Download Temperature Modeling and Control of Induction Furnace System

Temperature Modeling and Control of Induction Furnace System
Rina Ristiana1), Arief Syaichu Rochman2)
1) UPT. Balai Pengolahan Mineral Lampung – LIPI
Jl. Ir. Sutami Km 15 Tanjung Bintang Lampung Selatan
2) Course control and intelligent system - STEI ITB
Jl. Ganesa 10 Bandung
[email protected], [email protected]
Abstract – Coreless induction furnace is induction heating applicated which used for melting of metal.
Induction furnace system is modeled here in the electric system dynamics and thermal system dynamics. The
electric system dynamics is described by induction furnace system in form of electrical circuits, including the
equivalent circuit of the furnace system, parallel resonant circuits and fed current inverter. Meanwhile, the
dynamics of the thermal system, which is a conductive heat transfer process, is described by energy balance
principle, included heat energy generated and heat loss. Temperature controller of induction furnace system is
designed here by setting the PWM, by which the power of induction furnace can be adjusted, to have a
controlled temperature. Three cases of PI controllers presented, there were, case I, the linear model PI linear
controller, case II, the linear model PI linear saturated controller, and case III, linear model PI with anti wind-up
controller. Each case is simulated with Simulink to find the best output response that can be achieved. For
purposes of this paper, as a case, induction furnace is used by melting of metals with scrap into ferrite with
carbon yield 0.22 %. To get a product with these specifications, it is necessary to have the temperature
controlled at 9120C. After analyzing the output response in all three cases, the case III, has the best output
response with maximum overshoot of 7%, response time at 2.9 second. For realization, the controller of the
induction furnace system which has a high proportional gain value requires careful consideration.
Keywords: Modeling, Anti Wind-up PI Controller, Induction Furnace, Induction Heating, Parallel Resonance.
I. INTRODUCTION
Induction heating is a non-contact heating process which is used a power source to
generate heat conductor. The basic principle of induction heating have been discovered and
applied to industrial process since 1920. After world war of second, the technology
depelopments so quickly, many researchers have been studied and applied the induction of
technology in support of the industry.
In the early nineteenth century, the phenomenon of induction heating was applied to
experimental melting metals. The early furnace consisted of circular hearth or trough, which
contained the molten metal of an annular ring, then substituted a crucible containing the
metal charge in place of the channel surround with a multi-turn coil its called induction
furnace, that a new approach was made by E. F Northrup [1]. Along with the development of
science applications of induction heating is used for melting of metals, heated metal, the
crystallization of metal (brazing) and welding.
Several studies that examine the performance of such an induction furnace, Umar Shami
in 2006 found a technique to calculate the coil parameters for heating. Hsu Chun in 2007 to
analyzed the efficiency of the power at the serial resonance circuit. Atighechi et al calculate
the optimal capacitance with a genetic algorithm. In 2011 the Arab Markadeh determined the
optimal capacitor resonance method with genetic algorithm.
The process of induction furnace melting metal today still manual in temperature
controlled. Though temperature controlled can determine the success of the process in getting
the metal smelting and chemical composition of the molten metal structural shape. To
provide a solution, it is designed to control temperature induction furnace system without
core by performing mathematical modeling of the processes it controls, where it would be
interesting to study further.
1
Furnaces being modeled in this research can be seen in Figure 1 that used induction
furnace with capacity of 500 kg which functioned for engineering research of metal and nonmetalic mineral processing, there is in UPT Balai Pengolahan Mineral Lampung LIPI.
Fig. 1 Induction furnace with capacity of 500 Kg
II. CORELESS INDUCTION FURNACES SYSTEM
In this research used coreless induction furnace. The main components of induction
furnace system are furnace and power converter as can be seen in Figure 2.
Power Converter
Furnace
Fig. 2 The Main Components
a.
Coreless Induction Furnace
Coreless induction furnace, one of the electric induction furnaces, which utilized heat
sources from an electric element that is used to heat the furnace melting of metals. The
furnace of heat transfer occured by conduction, where the heat transfer propagated from the
heat source (coil) to the metal to be melted [1]. From Figure 1 it can be seen that the
induction furnace is an induction coil of copper tubing equipped with shell, cooling water and
tilting furnace.
The induction furnace of contruction includes geometric system, electrical systems,
thermal systems and heat loss. Geometric system is mechanical parameters of the furnace.
Electrical systems functioned to determine the total heat energy generated by the current
induced eddy current induction heating takes place. Thermal system is generated heat directly
by the materials that would be merged resulting in an increase in temperature. Heat loss is
energy lost in the melting process.
2
b. Power Converter
Power converter is voltage devices or power source. From figure 1 it can be seen, the
power converter includes AC to DC rectifier, DC-link, DC to AC inverter and a capacitor
bank. [1] and [7]
Induction furnace system use three-phase rectifier with 6-pulse bridge SCR that is
enabled for voltage. DC-link called DC filter such inductance which used to DC voltage filter
that provided constant voltage of output terminal DC variable when filter circuit get source
voltage from rectifier. Inverter used a single phase ful bridge with four thyristors. If input
voltage inverter is constant, then output voltage can get with setting PWM inverter variable.
And the capacitors bank connected with furnace coil in parallel circuit that is intended to
correct the power factor.
III. MODELING OF INDUCTION FURNACES SYSTEM
An important stage in the analysis and design of control systems is the mathematical
modeling from the controlled process. Modeling of induction furnace system includes
electrical system dynamics and thermal system dynamics.
a. The electrical systems dynamics is described by induction furnace system in form of
electrical circuits, as shown in Figure 3, included equivalent circuit of furnace (RET), parallel
resonant circuits (RSP) and current fed inverter (FCI). Induction furnace like a simple
transformer principle, wherein the primary is coil, and the secondary is raw material to be
melted. These furnaces are designed so that induced current changed to heat that can be
melted raw materials. Thus, induction furnace can be translated into the furnaces equivalent
circuit (RET) that consists of equivalent resistance and equivalent inductance in series.
Q1
Q3
A
IDC
B
Q4
Q2
RESR
Req
C
Leq
Vo
RSP
RET
FCI with RSP
Figure 3. The Electric System Dynamics
Equivalent resistance (Req) is the number of coil resistance (Rcoil) with shunt resistance
(Rch), can be written by the equation; [2], [4] and [8]
𝑅𝑒𝑞 = 𝑅𝑐𝑜𝑖𝑙 + 𝑅𝑐ℎ
(1)
Where, resistance coil is coil resistivity (ρc) times length of coil (𝑙𝑐 = 𝜋𝐷𝑖𝑛 𝑁) divided by the
cross sectional coil (𝐴𝑐 = 𝜋𝑥 2 ) where, 𝑥 = 12𝑑𝑚 , the equation can be written as follows :
3
𝑅𝑐𝑜𝑖𝑙 =
𝜌𝑐 𝑙𝑐
(2)
𝐴𝑐
Shunt resistance is the resistance containing mutual inductance, where the number of turns
squared multiplied by resistance weights or resistance of raw materials to be melted (RL),
written by the following equation:
𝑅𝑐ℎ = 𝑁 2 𝑅𝐿 with; 𝑅𝐿 =
𝜌𝑚 𝑙𝑚
(3)
𝐴𝑚
Where, iron resistivity [𝜌𝑚 ] of 2.83 x 10-8, the length of the tube cross section [𝑙𝑚 = 2𝜋𝑥]
with [𝑥 = 12𝑑𝑚] half of the distance axis vertical sectional area tube [𝐴𝑚 = 𝐻𝑚 𝑑𝑥] with 𝑑𝑥 =
𝜌
𝑚
𝑑𝑜 = √𝜋𝜇𝑓
.
Equivalent inductance is the self inductance reduced by mutual inductance, the equation;
𝐿𝑒𝑞 = 𝐿1 − 𝑁𝐿𝑀
(4)
Self inductance;
𝐿1 =
𝜇𝑁 2 𝜋𝐷𝑖𝑛 2
(5)
4𝐻𝑖𝑛
And mutual inductance ;
√2𝐼𝑚 𝑅𝑐ℎ 𝜇
𝑚 𝐻𝑖𝑛 2𝜋𝑓
𝑁𝐿𝑀 = 𝐵
(6)
Parallel resonant circuit (RSP) is the furnace equivalent circuit that is connected parallel
to the capacitor bank as shown in Figure 3. That it coud be described as impedance capacitor
1
bank, 𝑍1 = 𝑅𝐸𝑆𝑅 + 𝑠𝐶 and impedance load (furnace), 𝑍2 = 𝑅𝑒𝑞 + 𝑠𝐿𝑒𝑞 . So the total
impedance is obtained:
𝑍𝑇 = 𝑍1 //𝑍2 =
𝑍1 𝑍2
(7)
𝑍1 +𝑍2
Output inverter voltage (Vo) with Fourier series can be written;
𝑉𝑜 = 𝑉𝑠 cos(𝜔𝑡) + ℎ𝑎𝑟𝑚𝑜𝑛𝑖𝑐𝑠
(8)
Vs is amplitude (maximum output inverter voltage) and ω = 2πf is the magnitude frequency.
From Fig.3, we can write output inverter voltage (Vo) is;
𝑉𝑜 = 𝑍𝑇 𝐼𝑠
(9)
And coil current is;
𝑉
𝐼𝑐𝑜𝑖𝑙 = 𝑍𝑜
(10)
2
Inverter with constant current input is also called fed current inverter that is a devices
inverter switching such as PWM that used to regulate coil current. For driver PWM inverter
4
is needed for produced a voltage pulse, by comparing the amplitude modulation between sine
or reference signal (Vref) is larger than triangle signal or carrier signal (Vtri). If Vref changed,
then PWM can be adjusted. Maximum inverter voltage is;
𝑉
𝑉𝑠 = 𝑚𝑎 𝑉𝑑𝑐 = 𝑉𝑟𝑒𝑓 𝑉𝑑𝑐
(11)
𝑡𝑟𝑖
𝑉
If signal PWM Gain; 𝐾𝑃𝑊𝑀 = 𝑉 𝑑𝑐 then inverter voltage maximum become;
𝑡𝑟𝑖
𝑉𝑠 = 𝐾𝑃𝑊𝑀 𝑉𝑟𝑒𝑓
(12)
Fed inverter current with parallel resonant circuit, when power is connected, the electrical
energy stored in the inductance and sent to the capacitor. Resonance occurred when the
inductance and the capacitance exchanged energy. The total amount of energy stored in the
resonance circuit remains unchanged. This amount is equal to the amount of energy stored in
the inductance and capacitance.
Power flowing to the load is effective power, whereby effective power (Peff) is;
𝑃𝑒𝑓𝑓 = 𝑉𝑒𝑓𝑓 𝐼𝑒𝑓𝑓 cos 𝛼 = 12𝐼𝑐𝑜𝑖𝑙 2 𝑅𝑒𝑞
(13)
Or effective power equal to the power input of the induction furnace, the equation can be
written;
𝑃𝑒𝑓𝑓 = 𝑃𝑖𝑛 = 12𝐼𝑐𝑜𝑖𝑙 2 𝑅𝑒𝑞
(14)
Input power of induction furnace can be controlled by regulating PWM. The reference
voltage (Vref) can be changed with regulating the control signal (Vcon) as the output PI
controller which is multiplied by a generated gain signal voltage reference (KG);
𝑉𝑟𝑒𝑓 = 𝐾𝐺 𝑉𝑐𝑜𝑛
(15)
Substitusi equation (15) to (12);
𝑉𝑠 = 𝐾𝑃𝑊𝑀 𝐾𝐺 𝑉𝑐𝑜𝑛
(16)
Coil current (Icoil) is;
𝐼𝑐𝑜𝑖𝑙 = 𝐼𝑐𝑜𝑖𝑙,𝑚𝑎𝑥 cos(𝜔𝑡 + 𝜑)
(17)
𝑉
Where 𝐼𝑐𝑜𝑖𝑙,𝑚𝑎𝑥 = 𝑍𝑠
(18)
2
Substitution eq (7) to (9); and (9) to (10) ;
1
𝑉𝑜
+𝑠𝐿
𝑒𝑞
𝑒𝑞
𝐼𝑐𝑜𝑖𝑙 = 𝑅
(19)
The differensial form is;
5
𝑉𝑜 (𝑡) = 𝐿𝑒𝑞
𝑑 (𝐼𝑐𝑜𝑖𝑙 )
𝑑𝑡
+ 𝑅𝑒𝑞 𝐼𝑐𝑜𝑖𝑙
(20)
b. The thermal system dynamics is describing heat energy balance principle that occurs
during the process of induction heating takes place, from the heat generated until heat loss.
The simply, based principles of the energy balance can be expressed by the following
equation:
ℎ𝑒𝑎𝑡 𝑖𝑛𝑝𝑢𝑡 = ℎ𝑒𝑎𝑡 𝑜𝑢𝑡𝑝𝑢𝑡 + ℎ𝑒𝑎𝑡𝑙𝑜𝑠𝑠
(21)
Heat input is an input resource for induction heating. Heat output is the total heat energy
during a period of melting metal per unit time. And heat loss is energy lost in the process of
melting. Referring to [4], [6] and [7] the related equation can be seen in table 1.
Table 1. Heat equation
Heat Input
=
1
𝐼 2𝑅
2 𝑐𝑜𝑖𝑙 𝑒𝑞
Heat Output
𝑚 𝑐 (𝑇ℎ − 𝑇𝑎 ) + 𝑚 𝐿𝑝𝑡 + 𝑘𝑠 𝐺𝑠
𝑡
Heat massa
𝑄𝑚 = 𝑚 𝑐 (𝑇ℎ − 𝑇𝑎 )
Heat latent heat
𝑄𝐿 = 𝑚 𝐿𝑝𝑡
Heat slag
𝑄𝑠 = 𝑘𝑠 𝐺𝑠
=
Heat loss
1
(𝑇 − 𝑇𝑎 )
𝑅𝑇 ℎ
𝑃𝑐 = 𝐼𝑐𝑜𝑖𝑙 2 𝑅𝑐𝑜𝑖𝑙
(𝑇ℎ − 𝑇𝑎 )
𝑃𝑓 =
𝑅𝑇
𝑃𝑤 = 𝑚𝑤 𝑐𝑤 𝑇𝑤
= 𝐼𝑐𝑜𝑖𝑙 2 𝑅𝑐𝑜𝑖𝑙 + 𝑚𝑤 𝑐𝑤 𝑇𝑤 +
Heatloss coil
Heatloss wall furnace
Heatloss cooling water
With table 1, equation (21) in the differential form;
𝐼𝑐𝑜𝑖𝑙,𝑚𝑎𝑥 2 𝐾𝑅 − 𝑑 = 𝑚 𝑐
𝑑𝜃
𝑑𝑡
+ 𝐽1 𝜃
(22)
1
Where, 𝜃 = 𝑇ℎ − 𝑇𝑎 , 𝑑 = 𝑚𝑤 𝑐𝑤 𝑇𝑤 , 𝐽1 = 𝑅 and
𝑇
1
𝐾𝑅 = (2 𝑅𝑒𝑞 − 𝑅𝑐𝑜𝑖𝑙 ).
c. Dynamics models of induction furnace system
Obtained from the calculation of Leq and Req is 2.7211 x 10-4 H and 0.0481Ω. The
importance of the value of time constant coil is τc = 0.0056 ≈ 6 x 10-3 second. While the value
of m = 0.5 ton, c = 450 J / kg 0C and J1 = 38.5347 mho time constant value obtained is τT =
5.83 ≈ 6 second. It can be concluded that the time constant τc coil has a faster response time
compared to 1000 times τT constant temperature, so that the Leq can be neglected. So that if
𝑢 = 𝑉𝑐𝑜𝑛 then eq (18) become;
1
𝐼𝑐𝑜𝑖𝑙,𝑚𝑎𝑥 = 𝑅 𝐾𝑃𝑊𝑀 𝐾𝐺 𝑢
(23)
𝑒𝑞
6
Substitution eq (22) to (23) ;
2
1
𝑑𝜃
(𝑅 𝐾𝑃𝑊𝑀 𝐾𝐺 ) 𝑢2 𝐾𝑅 − 𝑑 = 𝑚 𝑐
𝑑𝑡
𝑒𝑞
+ 𝐽1 𝜃
(24)
2
̃ = ( 1 𝐾𝑃𝑊𝑀 𝐾𝐺 ) 𝐾𝑅
If 𝐾
𝑅
𝑒𝑞
Then eq (24) become;
̃𝑢2 – 𝑑 = 𝑚 𝑐
𝐾
𝑑𝜃
𝑑𝑡
+ 𝐽1 𝜃
(25)
Equation (25) is nonlinear system, linearization system, by making the following replacement
variables;
̃𝑢2
𝑣=𝐾
(26)
Substitution eq (26) to (25) ;
𝐽
1
1
𝜃̇ = − 𝑚𝑐1 𝜃 + 𝑚𝑐 𝑣 − 𝑚𝑐 𝑑
(27)
Equation (24) by entering values, obtained a = 0.1713, b = c = 0.0044, the equation can be
written;
𝜃̇ = −0.1713 𝜃 + 0.0044 𝑣 − 0.0044 𝑑
(28)
V. TEMPERATURE CONTROL DESIGN
In designing the good controller needs analysis for getting overview system responses
against control actions. Control system is needed to fix the system dynamic response in order
to obtain the desired output signal. Controller design can be seen in Figure 4.
θe
vd
θd
+
Sensor
Tranducer
u
Controller
-
Actuator
v
d
vT
θ
Sensor
Tranducer
Plant
Fig. 4 Block diagram of the control design
The PI controller is a combination of the Proportional controller and Integral controller,
which has the following equation;
𝑡
𝑣(𝑡) = 𝐾𝑃 𝑒(𝑡) + 𝐾𝐼 ∫0 𝑒(𝜏) 𝑑𝜏
(29)
7
Where the error state is determined from difference between the desired temperature and
actual temperature,
𝑒 = 𝜃𝑒 = 𝜃𝑑 − 𝜃
(30)
To determine the control signals, by substituting equations (26) and (29), obtained;
𝐾𝑃 𝜃𝑒
̃
𝐾
𝑢=√
𝑡
+
𝐾𝐼 ∫0 𝜃𝑒 (𝜏) 𝑑𝜏
(31)
̃
𝐾
In the implementation of the PI controller, PI controllers in this thesis are designed in
three cases. Case I, the linear model of PI control linear is plant ideal and controller ideal.
Case II, the linear model of PI control saturated is plant ideal and controller saturated. And
case III, the linear model of PI anti wind up control is plant saturated and controller saturated,
that can see in Figure 5.
θd
θe
KTKp
+
θ(s)
U(s)
-
Plant
++
𝐾𝑖
𝑠 + 𝐾𝑖
Fig. 5 Block diagram of case III
From Figure 5, closed loop system;
𝜃(𝑠)
𝜃𝑑 (𝑠)
𝑏 𝐾𝑇 𝐾𝑝 (𝑠+𝐾𝑖 )𝑠+
2
𝑠 +(𝑎+𝑏 𝐾𝑇 𝐾𝑝 )𝑠+𝑏𝐾𝑇 𝐾𝑝 𝐾𝑖
=
(32)
With characteristic order-2, so transient response get;
𝑠 2 + (𝑎 + 𝑏 𝐾𝑇 𝐾𝑝 )𝑠 + 𝑏𝐾𝑇 𝐾𝑝 𝐾𝑖 = 𝑠 2 + 2𝜁𝜔𝑛 𝑠 + 𝜔𝑛 2
(33)
Proportional gain is;
𝐾𝑝 =
2𝜁𝜔𝑛 −𝑎
(34)
𝑏𝐾𝑇
Integral gain is;
𝜔 2
𝐾𝑖 = 𝑏𝐾 𝑛𝐾
(35)
𝑇 𝑝
To achieved maximum overshoot Mp = 2% and ts = 2 second, then damping ratio ζ = 0.7797
and damping factor ωn = 2.5651 rad/s. the PI controller parameter are set to 72928 for Kp and
1.7185 for Ki.
VI. RESPONSE ANALYSIS SYSTEM
8
Analysis of system response in this research, observing the system's response by
comparing input response and output response in the time domain. The output response is
expected to follow the reference signal, starting at the beginning of the initial conditions, this
was done as performance system evaluation of control system design was designed.
For purposes of this research, as a case, induction furnace is used by melting of metals
with scrap into ferrit where the specification had carbon content 0.22%. In accordance with
the phase diagram that the success rate of the metal melting process to get a product with
these specification, it is necessary to have the temperature controller at 9120C. The
parameters used for the simulation can be seen in Table 2.
Vcon (u)
Vref
KG
Kpwm
Id
Vdc
Kp
Ki
Tabel 2. Parameter
Control signal
Reference voltage signal
Reference of gain generated signal
Generated PWM gain
DC output current
DC output voltage
Proporsional gain
Integral gain
0 – 15
4
0.267
28.02
57.363
140.115
72928
1.7185
Volt
Volt
Ampere
Volt
Output respon can seen in figure 6, explaining that for case I, the output response with
desired temperature θd(t) is given set point at 9120C, then the output response or actual
temperature θ(t) reaches 9120C on time contant 1.5detik, 11.13% maximum overshoot. The
output response of case I is considered good, but seen in figure 7, the control response in case
I happened wind-up that exceeded the maximum limit specified control signals. So it can be
seen that the maximum limit of the control signals with case I signal control worked nearly
double the approximately 37V. The error response of case 1 can be seen in figure 8, which
explained that the error response can be closed to zero (10C) at 3.27 second.
Case II is designed for resetting wind up that occurred of the control signal. As can be
seen in figure 7, the control sinyal was given value limit so wind up can be reset. However,
time response is slower about 12 second. Likewise output response is worse than case I,
where maximum overshoot is 53.51% and time response is 2.9 second, it was not as
expected, shown in figure 6. The explained that the error response can be close to zero (10C)
at 11 second.
Case III is designed to get well output. The output response can be seen in figure 6, the
output response of case III is better than other cases, where the maximum overshoot is
happening at 7% and time response at 2.9 second. For control response can be seen in figure
7, it can be constant at 5 second. And the explained that the error response can be close to
zero (10C) at 4.5 second.
The third of these cases can be made in table, the performance characteristics of the
designed control can be seen in table 3.
ature (0C)
Cases
I
II
III
Tabel 3. Preformansi karakteristik kendali
Models
The linear model of PI linear control
The linear model of PI saturated control
The linear model of PI anti windup control
Specification
9
Mp
11.13%
53.51%
7%
2%
ts
1.5 s
2.9 s
2.9 s
2s
Fig. 6 The output response
Voltage (V)
Case I
Case II
Case III
Time (second)
Fig.7 The Control Signal Response
Temperature (0C)
Case I
Case II
Case III
Time (second)
Fig.8 The Error Response
VII. CONCLUSION
Coreless induction furnace system is induction heating applications that is used to melting
of metals. The main components are induction furnace and power converter. The induction
furnace was constructed that included geometric system, electric system and thermal system.
The power converter included AC to DC rectifier, DC-link, DC to AC inverter ang capacitor
bank.
The performed modeling in this research that modeled system dynamic that included
electric system dynamic and thermal system dynamic. The time response of coil was faster
1000 times than the time response of temperature, then coil time response is ignored.
10
Dynamics system model is order-1 and nonlinearity, so it must be done linearization system,
by making replacement variable.
PI controller was designed in three cases, each cases simulated with Simulink. One of all
cases were explained that case III have the best output response been achieved. For
realization, the controller of the induction furnace system which has a high proportional gain
value requires careful consideration. This problem can be further realized.
REFERENCE
[1] Ahmed Mohmed El-Sharkawy, (2008): Industrial Electronics Applications (Design and
Simulation of Coreless Induction Furnace), Thesis Program Master, Alexandria
University.
[2] Arab Markadeh G. R., Daryabeigi E., (2011): An Optimal Selection of Induction Heating
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