chapter 4 pid controller based speed control of three phase induction...

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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

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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.

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(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 6.085

Stator inductance Ls 489.3e-3 H

Rotor inductance Lr 489.3e-3 H

Poles 4

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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.

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(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)

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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

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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.

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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.

4.3.1 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

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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

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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.

4.3.2 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

Kp : Proportional gain, a tuning parameter

Ki : Integral gain, a tuning parameter

Kd : Derivative gain, a tuning parameter

e : Error = SP PV

t : Time or instantaneous time (the present)

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Figure 4.4 shows the PID controller.

Figure 4.4 PID controller

4.3.2.1 Proportional term

Figure 4.5 Process variables for different Kp values

Process variables for different Kp values (Ki and Kd 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 Kp, called the proportional

gain.

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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 / Kp (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.

4.3.2.2 Integral term

Process variables for different Ki values (Kp and Kd 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

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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 (Ki)

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.

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4.3.2.3 Derivative term

Process variables for different Kd values (Ki and Kp 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 Kd. The magnitude of the contribution of the

derivative term to the overall control action is termed the derivative gain Kd.

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 controller-

process 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

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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.

4.3.2.4 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

No math required. Online method Requires experienced personnel

Ziegler�Nichols

Proven Method. Online method Process upset, some trial-and-error, very aggressive tuning

Software Tools

Consistent tuning. Online or offline method. May include valve and sensor analysis. Allow simulation before downloading. Can support Non-Steady State (NSS) Tuning

Some cost and training involved

Cohen-Coon

Good process models 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

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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.

4.3.2.5 Manual tuning

If the system must remain online, one tuning method is to first set

Ki and Kd values to zero. Increase the Kp until the output of the loop

oscillates, then the Kp should be set to approximately half of that value for a

"quarter amplitude decay" type response. Then increase Ki until any offset is

corrected in sufficient time for the process. However, too much Ki will cause

instability. Finally, increase Kd, if required, until the loop is acceptably quick

to reach its reference after a load disturbance. However, too much Kd will

cause excessive response and overshoot.

Table 4.3 Effects of increasing a tuning parameter

Parameter Rise time Overshoot Settling time

Steady-state error

Stability

Kp Decrease Increase Small change

Decrease Degrade

Ki Decrease Increase Increase Decrease significantly

Degrade

Kd Minor decrease

Minor decrease

Minor decrease

No effect in theory

Improve if Kd 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

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require a Kp setting significantly less than half that of the KP setting causing

oscillation. The effects of increasing a tuning parameters KP, Kd and Ki is

given in Table 4.3.

4.3.2.6 Ziegler–Nichols method

For more details on this topic, see Ziegler�Nichols method.

Another heuristic tuning method is formally known as the Ziegler�Nichols

method, introduced by John G. Ziegler and Nathaniel B. Nichols in the 1940s.

As in the method above, the Ki and Kd gains are first set to zero. The P gain is

increased until it reaches the ultimate gain, Ku, at which the output of the loop

starts to oscillate. Ku and the oscillation period Pu 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 Kp Ki Kd

P 0.50Ku - -

PI 0.45Ku 1.2Kp / Pu -

PID 0.60Ku 2Kp / Pu KpPu / 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 Ti and Td are only dependent on the oscillation period Pu.

4.3.3 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

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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

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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 proportional�integral�derivative 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) =KP e(t) + Ki e( ) d( ) + Kd. de(t)/dt (4.16)

Where

Kp : Proportional gain,

Ki : Integral gain,

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Integral limit is from 0 to t.

Kd : Derivative gain

e : Error

t : Time or instantaneous time (the present)

The Ziegler�Nichols 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 = 1.5750

and Kd = 11.52. 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.

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

Figure 4.9 PID controller

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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 4.10. When the first pulse comes the counter output in SVM module

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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 4.11.

Figure 4.11 PID controller reference frequency

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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 4.20. PID responses under full load are shown in Figures 4.21,

4.22, 4.23 and 4.24.

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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

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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.

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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

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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

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PID controller�s performance parameters for various speeds without

load are given in Table 4.5.

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Table 4.5 PID controller based system performance parameters for various speeds without load

Set Speed (RPM)

ActualSpeed (RPM)

Peak Speed (RPM)

Rise Time(Sec)

Settling Time (Sec)

OverShoot(%)

Steady State Error

(%)

1350 1400 1500 0.4 3 11.1 3.7

1440 1500 1550 0.52 2.5 7.6 4.2

1850 1860 1870 0.8 2 1.1 0.5

2000 2020 2070 1.2 2 3.5 1

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 controller�s

performance parameters for various speeds with 50% load are given in Table

4.6. PID controller�s 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 (%)

1350 1390 1450 7.4 3

1440 1500 1590 10.4 4.2

1850 1850 1890 2.2 0

2000 2000 2100 5 0

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.

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Table 4.7 PID controller based system performance parameters for various speeds with full load

Set Speed

(RPM)

Actual Speed

(RPM) Peak Speed

(RPM) Over Shoot

(%)Steady State Error (%)

1350 1375 1510 11.9 1.9

1440 1450 1620 12.5 0.7

1850 1850 2000 8.1 0

2000 2000 2200 10 0

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 d�q reference frame

concept. PID controller is designed and it is tuned by Ziegler�Nichols 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.

chapter 4 pid controller based speed control of three phase induction...

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