CHAPTER 4 PID CONTROLLER BASED SPEED CONTROL OF THREE PHASE INDUCTION MOTOR

Transcription

1 36 CHAPTER 4 PID CONTROLLER BASED SPEED CONTROL OF THREE PHASE INDUCTION MOTOR 4.1 INTRODUCTION Now a day, a number of different controllers are used in the industry and in many other fields. In a quite general way those controllers can be divided into two main groups: a) Conventional controllers b) Non-conventional controllers Under the conventional controllers it is possible to count the controllers known for years now, such as P, PI, PD, PID, Otto-Smith, all their different types and realizations, and other controller types. It is a characteristic of all conventional controllers that one has to know a mathematical model of the system in order to design a controller. Unconventional controllers utilize a new approach to the controller design in which knowledge of a mathematical model of a process is not required. Examples of unconventional controller are a fuzzy controller and Neuro or Neuro-fuzzy controllers. Many industrial processes are nonlinear and thus complicate to describe mathematically. However, it is known that a good many nonlinear processes can satisfactorily controlled using PID controllers

2 37 providing that controller parameters are tuned well. Practical experience shows that this type of control has a lot of sense since it is simple and based on 3 basic behavior types: proportional (P), integrative (I) and derivative (D). Instead of using a small number of complex controllers, a larger number of simple PID controllers are used to control simpler processes in an industrial assembly in order to automate the certain more complex process. PID controller and its different types such as P, PI and PD controllers are the basic building blocks in the control of various processes. In spite their simplicity; they can be used to solve even a very complex control problems, especially when combined with different functional blocks, filters (compensators or correction blocks), selectors etc. A continuous development of new control algorithms insures that the time of PID controller has not passed and that this basic algorithm will have its part to play in process control in foreseeable future. It can be expected that it will be a backbone of many complex control systems. In this chapter mathematical modeling and PID controller based induction motor speed control are discussed. 4.2 MATHEMATICAL MODELING OF THREE PHASE INDUCTION MOTOR. In the control of any power electronics drive system to start with a mathematical model of the plant is important. To design any type of controller to control the process of the plant, the mathematical model is required. Lashok Kusagur, Kodad & Sankar ram (2009) have analyzed the mathematical modeling of induction motor. The equivalent circuit used for obtaining the mathematical model of the induction motor is shown in the Figure 4.1.

3 38 (a) d-axis (b) q-axis Figure 4.1 Equivalent circuit of induction motor in d-q frame The induction motor model is established using d, q field reference concept. The Motor parameters are given in Table 4.1. Table 4.1 Motor parameters Parameters Value Power 0.5 HP Voltage 415 V Current 0.9 A Frequency 50Hz Speed 1440 RPM Stator resistance Rs 6.03 Rotor resistance Rr Stator inductance Ls 489.3e-3 H Rotor inductance Lr 489.3e-3 H Poles 4

4 39 An induction motor model is then used to predict the voltage required to drive the flux and torque to the demanded values within a fixed time period. This calculated voltage is then synthesized using the space vector modulation. Direct axes and quadrature axes stator and rotor voltages are given in Equations (4.1), (4.2), (4.3) and (4.4). (4.1) (4.2) (4.3) (4.4) Vsd and Vsq, Vrd and Vrq are the direct axes & quadrature axes stator and rotor voltages. The flux linkages to the currents are given by the Equation (4.5) (4.5) The electrical part of an induction motor can thus be described, by combining the above equations the Equation (4.6) is obtained.

5 40 (4.6) Where A is given by the Equation (4.7) (4.7) The instantaneous torque produced is given by the Equation (4.8) (4.8) The electromagnetic torque expressed in terms of inductances is given by the Equation (4.9) (4.9) The mechanical part of the motor is modeled by the Equation (4.10) (4.10)

6 41 D and Q blocks are shown in Figure 4.2. The mathematical modeling of three phase induction motor is shown in Figure 4.3. (a) D block (b) Q block Figure 4.2 D-Q blocks

7 42 Figure 4.3 Three phase induction motor mathematical modeling This induction motor model is further used to design a controller using fuzzy control strategy. 4.3 PID CONTROLLER A Proportional integral derivative controller (PID controller) is a generic control loop feedback mechanism (controller) widely used in industrial control systems. The PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measured process variable and a desired set point. The controller attempts to minimize the error by adjusting the process control inputs. The PID controller calculation (algorithm) involves three separate constant parameters, and is accordingly sometimes called three-term control: Proportional, Integral and Derivative values, denoted P, I, and D.

8 43 Heuristically, these values can be interpreted in terms of time: P depends on the present error, I on the accumulation of past errors, and D is a prediction of future errors, based on current rate of change. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve, or the power supplied to a heating element. In the absence of knowledge of the underlying process, a PID controller is the best controller. By tuning the three parameters in the PID controller algorithm, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the set point and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability. Some applications may require using only one or two actions to provide the appropriate system control. This is achieved by setting the other parameters to zero. A PID controller will be called as PI, PD, P or I controller in the absence of the respective control actions. PI controllers are fairly common, since derivative action is sensitive to measurement noise, whereas the absence of an integral term may prevent the system from reaching its target value due to the control action Control Loop Basics A familiar example of a control loop is the action taken when adjusting hot and cold faucets (valves) to maintain the water at a desired temperature. This typically involves the mixing of two process streams, the hot and cold water. The person touches the water to sense or measure its temperature. Based on this feedback they perform a control action to adjust the hot and cold water valves until the process temperature stabilizes at the desired value. The sensed water temperature is the process variable or process value (PV). The desired temperature is called the set point (SP). The input to

9 44 the process (the water valve position) is called the manipulated variable (MV). The difference between the temperature measurement and the set point is the error(e) and quantifies whether the water is too hot or too cold and by how much. After measuring the temperature (PV), and then calculating the error, the controller decides when to change the tap position (MV) and by how much. When the controller first turns the valve on, it may turn the hot valve only slightly if warm water is desired, or it may open the valve all the way if very hot water is desired. This is an example of a simple proportional control. In the event that hot water does not arrive quickly, the controller may try to speed-up the process by opening up the hot water valve more and more as time goes by. This is an example of an integral control. Making a change that is too large when the error is small is equivalent to a high gain controller and it will lead to overshoot. If the controller were to repeatedly make changes that were too large and repeatedly overshoot the target, the output would oscillate around the set point in a constant, growing, or decaying sinusoid. If the oscillations increase with time, then the system is unstable, whereas if they decrease the system is stable. If the oscillations remain at a constant magnitude the system is marginally stable. In the interest of achieving a gradual convergence at the desired temperature (SP), the controller may wish to damp the anticipated future oscillations. So in order to compensate this effect, the controller may elect to temper their adjustments. This can be thought of as a derivative control method. If a controller starts from a stable state at zero error (PV = SP), then further changes by the controller will be in response to the changes in other measured or unmeasured inputs to the process that impact on the process, and hence on the PV. Variables impact on the process, other than the MV is known as disturbances. Generally controllers are used to reject disturbances and / or implement set point changes. The variations in feed water temperature constitute a

10 45 disturbance to the faucet temperature control process. In theory, a controller can be used to control any process which has a measurable output (PV), a known ideal value for that output (SP) and an input to the process (MV) that will affect the relevant PV. Controllers are used in the industry to regulate temperature, pressure, flow rate, chemical composition, speed and practically every other variable for which a measurement exists PID Controller Theory The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining u (t) as the controller output, the final form of the PID algorithm is: (4.11) Where K p : K i : K d : Proportional gain, a tuning parameter Integral gain, a tuning parameter Derivative gain, a tuning parameter e : Error = SP PV t : Time or instantaneous time (the present)

11 46 Figure 4.4 shows the PID controller. Figure 4.4 PID controller Proportional term Figure 4.5 Process variables for different Kp values Process variables for different Kp values (K i and K d held constant) are shown in Figure 4.5. The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant K p, called the proportional gain.

12 47 The proportional term is given by the Equation (4.12) (4.12) A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (see the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive or less sensitive controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances. Tuning theory and industrial practice indicate that the proportional term should contribute the bulk of the output change. A pure proportional controller will not always settle at its target value, but may retain a steady-state error. Specifically, drift in the absence of control, such as cooling of a furnace towards room temperature, biases a pure proportional controller. If the drift is downwards, as in cooling, then the bias will be below the set point, hence the term "droop. Droop is proportional to the process gain and inversely proportional to proportional gain. Specifically the steady-state error is given by the Equation (4.13) e = G / K p (4.13) Droop is an inherent defect of purely proportional control. Droop may be mitigated by adding a compensating bias term (setting the set point above the true desired value), or corrected by adding an integral term Integral term Process variables for different Ki values (Kp and K d held constant) are shown in figure 4.6. The contribution of the integral term is proportional to both the magnitude of the error and the duration of the error. The integral in

13 48 a PID controller is the sum of the instantaneous error over time and gives the accumulated offset that should have been corrected previously. Figure 4.6 Process variables for different Ki values The accumulated error is then multiplied by the integral gain (K i ) and added to the controller output. The integral term is given by the Equation (4.14) (4.14) The integral term accelerates the movement of the process towards set point and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral term responds to accumulated errors from the past, it can cause the present value to overshoot the set point.

14 Derivative term Process variables for different K d values (K i and K p held constant) are shown in Figure 4.7. The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain K d. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain K d. Figure 4.7 Process variable for different Kd values The derivative term is given by the Equation (4.15) (4.15) The derivative term slows the rate of change of the controller output. Derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controllerprocess stability. However, the derivative term slows the transient response of the controller. Also, the differentiation of a signal, amplifies noise and thus this term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are

15 50 sufficiently large. Hence an approximation to a differentiator with a limited bandwidth is more commonly used. Such a circuit is known as a phase-lead compensator Overview of methods There are several methods for tuning a PID loop. Controller tuning methods with its advantages and disadvantages are given in Table 4.2. Table 4.2 Tuning methods Method Advantages Disadvantages Manual Tuning Ziegler Nichols Software Tools Cohen- Coon No math required. Online method Proven Method. Online method Consistent tuning. Online or offline method. May include valve and sensor analysis. Allow simulation before downloading. Can support Non-Steady State (NSS) Tuning Good process models Requires experienced personnel Process upset, some trialand-error, very aggressive tuning Some cost and training involved Some math. Offline method. Only good for first order processes The most effective methods generally involve in the development of some form of the process model, and then choosing P, I, and D based on the dynamic model parameters. Manual tuning methods can be relatively inefficient, particularly if the loops have response times on the order of minutes or longer. The choice of method will depend largely on whether or

16 51 not the loop can be taken "offline" for tuning, and the response time of the system. If the system can be taken offline, the best tuning method often involves subjecting the system to a step change in input, measuring the output as a function of time, and using this response to determine the control parameters Manual tuning If the system must remain online, one tuning method is to first set K i and K d values to zero. Increase the K p until the output of the loop oscillates, then the K p should be set to approximately half of that value for a "quarter amplitude decay" type response. Then increase K i until any offset is corrected in sufficient time for the process. However, too much K i will cause instability. Finally, increase K d, if required, until the loop is acceptably quick to reach its reference after a load disturbance. However, too much K d will cause excessive response and overshoot. Table 4.3 Effects of increasing a tuning parameter Parameter Rise time Overshoot Settling time K p Decrease Increase Small change Steady-state error Decrease Stability Degrade K i Decrease Increase Increase Decrease significantly Degrade K d Minor Minor Minor No effect in Improve if decrease decrease decrease theory K d small A fast PID loop tuning usually overshoots slightly to reach the set point more quickly; however, some systems cannot accept overshoot, in which case an over-damped closed-loop system is required, which will

17 52 require a K p setting significantly less than half that of the K P setting causing oscillation. The effects of increasing a tuning parameters K P, K d and Ki is given in Table Ziegler Nichols method For more details on this topic, see ZieglerNichols method. Another heuristic tuning method is formally known as the ZieglerNichols method, introduced by John G. Ziegler and Nathaniel B. Nichols in the 1940s. As in the method above, the K i and K d gains are first set to zero. The P gain is increased until it reaches the ultimate gain, K u, at which the output of the loop starts to oscillate. K u and the oscillation period P u are used to set the gains. The formulas to calculate tuning parameters in P, PI, and PID controllers are given in Table 4.4. Table 4.4 Ziegler Nichols method Control Type K p K i K d P 0.50K u - - PI 0.45K u 1.2K p / P u - PID 0.60K u 2K p / P u K p P u / 8 These gains apply to the ideal, parallel form of the PID controller. When applied to the standard PID form, the integral and derivative time parameters T i and Td are only dependent on the oscillation period P u Limitations of PID Control While PID controllers are applicable to many control problems, and often perform satisfactorily without any improvements or even tuning, they can perform poorly in some applications, and do not in general provide

18 53 optimal control. The fundamental difficulty with PID control is that a feedback system, with constant parameters, and no direct knowledge of the process, and thus overall performance is reactive and a compromise while PID control is the best controller with no model of the process, better performance can be obtained by incorporating a model of the process. The most significant improvement is to incorporate feed-forward control with knowledge about the system, and using the PID only to control error. Alternatively, PIDs can be modified in more minor ways, such as by changing the parameters (either gain scheduling in different use cases or adaptively modifying them based on performance), improving measurement (higher sampling rate, precision, and accuracy, and low-pass filtering if necessary), or cascading multiple PID controllers. PID controllers, when used alone, can give poor performance when the PID loop gains must be reduced so that the control system does not overshoot, oscillate or hunt about the control set point value. They also have difficulties in the presence of non-linearities, may trade-off regulation versus response time, do not react to changing process behavior and lag in responding to large disturbances. Another problem faced with PID controllers is that they are linear, and in particular symmetric. So, the performance of PID controllers in non-linear systems (such as HVAC systems) is varies. For example, in temperature control, a common use case is active heating (via a heating element) but passive cooling (heating off, but no cooling), so overshoot can only be corrected slowly it cannot be forced downward. In this case the PID should be tuned to be over damped, to prevent or reduce overshoot, though this reduces performance (it increases settling time). A problem with the derivative term is that small amounts of measurement or process noise can cause large amounts of change in the output. It is often helpful to filter the measurements with a low-pass filter in

19 54 order to remove higher-frequency noise components. However, low-pass filtering and derivative control can cancel each other out. So reducing noise by instrumentation means is a much better choice. Alternatively, a nonlinear median filter may be used, which improves the filtering efficiency and practical performance. In some case, the differential band can be turned off in many systems with little loss of control. This is equivalent to using the PID controller as a PI controller. 4.4 PID CONTROLLER BASED INDUCTION MOTOR V/F CONTROL The design of PID controller for SVPWM inverter for induction motor V/f speed control is in Figure 4.8. Space Vector Modulation (SVM) was originally developed as a vector approach to Pulse Width Modulation (PWM) for three phase inverter (Munira Batool 2013). It is a well sophisticated technique for generating the sine wave that provides a higher voltage to the motor with lower total harmonic distortion. The Space Vector Pulse Width Modulation (SVPWM) method is an advanced PWM method and possibly the best among all the PWM techniques for variable frequency drive application. A proportionalintegralderivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. PID is the most commonly used feedback controller. Defining u (t) as the controller output, the final form of the PID algorithm is u(t) =MV(t) =K P e(t) + K i e( ) d( ) + K d. de(t)/dt (4.16) Where K p : K i : Proportional gain, Integral gain,

20 55 Integral limit is from 0 to t. K d : e : Derivative gain Error t : Time or instantaneous time (the present) The ZieglerNichols tuning method is used to tune the PID controller. The proportional, integral, and derivative terms are summed up in order to calculate the output from the PID controller. A PID controller calculates an "error" value as the difference between the measured process variable and the desired set point. The controller is tuned based on Integral of absolute value of the error (IAE). IAE= e(t).dt (4.17) The limits are from 0 to. The controller is tuned and calculated the values Kp, Ki and Kd are calculated based on IAE to produce reference frequency output proportional to the absolute error. Kp= 0.01, Ki = and Kd = The controller attempts to minimize the error by adjusting the process control inputs (Kumar et al 2013). PID controller is shown in Figure 4.9. A PID controller calculates an "error" value as the difference between the measured process variable and the desired set point. The controller attempts to minimize the error by adjusting the process control inputs.

21 56 Figure 4.8 Simulink model of PID controller based three phase Induction motor V/f speed control Figure 4.9 PID controller

22 57 Figure 4.10 SVM module The three phase AC is converted into DC by the rectifier. The DC is converted into controlled AC by three phase inverter. Inverter output is given to three phase induction motor. The PID controller is designed to give control signals to IGBTs present in the inverter module. The inverter output is V/f controlled, and it is given as input to three-phase induction motor. The speed is taken as feedback, and it is compared with the set speed. The error signal is given as input to the PID controller (Kumar et al 2013). The controller output is the reference frequency. The reference frequency and actual frequency are compared. It is learnt that the frequency error creeps in.v/f ratio maintains constant, based on reference speed Nref and frequency error. The V/f ratio is maintained constant as per the following Equation (4.18) V/f ratio = [Nref/5+ [frequency error x 0.5]. (4.18) Oscillator (OSC) is used to produce the frequency output based on V/f ratio. The frequency of the OSC is chosen to produce 6 pulses. OSC outputs are given as input to the SVM module. SVM module is shown in Figure When the first pulse comes the counter output in SVM module

23 58 will be 0. For the second pulse the counter output will be 1 and so on. Based on the V/f ratio OSC frequency is changed. The pulse widths of the OSC pulses are based on the V/f ratio. For example, when the pulse width is increased for pulse 1, the counter output remains 0 and the switches 1 and 4 will be in the conduction state till the counter output changes to 1. When the reference speed is increased the V/f ratio is constant (Soni et al 2013). But all 6 pulses widths of OSC will be reduced. So the counter value will be changed quickly. Due to this the duration of on time of IGBTs are reduced. T=Ton+Toff (4.19) F= 1/T (4.20) So the inverter output voltage is controlled based on Ton. When the on time is varied, frequency will also be varied. V/f control is achieved. 4.5 SIMULATION RESULTS AND INFERENCES In this part PID controller performance is analyzed. The nominal speed of the motor is 1440 RPM. PID controller reference frequency output of step change speed from 1500 RPM to 1800 RPM is shown in Figure Figure 4.11 PID controller reference frequency

24 59 The PID controller speed and frequency output have more overshoot. Settling time and rise time are also more for a set speed of 2000 RPM. Figure 4.12 shows the PID controller speed change from 1600 RPM to 1800 RPM. Figure 4.12 PID controller step response The speed reaches the 1800 RPM after 2.5 Sec. PID responses for various set speeds 1350 RPM, 1440 RPM, 1850 RPM and 2000 RPM under no load are shown in Figures 4.13, 4.14, 4.15 and 4.16 in chapter 4. PID responses for the same speeds under 50% loads are shown in Figures 4.17, 4.18, 4.19 and PID responses under full load are shown in Figures 4.21, 4.22, 4.23 and 4.24.

25 60 Figure 4.13 PID response for a set speed of 1350 RPM under no load Figure 4.14 PID responses for a set speed of 1440 RPM under no load

26 61 Figure 4.15 PID response for a set speed of 1850 RPM under no load Figure 4.16 PID responses for a set speed of 2000 RPM under no load A PID controller speed response to a set speed of 2000 RPM (above rated) at no load is analyzed.

27 62 Figure 4.17 PID response for a set speed of 1350 rpm under 50% load Figure 4.18 PID response for a set speed of 1440 RPM under 50% load

28 63 Figure 4.19 PID response for a set speed of 1850 RPM under 50% load Figure 4.20 PID response for a set speed of 2000 RPM under 50% load

29 64 Figure 4.21 PID response for a set speed of 1350 RPM under full load Figure 4.22 PID responses for a set speed of 1440 RPM under full load

30 65 Figure 4.23 PID response for a set speed of 1850 RPM under full load Figure 4.24 PID response for a set speed of 2000 RPM under full load PID controllers performance parameters for various speeds without load are given in Table 4.5.

31 66 Table 4.5 PID controller based system performance parameters for various speeds without load Set Speed (RPM) Actual Speed (RPM) Peak Speed (RPM) Rise Time (Sec) Settling Time (Sec) Over Shoot (%) Steady State Error (%) In no load the overshoot is more in 1350 RPM and steady state error is more for the set speed of 1440 RPM. PID controller performance is good for the set speed of 1850 RPM under noload.pid controllers performance parameters for various speeds with 50% load are given in Table 4.6. PID controllers performance parameters for various speeds with full load are given in Table 4.7. Table 4.6 PID controller based system performance parameters for various speeds with 50% load Set Speed (RPM) Actual Speed (RPM) Peak Speed (RPM) Over Shoot (%) Steady State Error (%) The overshoot and steady state error are more for the set speed of 1440 RPM under 50% load, compared to other set speeds. PID controller performance is good for the set speed of 2000 RPM, compared to other set speeds for this load.

32 67 Table 4.7 PID controller based system performance parameters for various speeds with full load Set Speed Actual Speed Peak Speed Over Shoot Steady State (RPM) (RPM) (RPM) (%) Error (%) While comparing other set speeds, the overshoot is more for 1440 RPM and the steady state error is higher for 1350 RPM under full load. The PID controller response is good for the set speed of 1850 RPM compared to other set speeds for the same load. 4.6 CONCLUSION In this chapter the three parts namely mathematical modeling of induction motor, PID controller design and application of PID controller for three phase induction motor speed control are discussed. The mathematical model is established for 0.5HP induction motor using dq reference frame concept. PID controller is designed and it is tuned by ZieglerNichols method for V/f control of induction motor. An SVM based inverter is used for this purpose. The controller gives control signals to inverter to maintain V/f ratio constant. It is understood from the simulation results that, in general the PID controller responses have high overshoot and steady state error in various speeds and loads.