remain fixed while the controller is in operating condition, infect it is updating at each sample by updating factors and. In this work, neuro fuzzy controller has been used to tune the PI controller parameters in the off line mode. An Offline mode tuning of PI controller using ANFIS is proposed for the tuning of input SAPFs by developing the adjustment rules defined in terms of and for updating the SAPFs, in dependence on the performance of the closed loop system. In this work, offline mode tuning mechanism is carried out by tuning the input gains based on error and change in error. Initially, the system is trained for the actual values of THD and settling voltage. Then, the PI controller parameters values are tuned based on the neuro-fuzzy system which is trained to adopt gain setting of PI

22 113 controller. When the ANFIS has been trained, it will yield the optimal PI gains for any operating conditions and time delays even if they are not in the train data. The fuzzy inference system under consideration has the DC Link Voltage error as input and produces the output values of. The range of the input values to the neuro fuzzy system is given based on the capacitor voltage selection formula Here, since the capacitor voltage should be greater than 600 V, the range considered here is (-800, 800) in which 20 neurons have to be used as hidden neurons in this proposed neuro fuzzy system. Table 1 shows the tested values which give the equivalent THD and Based on this response, the range of the output considered as (0, 10) and (0, 1) respectively. Settling time. values are The training time is about 4 times longer for the networks with 20 neurons in the hidden layers. Triangular fuzzy membership function is used to generate the fuzzy rules. Here the fuzzy values are chosen between (-1, 1) for the triangular membership function. Here, the (7x7=49 rules) are generated with around 800 epochs. Table 4.1: & Range Values Range THD Settling

23 Based on the input and output range values, the data is trained and then Fuzzy Inference System (FIS) is generated through fuzzy membership functions and rules. With this FIS information, ANFIS function is carried out through epochs and trained data. Based on the above given input sets, an optimal value is obtained as output which in turn is used in the PI controller for minimizing the THD. In the PI controller, the saturation limit range is considered as (-25, 25) based on the load current magnitude. 4.5 SUMMARY The present work proposed an efficient harmonic mitigation framework using control strategy with SVPWM approach. The main factor in this present research work is the usage of Adaptive Neuro Fuzzy Inference System (ANFIS) network for tuning and parameters of the PI controllers to improve the steady state control result in motor.