Sage Journals: Discover world-class research

Abstract

In order to improve quality in manufacturing, the measuring instruments used in production process should regularly be monitored and corrected according to international or national standards. Calibration of high-voltage equipment and precise measurements of DC high voltages are accomplished by standard voltage divider. Self-heating effect is the main error source of measurement in high-voltage DC resistive dividers. Therefore, precise control systems should be designed to keep stability of the ambient temperature and to regulate the heat distribution along the high-voltage DC resistive divider. For this purpose, a heat controlled resistive divider whose input voltage (V_in) is up to 5 kV was designed. This study is focused on heat convention and the dissipation model of the resistive divider and executes some control simulations under various conditions that aim to find the appropriate control method. Responses of the high-voltage DC resistive divider model are compared with and are validated by the responses of the designed actual system. The model provides us faster analyze and design solutions for novel methods. In this way, analyzing and controlling higher voltage dividers, such as 100 KV, will reduce just into a parameter change on the model. The fuzzy control method is suggested since the system dynamic has non-linear characteristics. Fuzzy temperature difference controller keeps temperature at a certain degree where fuzzy vertical temperature gradient controller keeps vertical temperature gradient around zero. Actual system and model responses for the fuzzy control are compared and interpreted.

Keywords

High-voltage DC resistive divider self-heating measurement uncertainty heat control fuzzy controller heat and mass transfer model

Introduction

Resistive self-heating effects cause measurement uncertainties in high-voltage DC resistive divider (HVDC-ResDiv) systems.^1,2 Calibration of equivalent measurement devices and precise measurements of DC HVs are realized by standard HVDC-ResDiv.^3–⁵ HVDC-ResDiv designed for calibration of other reference systems should have a measurement uncertainty of less than 0.002%. Therefore, resistor modules’ thermal coefficients should be lowest, and temperature gradient caused by the self-heating should be reduced to reasonable levels.⁶ Power dissipation results in temperature difference (TD) in overall resistors and vertical temperature gradient (VTG) along the resistor block and significantly threatens precise measurements.^7,8 Individual contribution of TD and VTG to the measurement uncertainties is emphasized in literature.^1,3,7–10

In this purpose, several TD control techniques are proposed in order to reduce TD errors.^1,8,10 By means of a shielded structure of the resistors and TD control, a 2-ppm measurement uncertainty is achieved.¹⁰

Effects of noise rejection techniques and partial discharge measurements on HVDC cables should be investigated.¹¹ Small-signal model of an HVDC system with angle control and an HVDC system with constant DC voltage control are proposed.¹² Improvement in stability of HVDC systems by convenient control strategies is essential for providing standards.^13,14

Mathematical modeling and simulation of heat and mass transfer offers definitely appropriate controller designs.^15,16

Control method is getting important to maintain uncertainty arisen from heating around ppm levels.^1,10 However, literature mention on neither TD nor VTG control methods. For this reason, an HVDC-ResDiv, which adjusts its TD and VTG by controllers and provides a consistent measurement, is realized.¹⁷

Physical description and numerical calculation of a system’s heat and mass transfer characteristics under influence of temperature gradients allow us critical values of the temperature gradients and critical velocities of the convection. In the organization of complex systems, mathematical modeling and simulation are required.²¹ In this way, analyzing dissimilar operational modes could be possible, and statement of potential errors and their adjustment is achieved without additional costs of real conditions.²² Heat conduction model is commonly employed in analysis and designing of nano transistors. Simulations and improvements in the layer parameters could allow reducing self-heating effect and provide a considerable enhancement in the temperature performance of metal–oxide–semiconductor field-effect transistors (MOSFETs).²³ Modeling HVDC systems provides more rapid and accurate designs that overcome the deficiencies, or combining and observing the advantages, of the previous systems.^24–26

Artificial intelligence is widely used as a method of self-heating control, feature extraction or classification of internal-corona-surface discharges in HVDC applications.²⁷ Fuzzy-based systems provide effective controllers for uncertain non-linear systems.²⁸ Application of fuzzy logic–based controllers on first- and second-order unknown non-linear systems is one of the state-of-art research studies on control theory.^29,30

In this study, heat conduction–convention and mass transfer model of the 5-kV HVDC-ResDiv is extracted: Heat conduction and energy equations are determined. General expression of system’s dynamic equations is given in terms of state-space equations and is converted to Simulink equivalent block model. A series of control simulations are realized under various conditions that aim to find the appropriate control method.

In the control of TD and VTG, fuzzy and proportional–integral–derivative (PID) methods are addressed. The study lies heavily on fuzzy inference systems (FISs) in order to satisfy non-linear relations³¹ of the HVDC-ResDiv control system. Fuzzy TD and fuzzy VTG controllers are developed, and the results for the actual system and the model are compared and interpreted.

Because of the non-linear system structure, proposed fuzzy control provides a contribution to accurate measurement of HVDCs. Each control experiment and returning of the system to initial conditions takes half of the hours of attendance. Modeling and simulation of the system accelerates design and parameter changes and allows rapid application of new control methods.

Heat distribution controlled HVDC-ResDiv platform

Heat distribution control of the system is ensured by keeping TD at a constant degree and by reducing VTG around zero. In this purpose, a system composed of two chambers filled with transformer oil, a motor pump, Peltier effect heat pump (PHT) and their control software is designed (Figure 1). Transfer of the heat through the HVDC-ResDiv is carried out with both forced convection and conduction.³² Total resistors (TRs) are kept in heat producer cylinder tank (HPCyT) on the left, where Q_P (W), origin of TD (°C) and VTG (°C), is removed from cooling tank (CoT) on the right: circulation pump (CP) forces heated oil and convects Q_P to CoT and PHTs and cooling fans on the CoT transfer it to the outer ambiance.

Figure 1.

HVDC-ResDiv (5 KV).

T_upper and T_lower sensors of the HPCyT determine TD and VTG values (Figure 2), (equation (2)).

Features of the mentioned HVDC-ResDiv system are listed:¹⁷

V_in is up to 5 kV

TR is 10 MΩ: 9 × 1 MΩ + 9 × 100 kΩ + 10 × 10 kΩ

HV arm resistors: 9990 kΩ

Low-voltage (LV) arm (output) resistor: R_low: 10 kΩ

Voltage division ratio (VDR): 1000:1

Max V_o: 5 V.

One of the most critical assisting tools in the analysis and design of measurement systems and measuring instrumentation is the mathematical modeling.³³

Figure 2.

Heat distribution in the temperature controlled 5-kV HVDC-ResDiv system.

Heat conduction and convention model of the 5-kV HVDC-ResDiv

Extraction of mathematical model and its simulation, which allows analysis and evolution of various operating modes and behaviors of systems, is essential in the design of complex installations.^34,35

Heat conduction and energy equations

Q_P is about 2.5 W, and a trivial amount is stored in the resistors, and the rest is transferred to the liquid (Figure 1). When the temperature of the outer ambiance is cooler than the inner ambiance of the HPCyT, a certain part of the heat is transferred to the outer ambiance through the lateral surface of the tank. Another part of the heat is stored on the cylinder tank (CyT) and on the liquid. The rest is brought to the CoT by means of the CP. Similarly, the heat which arrived is stored in the liquid of the CoT and on its aluminum body. The rest is transferred from the system by means of PHTs. Heat is thrown from the heated surface of the PHT by forced convection which uses cooling fans. Related temperatures are given in the nomenclature.

The temperatures and heat conduction in the temperature controlled 5-kV HVDC-ResDiv system are illustrated in Figure 2.

General heat equation describes energy exchanges as follows (equation (1))

E_{s} = E_{i} - E_{o} + E_{p}

(1)

Assumptions:

1. In the experiments, $T_{OCyT}$ is accepted as average of $T_{Upper}$ and $T_{Lower}$ sensorial results (Figure 1). Thus

T_{Upper} + T_{Lower} = 2 * T_{OCyT}

(2)

2. Referring to the experimental observations of the system (Figure 3), relation between difference of $T_{Upper}$ and $T_{Lower}$ and V_m and V_p is as follows (Figure 3).

According to observations given in Figure 3, a linear empiric approximation of $T_{Upper} - T_{Lower}$ as a function of $V_{m} and V_{P}$ which provides the behavior of the $T_{Upper} - T_{Lower}$ dynamic is as follows

\begin{matrix} T_{Upper} - T_{Lower} ≅ f (V_{m}, V_{P}) \\ = \frac{[(- 0.15 * V_{m} + 1) + (0.08 * V_{P} - 0.56)]}{2} \end{matrix}

(3)

Solution of equations 2 and 3 yields

T_{Upper} = T_{OCyT} + \frac{f (V_{m}, V_{P})}{2}

(4)

T_{Lower} = T_{OCyT} - \frac{f (V_{m}, V_{P})}{2}

(5)

Figure 3.

Variation of $T_{Upper} - T_{Lower}$ in the operation ranges of V_m and V_p.

Heat distribution in the HPCyT

All heat expressions in this section refer to the quantity of heat in a unit time (W)

Q_{P} ≅ 2.5

(6)

Q_{C 1} = \frac{T_{S 1} - T_{OCyT}}{R_{1}}

(7)

Referring to the direction of the motor pump, HPCyT will remove the heat, $Q_{EOCyT}$ and receives a heat of colder oil mass, $Q_{IOCyT}$ (Figure 2). Let us define the net heat as

Q_{net} = Q_{EOCyT} - Q_{ICyT}

(8)

Q_{C 2} = \frac{T_{OCyT} - T_{S 2}}{R_{2}}

(9)

Q_{EOCyT} = mc (T_{Lower} - T_{OCoT})

(10)

Q_{EOCyT} = mc (T_{OCyT} - \frac{f (V_{m}, V_{P})}{2} - T_{OCoT})

(11)

Q_{ICyT} = mc (T_{OCoT} - T_{Upper})

(12)

Q_{ICyT} = mc (T_{OCoT} - T_{OCyT} - \frac{f (V_{m}, V_{P})}{2})

(13)

\begin{matrix} Q_{net} = mc (T_{OCyT} - \frac{f (V_{m}, V_{P})}{2} - T_{OCoT}) \\ - mc (T_{OCoT} - T_{OCyT} - \frac{f (V_{m}, V_{P})}{2}) \end{matrix}

(14)

Q_{net} = 2 mc (T_{OCyT} - T_{OCoT})

(15)

Q_{C 3} = \frac{T_{S 3} - T_{\infty}}{R_{3}}

(16)

Q_{t 4} = \frac{T_{OCoT} - T_{S 4}}{R_{4}}

(17)

Q_{t 5} = \frac{T_{k} - T_{\infty}}{R_{k}}

(18)

Q_{t, c} = \frac{T_{S 5} - T_{C}}{R_{2 t, c}}

(19)

Q_{h} = P_{in} + Q_{C}

(20)

Q_{c} = Q_{max} (1 - \frac{T_{h} - T_{c}}{Δ T_{max}})

(21)

Heat distribution could be grouped into two sections: in HPCyT and in CoT.

Heat distribution in the CyT

Heat stored on the resistors is

C_{TR} \frac{d T_{TR}}{dt} = Q_{P} - Q_{C 1}

(22)

C_{TR} \frac{d T_{TR}}{dt} = Q_{P} - \frac{T_{S 1} - T_{OCyT}}{R_{1}}

(23)

Heat stored in the fluid (oil) of HPCyT (W): Heat stored in the oil of HPCyT is equal to the difference of incoming and outgoing heats to the fluid in the tank (equation (24))

C_{OCyT} \frac{d T_{OCyT}}{dt} = Q_{C 1} - Q_{C 2} - Q_{net}

(24)

\begin{matrix} C_{OCyT} \frac{d T_{OCyT}}{dt} = \frac{T_{S 1} - T_{OCyT}}{R_{1}} - \frac{T_{OCyT} - T_{S 2}}{R_{2}} \\ - 2 mc (T_{OCyT} - T_{OCoT}) \end{matrix}

(25)

Heat stored in the side walls of HPCyT

C_{CyT} \frac{d T_{CyT}}{dt} = Q_{C 2} - Q_{C 3}

(26)

Therefore

C_{CyT} \frac{d T_{CyT}}{dt} = \frac{T_{OCyT} - T_{S 2}}{R_{2}} - \frac{T_{S 3} - T_{\infty}}{R_{3}}

(27)

Heat distribution in the CoT

The net heat entering CoT will be equal to heat stored in the CoT and the heat removed from the HPCyT by forced convection. Stored heat of the fluid inside the CoT is described by the equation (28)

C_{OCoT} \frac{d T_{OCoT}}{dt} = Q_{net} - Q_{t 4}

(28)

C_{OCoT} \frac{d T_{OCoT}}{dt} = 2 mc (T_{OCyT} - T_{OCoT}) - \frac{T_{OCoT} - T_{S 4}}{R_{4}}

(29)

Heat stored in the body of the CoT is expressed by equation (30)

C_{CoT} \frac{d T_{CoT}}{dt} = Q_{t 4} - Q_{t, c}

(30)

where

C_{CoT} \frac{d T_{CoT}}{dt} = \frac{T_{OCoT} - T_{S 4}}{R_{4}} - \frac{T_{S 5} - T_{C}}{R_{2 t, c}}

(31)

Because $R_{t, c} \frac{\partial^{2} Ω}{\partial u^{2}}$ is too low, $T_{S 5} ≅ T_{C}$ assumption could be acceptable. All the heat in the outer surface of CoT is completely transferred from T_c to T_h. For this reason, $Q_{C}$ can be used directly instead of the thermal load $Q_{t, c}$ drawn by the Peltier from the system.

$Q_{t, c}$ is found in the catalog details of the Peltier (W).³⁶

Thus, the modified form of equation (30) is as follows

C_{CoT} \frac{d T_{CoT}}{dt} = Q_{t 4} - Q_{C}

(32)

C_{CoT} \frac{d T_{CoT}}{dt} = \frac{T_{OCoT} - T_{S 4}}{R_{4}} - Q_{max} (1 - \frac{T_{h} - T_{c}}{Δ T_{max}})

(33)

Heat stored in the cooling fins

C_{k} \frac{d T_{k}}{dt} = Q_{h} - Q_{t 5}

(34)

C_{k} \frac{d T_{k}}{dt} = P_{in} + Q_{max} (1 - \frac{T_{h} - T_{c}}{Δ T_{max}}) - \frac{T_{k} - T_{\infty}}{R_{k}}

(35)

Expression of heat transitions in terms of state equations for the system is as follows.

Assumptions

x_{1} = T_{TR} ≅ T_{S 1}

(36)

x_{2} = T_{OCyT}

(37)

x_{3} = T_{CyT} ≅ T_{S 2} ≅ T_{S 3}

(38)

x_{4} = T_{OCoT}

(39)

x_{5} = T_{CoT} ≅ T_{S 4} ≅ T_{S 5} ≅ T_{c}

(40)

x_{6} = T_{h} ≅ T_{k}

(41)

and the empiric equation for the temperature gradient is

\begin{matrix} x_{7} = T_{upper} - T_{lower} \\ = \frac{[(- 0.15 * V_{m} + 1) + (0.08 * V_{P} - 0.56)]}{2} \end{matrix}

(42)

HPCyT and CoT equations are expressed in terms of state variables, respectively

{\overset{\cdot}{x}}_{1} = \frac{Q_{P}}{C_{TR}} - \frac{x_{1}}{R_{1} C_{TR}} + \frac{x_{2}}{R_{1} C_{TR}}

(43)

\begin{matrix} {\overset{\cdot}{x}}_{2} = \frac{x_{1}}{R_{1} C_{OCyT}} - \frac{1}{C_{OCyT}} (\frac{1}{R_{1}} + \frac{1}{R_{2}} + 2 m c_{O}) x_{2} \\ + \frac{x_{3}}{R_{2} C_{OCyT}} + \frac{2 m c_{O} x_{4}}{C_{OCyT}} \end{matrix}

(44)

{\overset{\cdot}{x}}_{3} = \frac{x_{2}}{R_{2} C_{CyT}} - (\frac{1}{R_{2}} + \frac{1}{R_{3}}) \frac{x_{3}}{C_{CyT}} + \frac{T_{\infty}}{R_{3} C_{CyT}}

(45)

And CoT equations become

{\overset{\cdot}{x}}_{4} = \frac{2 m c_{O} (x_{2} - x_{4})}{C_{OCoT}} - \frac{x_{4}}{R_{4} C_{OCoT}} + \frac{x_{5}}{R_{4} C_{OCoT}}

(46)

{\overset{\cdot}{x}}_{5} = \frac{x_{4}}{R_{4} C_{CoT}} - \frac{x_{5}}{R_{4} C_{CoT}} - \frac{Q_{max}}{C_{CoT}} (1 - \frac{x_{6} - x_{5}}{Δ T_{max}})

(47)

{\overset{\cdot}{x}}_{6} = \frac{P_{in}}{C_{k}} + \frac{Q_{max}}{C_{k}} (1 - \frac{x_{6} - x_{5}}{Δ T_{max}}) - (\frac{x_{6} - T_{\infty}}{C_{k} R_{k}})

(48)

Here, general expression of system’s dynamic equations is

\begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + Bu (t) \\ y (t) = Cx (t) + Du (t) \end{matrix}

(49)

For convenience in representation of the Matlab Simulink blocks, state-space equations of the heat model are modified as follows

{\overset{\cdot}{x}}_{1} = \frac{1}{C_{TR}} [Q_{P} - \frac{1}{R_{1}} (x_{1} - x_{2})]

(50)

\begin{matrix} {\overset{\cdot}{x}}_{2} = \frac{1}{C_{OCyT}} \\ [\frac{1}{R_{1}} (x_{1} - x_{2}) - \frac{1}{R_{2}} (x_{2} - x_{3}) - 2 m c_{O} (x_{2} - x_{4})] \end{matrix}

(51)

{\overset{\cdot}{x}}_{3} = \frac{1}{C_{CyT}} [\frac{1}{R_{2}} (x_{2} - x_{3}) + \frac{1}{R_{3}} (T_{\infty} - x_{3})]

(52)

{\overset{\cdot}{x}}_{4} = \frac{1}{C_{OCoT}} [2 m c_{O} (x_{2} - x_{4}) - \frac{1}{R_{4}} (x_{4} - x_{5})]

(53)

{\overset{\cdot}{x}}_{5} = \frac{1}{C_{CoT}} [\frac{1}{R_{4}} (x_{4} - x_{5}) + Q_{max} (1 - \frac{x_{6} - x_{5}}{Δ T_{max}})]

(54)

\begin{matrix} {\overset{\cdot}{x}}_{6} = \frac{1}{C_{k}} \\ [\frac{- Q_{max}}{Δ T_{max}} (x_{6} - x_{5}) - \frac{1}{R_{k}} (x_{6} - T_{\infty}) + P_{in} + Q_{max}] \end{matrix}

(55)

The P_in power is applied to the Peltiers. Resistors of the Peltiers are about 5.57Ω. V_p varies between 0 and 14 V. Thus, P_in is between 0 and 78 W. The motor’s mass flow rate $m$ is taken as the input variables,³⁷ and the outer ambiance temperature $T_{\infty}$ and the heat generated in the resistances Q_p are constants. In addition, $T_{Upper} - T_{Lower}$ is defined by the empiric equation (equation (3)).

$V_{m}$ range is between 0 and 12 V and the corresponding $m$ is between 0 and 0.228 kg/sn. The transformation coefficient is as follows:

K18 = 0.228/12 = 0.019. Thus, m = 0.019 $V_{m}$

2 m c_{O} = 0.038 c_{O} V_{m}

(56)

P_{in} = 7.84 V_{p}

(57)

\begin{matrix} Q_{max} = 0.52 P_{in} = 4.0768 V_{p} [38] \\ Q_{c} = 4.0768 V_{p} (1 - \frac{x_{6} - x_{5}}{Δ T_{max}}) \end{matrix}

(58)

Thus, equations are reinterpreted as

{\overset{\cdot}{x}}_{1} = \frac{1}{C_{TR}} [Q_{P} - \frac{1}{R_{1}} (x_{1} - x_{2})]

(59)

\begin{matrix} {\overset{\cdot}{x}}_{2} = \frac{1}{C_{OCyT}} \\ [\frac{1}{R_{1}} (x_{1} - x_{2}) - \frac{1}{R_{2}} (x_{2} - x_{3}) - 0.038 c_{O} V_{m} (x_{2} - x_{4})] \end{matrix}

(60)

{\overset{\cdot}{x}}_{3} = \frac{1}{C_{CyT}} [\frac{1}{R_{2}} (x_{2} - x_{3}) + \frac{1}{R_{3}} (T_{\infty} - x_{3})]

(61)

{\overset{\cdot}{x}}_{4} = \frac{1}{C_{OCoT}} [0.038 c_{O} V_{m} (x_{2} - x_{4}) - \frac{1}{R_{4}} (x_{4} - x_{5})]

(62)

{\overset{\cdot}{x}}_{5} = \frac{1}{C_{CoT}} [\frac{1}{R_{4}} (x_{4} - x_{5}) + 4.0768 V_{p} (1 - \frac{x_{6} - x_{5}}{Δ T_{max}})]

(63)

\begin{matrix} {\overset{\cdot}{x}}_{6} = \frac{1}{C_{k}} \\ [\frac{- 4.0768 V_{p}}{Δ T_{max}} (x_{6} - x_{5}) - \frac{1}{R_{k}} (x_{6} - T_{\infty}) + 7.84 V_{p} + 4.0768 V_{p}] \end{matrix}

(64)

As output, the temperature of the oil in HPCyT where the resistances are found is taken as T₂, and the TD along the resistances is taken as T_upper − T_lower. The Simulink model based on the equations provides viability of the model in terms of prescription blocks that illustrate the physical relations between the variables.³⁸ The Simulink equivalent block model is shown in Figure 4.

Figure 4.

Equivalent Simulink block diagram of the system.

The coefficient values of the equivalent block diagram are given in Table 1.

Table 1.

Coefficient values in the equivalent block model.¹⁷

K1	1/126.06	K5	1/661.47	K9	1/0.797	K13	0.52
K2	1/22,019.26	K6	1/514.71	K10	1/0.1096	K14	1/0.1248
K3	1/553.113	K7	1/0.44248	K11	1/67	K15	2307
K4	−1/4071.52	K8	1/0.0415	K12	1	K16	4
						$Δ T_{max}$	67

Control methods

PID and fuzzy methods carry out the process to determine the suitable control for the system.

PID (three terms) control method

The PID control algorithm is composed of three parameters: Proportional (P) + Integral (I) + Derivative (D) terms.³⁹ The P parameter defines return to mentioned error where the I defines the response to the sum of recent errors and the D defines the reaction to the error rate. Total reaction of the PID adjusts the output of the system to the desired value.

Fuzzy control method

Fuzzy control generally deals with control of complex non-linear systems, which are hard to express by accurate mathematical models.^40,41 FIS is a linguistic instruction–oriented system which is composed of fuzzification, fuzzy decision-making (conjunction of the antecedent propositions refers to their supporting degrees and implication of consequent results), aggregation of the results in a output set and defuzzification of the output fuzzy set (Figure 5). Outputs of the FIS systems command both the V_p and the V_m.

Figure 5.

Graphical representation of an example of Mamdani-based FIS.⁴¹

Design of the controllers

TD controller adjusts V_p between 0 and 14.5 V and yields an amount of Q_c between 0 and 136 W. VTG controller regulates V_m between 0 and 12 V and convects oil within a rate of v between 0 and 0.228 kg/s. Reduced VTG and TD allows appropriate conditions for precise measurements.¹⁰

Design of TD controller

In the first stage, η_c should be determined: η_c (equation (2)) could vary within 0–0.65 rates, and together with TD, they determine V_p¹⁷

η_{c} = \frac{Q_{C}}{P_{Peltiers}}

(65)

Since R_Peltiers≅ 1087 Ω then

P_{Peltiers} ≅ V_{p}^{2}

(66)

In order to reduce TD around zero, necessary voltage will be about

V_{p} = \sqrt{\frac{(m c_{o} TD)}{η_{c}}}

(67)

η_c is a function of the Peltiers T_h − T_c, and ΔT_max is 67 °C³⁶

η_{c} = 0.65 (1 - \frac{T_{h} - T_{c}}{Δ T_{max}})

(68)

PID TD controller design for the system is proposed in detail.¹⁷

Fuzzy TD controller design

The empirical interpretations of the experiments point out that η_c is a non-linear characterization of v and T_h − T_c where T_∞ is 24 °C and Q_R = 2.5 W.¹⁷ Processing regions of V_m and its corresponding v are 4–8 V and 0.076–0.152 kg/s, respectively. Higher speed results in rising of heat caused by increasing friction. η_c should be revised as η_cTotal after taking Q_R and Q_m effects into account (Figure 6).

Figure 6.

Experimental observations that illustrate relation between T_h − T_c and $η_{cTotal}$ . under various V_P voltages.

Instead of the linear relation given in equation (26), the empirical relation can be realized by means of fuzzy rules.

Therefore, a fuzzy rule base, which describes the non-linear relation, could be derived based on Figure 6 as shown in Table 2.

Table 2.

Rule base that determines relation between $η_{cTotal}$ and its factors.

1	If V_m is L ANDT_h − T_c is ZG THEN $η_{c}$ is H
2	If V_m is L ANDT_h − T_c is L THEN $η_{c}$ is H
3	If V_m is L ANDT_h − T_c is M THEN $η_{c}$ is M
4	If V_m is L ANDT_h − T_c is B THEN $η_{c}$ is L
5	If V_m is H ANDT_h − T_c is ZG THEN $η_{c}$ is ZG
6	If V_m is H ANDT_h − T_c is L THEN $η_{c}$ is H
7	If V_m is H ANDT_h − T_c is M THEN $η_{c}$ is M
8	If V_m is H ANDT_h − T_c is B THEN $η_{c}$ is L

L: low; H: high; M: medium; B: big; ZG: zero grade.

The relation between controller output V_p, and inputs $η_{cTotal}$ and TD is given in Table 3.

Table 3.

Rule table of changes in V_p due to TD and $η_{c}$ .

V_p
TD	ZG	S	M	B
$η_{c}$
H	ZG	S	M	B
M	ZG	S	M	B
L	ZG	S	M	B
ZG	S	M	B	B

L: low; H: high; M: medium; B: big; TD: temperature difference; ZG: zero grade.

When we interpret the first rule, “IF TD is Zero Grade AND η_cTotal is High (This means V_m is Low AND T_h − T_c is Zero Grade) THEN Cool around Zero”

Table 5 is derived by means of locating Table 2 in Table 3. Input variables concerned with the TD controller are given in Figure 7. Corner membership functions (MFs) are represented by means of trapezoidal functions, where interiors are represented by means of triangles. Input variables TD, V_m and T_h − T_c are composed of four, two and four MFs, respectively. Operating band, B, for the controller is optional but generally selected as 1 °C out of where it operates in on-off mode.

Figure 7.

Fuzzy TD controller.

Shape of the output MFs plays fewer roles on the output than input MFs, rules and defuzzification methods.⁴² Therefore, output variable is composed of four singleton MFs which make our program⁴³ simpler.

Rules agreed in Table 5 given in “Appendix 2” are set by means of experimental try-and-faults to accomplish the results of the non-linear behavior. In this way, the fuzzy TD controller determines a convenient control V_p (Figure 7).

Fuzzy VTG controller design

Length of the TR block and amount of the self-heating effect determines the VTG.^7,10 In this study, VTG is reduced by controlling the V_m which provides a forced convection against natural convection of the O_CyT caused by the Q_R. Lesser v below 0.076 kg/s, natural convection dominates the forced convection and rises T_upper (Figure 8). Contrariwise, v over 0.152 kg/s reverses the convection and O_CoT cooled by the Peltiers dominates on T_upper and brings VTG about negative values.

Figure 8.

Illustration of convection loop of heat and its corresponding VTG refer to v.

The v should operate within the mentioned values, and their corresponding voltages are within 4–8 V. Variation curves for VTG versus V_m and V_p are depicted in Figure 3 which commented and guided to fuzzy rules. For example, for lower V_ps around 2–4 V, V_m is reduced to maintain VTG around zero. On the other hand, for increased V_p within 6–14 V range, just medium V_m could minimize the VTG.¹⁷ Proposed input and output fuzzy variables for the VTG controller are shown in Figures 9 –11.

Figure 9.

Membership functions of the input variable VTG.

Figure 10.

Membership functions of the input variable $Δ VTG$ .

Figure 11.

Membership functions of the output variable V_m.

Fuzzy rules are established due to status of the variables: VTG and its derivative $(Δ VTG)$ . VTG input has four and $Δ VTG$ input has three MFs. Output variable is composed of two singletons (Figure 17).

Rules are given in Table 4.

Table 4.

Rule base of the VTG controller.

1	If VTG is NG AND $Δ VTG$ is NG THENV_m is L
2	If VTG is NG AND $Δ VTG$ is ZG THENV_m is L
3	If VTG is NG AND $Δ VTG$ is P THENV_m is L
4	If VTG is ZG AND $Δ VTG$ is NG THENV_m is L
5	If VTG is ZG AND $Δ VTG$ is ZG THENV_m is L
6	If VTG is ZG AND $Δ VTG$ is P THENV_m is M
7	If VTG is PM AND $Δ VTG$ is NG THENV_m is L
8	If VTG is PM AND $Δ VTG$ is ZG THENV_m is M
9	If VTG is PM AND $Δ VTG$ is P THENV_m is M
10	If VTG is PB AND $Δ VTG$ is NG THENV_m is M
11	If VTG is PB AND $Δ (T_{ü st} - T_{alt})$ is ZG THENV_m is M
12	If VTG is PB AND $Δ (T_{ü st} - T_{alt})$ is P THENV_m is M

M: medium; L: low; P: positive; PM = positive medium ; PB = positive big; VTG: vertical temperature gradient; NG: negative; ZG: zero grade.

Programs developed for the system

Fuzzy and PID control program is developed in Delphi 5.0.⁴⁴ Programs and their flowcharts are explained in previous studies in detail.^43,45 Simulation programs are developed in Matlab Simulink. In the fuzzy program, min functions are used for AND and IMPLICATION operators. Sum function is preferred for simplicity in AGGREGATION operations⁴⁶ and center of gravity is used as DEFUZZIFICATION method.⁴²

Simulation model of the control process

The experimental results of the control processes are thoroughly discussed.⁴⁵ Scope of the study is heat and mass transfer modeling of the system and comparison of its performance with that of the real-time results: Both PID and fuzzy control methods are applied to the system and to the model. Simulink block diagram (Figure 4) is embedded as a subsystem to the control system model given in Figure 12. The model allows switching between PID and fuzzy control methods. However, the study is focused on fuzzy control results of the actual system⁴⁵ and the simulation results of the model. Fuzzy control blocks are essentially attached to Matlab Simulink models and allow design process being shorter.⁴⁷

Figure 12.

Simulink block diagram of the general feedback control system.

Results and discussion

Fuzzy control software prepared uses the sum-min method as a fuzzy inference method (OR and AND methods, respectively) and the center of gravity method for defuzzification processes. Minimum and maximum functions are utilized for implication and aggregation methods, respectively.⁴⁶ The same methods and parameters are used both in experimental and simulation applications.

In the fuzzy TD controller; eight rules are inherently active at the same time, and their defuzzification results determines and drives V_p. Similarly, four rules are simultaneously active for the VTG controller and drives V_m. The set value of the TD is 19.5 °C, and set value of the VTG is 0 °C.

Actual and simulation of fuzzy control results of the TOCyT is shown in Figure 13. Actual system and the model reach to the set value in 60 and 35 min, respectively. Around 0.2 °C of steady-state error occurred in the model response.

Figure 13.

TOCyT time response for actual⁴⁵ and simulation of the system control.

Actual system yields an undershoot around 82nd minute, which is arisen from unpredictable turbulences in the circulation of the oil against natural convection. However, model follows general trend of the actual system. Actual TOCyT is obtained by average of measured data from T_upper (Figure 18) and T_lower (Figure 19) sensors (equation (2)) where the simulation model yields TOCyT in reference to its differential estimation, x₂ (integration of equation (60) in Figure 4).

Since TOCyT should follow an average trend similar to as that of T_upper and T_lower, their graphics are given in the Appendix 2: Information acquired from T_upper and T_lower sensors is monitored in Figures 18 and 19, respectively. It is clear that the turbulence is originated around the T_upper sensor, which forced convected OCoT from the CoT, faced of heated OCyT risen by natural convection in the CyT.

Simulation results of T_upper and T_lower are shown in Figures 18 and 19. T_upper and T_lower values are important for determination of VTG and for adjusting the V_m. The results, which indicate the heat and mass transfer behavior, are approximately imitated by the proposed model.

In the controller section, V_p determines T_OCoT. In this purpose, efficiency, $η_{c}$ , of the Peltiers as a function of T_h − T_c is required. T_h of the actual system is acquired by sensors mounted between the hot surface of the Peltiers and the fins (Figure 14).

Figure 14.

Time response of T_h for actual⁴⁵ and simulation of the system control.

T_h in the simulation model is derived from equation (60) (Figure 14(b)). Following an equilibrium situation, T_h follows a trend corresponding the TD fuzzy logic controller (FLC) voltage, V_p, which is acquired from DAC card of the computer (Figure 15(a)) and drives the Peltiers. When TD exceeds 0–1 °C band, the controller behaves as on-off.

Figure 15.

V_p time response for actual⁴⁵ and simulation of the system control.

TD FLC of the model behaves similar to the actual system. Second FLC controller feeds back T_upper − T_lower VTG (Figure 16(a)). Referring to the acquired VTG and its derivative, V_m is controlled (Figure 17(a)). By means of the FLC VTG controller, VTG does not exceed ±0.5 °C.

Figure 16.

T_upper − T_lower VTG time response for actual⁴⁵ and simulation of the system control.

Figure 17.

V_m time response for actual⁴⁵ and simulation of the system control.

Similarly, simulation model keeps VTG around zero (Figure 16(b)). Actual and simulation outputs of V_m are given in Figures 17(a) and (b), respectively.

The comparison of experimental results and modeling outputs is shown in the figures above. When these mappings are examined carefully, their results signify that the system is modeled considerably realistic.

Conclusion

Heat and mass transfer model interprets high voltages in terms of heat, which means higher voltages correspond to higher heats. Once the model is validated, it could be directly applied to higher voltages such as 100 and 200 kW. Since the backbone of the system is extracted, changes or annexes on the hardware could easily be adapted to the model as a mathematical expression.

Uncertainties arisen from both TD and VTG could be determined by a certain function.⁴⁵ Minimization of TD and VTG directly results in decreasing uncertainty and allows accurate HVDC measurements.

Experimental tests for HVDC systems with temperature control last for hours. Cooling of the oil and then returning of the system to initial conditions takes almost 4–5 h. Hence, modeling and simulation of such systems are unavoidable.

In this study, fuzzy control method is applied to the model. However, the proposed model could be used in performance comparison of new control methods. Application of new control methods and parameter changes will take considerably short time in the simulations compared to actual control applications.

Footnotes

Appendix 1

Appendix 2

Table 5.

Rule base of the TD controller.

1	If TD is ZG ANDV_m is L ANDT_h − T_c is ZGTHENV_p is ZG
2	If TD is ZG ANDV_m is L ANDT_h − T_c is L THENV_p is ZG
3	If TD is ZG ANDV_m is L ANDT_h − T_c is M THENV_p is ZG
4	If TD is ZG ANDV_m is L ANDT_h − T_c is B THENV_p is ZG
5	If TD is ZG ANDV_m is M ANDT_h − T_c is ZG THENV_p is S
6	If TD is ZG ANDV_m is M ANDT_h − T_c is L THENV_p is ZG
7	If TD is ZG ANDV_m is M ANDT_h − T_c is M THENV_p is ZG
8	If TD is ZG ANDV_m is M ANDT_h − T_c is B THENV_p is ZG
9	If TD is S ANDV_m is L ANDT_h − T_c is ZG THENV_p is S
10	If TD is S ANDV_m is L ANDT_h − T_c is L THENV_p is S
11	If TD is S ANDV_m is L ANDT_h − T_c is M THENV_p is S
12	If TD is S ANDV_m is L ANDT_h − T_c is B THENV_p is S
13	If TD is S ANDV_m is M ANDT_h − T_c is ZG THENV_p is M
14	If TD is S ANDV_m is M ANDT_h − T_c is L THENV_p is S
15	If TD is S ANDV_m is M ANDT_h − T_c is M THENV_p is S
16	If TD is S ANDV_m is M ANDT_h − T_c is B THENV_p is S
17	If TD is M ANDV_m is L ANDT_h − T_c is ZG THENV_p is M
18	If TD is M ANDV_m is L ANDT_h − T_c is L THENV_p is M
19	If TD is M ANDV_m is L ANDT_h − T_c is M THENV_p is M
20	If TD is M ANDV_m is L ANDT_h − T_c is B THENV_p is M
21	If TD is M ANDV_m is M ANDT_h − T_c is ZG THENV_p is B
22	If TD is M ANDV_m is M ANDT_h − T_c is L THENV_p is M
23	If TD is M ANDV_m is M ANDT_h − T_c is M THENV_p is M
24	If TD is M ANDV_m is M ANDT_h − T_c is B THENV_p is M
25	If TD is B ANDV_m is L ANDT_h − T_c is ZG THENV_p is B
26	If TD is B ANDV_m is L ANDT_h − T_c is L THENV_p is B
27	If TD is B ANDV_m is L ANDT_h − T_c is M THENV_p is B
28	If TD is B ANDV_m is L ANDT_h − T_c is B THENV_p is B
29	If TD is B ANDV_m is M ANDT_h − T_c is ZG THENV_p is B
30	If TD is B ANDV_m is M ANDT_h − T_c is L THENV_p is B
31	If TD is B ANDV_m is M ANDT_h − T_c is M THENV_p is B
32	If TD is B ANDV_m is M ANDT_h − T_c is B THENV_p is B

B: big; M: medium; S: small; L: low; TD: temperature difference; ZG: zero grade.

Acknowledgements

Authors have special thanks for Hasan DİNÇER, İbrahim EKSİN, Özcan KALENDERLİ, Dilber ÖZBAY, Serkan YAMAN and Reem ALMADHAJI for their valuable assistance on former studies that set light and refer to our current researches.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Sadettin Burak Kilci

References

Ziegler

NF.

Dual highly stable 150-kV divider. IEEE Trans Instrum Meas 1970; 19(4): 281–285.

D’Emilio

Gabbana

La Paglia

, et al. Calibration of DC voltage dividers up to 100 kV. IEEE Trans Instrum Meas 1985; 34(2): 224–227.

Park

JH.

Special shielded resistor for high-voltage DC measurements. J Natl Bureau Stand-C Eng Instrument 1962; 66C: 19–24.

Deacon

TA.

Intercomparison measurement of the ratios of a 100 Kilovolt DC voltage divider (BCR Information Applied Metrology, EUR10178EN), Brussels; pp. 1–23.

Rungis

Kim

, et al. Interlaboratory comparison of high direct voltage resistor dividers. IEEE Trans Instrum Meas 1999; 48(2): 158–161.

Elg

A-P

Bergman

Hällström

, et al. Traceability and characterization of a 1000 kV HVDC reference divider. IEEE Trans Instrum Meas 2015; 64(6): 1709–1715.

Schon

Investigation into the self-heating effect of HV DC dividers. In: Proceedings of the fifth international symposium on high voltage engineering, Braunschweig, 24–28 August 1987.

Childers

Dziuba

Lee

A resistive ratio standard for measuring direct voltages to 10 kV. IEEE Trans Instrum Meas 1976; 25(4): 505–508.

Peier

Graetsch

A 300 KV DC measuring device with high accuracy. In: Proceedings of the third international symposium on high voltage engineering, Milan, 28–31 August 1979.

10.

Marx

New concept of PTBs standard divider for direct voltages of up to 100 kV. IEEE Trans Instrum Meas 2001; 50: 426–429.

11.

Elben

Fechner

Lei

, et al. Modern noise rejection methods and their applicability in partial discharge measurements on HVDC cables. In: Proceedings of the IEEE international conference on high voltage engineering and application (ICHVE), 10–13 September 2018, Athens, pp. 1–4. New York: IEEE.

12.

Zheng

Guo

Cui

, et al. Comparative study on small-signal stability of LCC-HVDC system with different control strategies at the inverter station. IEEE Access 2019; 7: 34946–34953.

13.

Zhao

Sun

Study on control strategies to improve the stability of multi-infeed HVDC systems applying VSC-HVDC. In: Canadian conference on electrical and computer engineering, Ottawa, ON, Canada, 7–10 May 2006, pp. 2253–2257. New York: IEEE.

14.

Guo

Cai

, et al. Operating characteristic analysis of multi-terminal hybrid HVDC transmission system with different control strategies. In: International conference on power system technology, Gunagzhou, 6–8 November 2018, pp. 2616–2621. New York: IEEE.

15.

Lalouni

Rekioua

Modeling and simulation of a photovoltaic system using fuzzy logic controller. In: Second international conference on developments in eSystems engineering, Abu Dhabi, 14–16 December2009, pp. 23–28. New York: IEEE.

16.

Shvidkiy

Fatkhutdinov

Spirin

NA.

Automatic control system thermal operation of mine fuel furnace. In: International Russian automation conference (RusAutoCon), Sochi, 9–16 September 2018, pp. 1–4. New York: IEEE.

17.

Yilmaz

Dincer

Eksin

, et al. Heat control in HVDC resistive divider by PID and NN controllers. Energ Convers Manage 2007; 48: 2739–2748.

18.

Prokudina

LA.

Numerical simulation of effect of temperature gradients on flow of liquid film in heat and mass transfer devices. In: Global smart industry conference (GloSIC), Chelyabinsk, 13–15 November 2018, pp. 1–6. New York: IEEE.

19.

Rassudov

Non-adiabatic heating effects of a variable speed electric drive. In: 26th international workshop on electric drives: improvement in efficiency of electric drives (IWED), Moscow, 30 January–2 February 2019, pp. 1–3. New York: IEEE.

20.

Horyna

Hanuš

Smid

Virtual mass flow rate sensor using a fixed-plate recuperator. IEEE Sens J 2019; 19: 5760–5768.

21.

Prakash

Ilangkumaran

An investigation of various actuation mechanisms in robot arm. Meas Control 2019; 52: 1299–1307.

22.

Mandel

Arie

Shooshtari

, et al. A heat spreading model for double-sided cross-flow manifold-microchannel heat exchangers. In: 17th IEEE intersociety conference on thermal and thermomechanical phenomena in electronic systems (ITherm), San Diego, CA, 29 May–1 June 2018, pp. 147–155. New York: IEEE.

23.

Belkhiria

Echouchene

Mejri

Nanoscale heat transfer in MOSFET transistor with high-k dielectrics using a nonlinear DPL heat conduction model. In: The 9th international renewable energy congress (IREC), Hammamet, Tunisia, 20–22 March 2018, pp. 1–6. New York: IEEE.

24.

Gao

Zhu

Zhang

, et al. Modeling and simulation analysis of hybrid bipolar HVDC system based on LCC-HVDC and VSC-HVDC. In: IEEE 3rd advanced information technology electronic and automation control conference (IAEAC), Chongqing, 12–14 October 2018, pp. 1448–1452. New York: IEEE.

25.

Nanou

Papathanassiou

Generic average-value modeling of MMC-HVDC links considering submodule capacitor dynamics. In: 2nd international universities power engineering conference (UPEC), Heraklion, 28–31 August 2017, pp. 1–5. New York: IEEE.

26.

Gouda

Ibrahim

Soliman

Parameters affecting the arcing time of HVDC circuit breakers using black box arc model. IET Gener Transm Dis 2019; 13(4): 461–467.

27.

Karimi

Majidi

MirSaeedi

, et al. A novel application of deep belief networks in learning partial discharge patterns for classifying corona surface and internal discharges. IEEE Trans Indus Elec. Epub ahead of print 9 April 2019. DOI: 10.1109/TIE.2019.2908580.

28.

Jing

Y-H

Yang

G-H.

Adaptive fuzzy output feedback fault-tolerant compensation for uncertain nonlinear systems with infinite number of time-varying actuator failures and full-state constraints. IEEE Trans Cybernetics. Epub ahead of print 29 March 2019. DOI: 10.1109/TCYB.2019.2904768.

29.

Chen

Yuan

Global fuzzy adaptive consensus control of unknown nonlinear multi-agent systems. IEEE Trans Fuzzy Syst. Epub ahead of print 1 April 2019. DOI: 10.1109/TFUZZ.2019.2908771.

30.

Hossen

Ghosh

BC.

Performance analysis of a PMBLDC motor drive based on ANFIS controller and PI controller. In: International conference on electrical computer and communication engineering (ECCE), Cox’sBazar, Bangladesh, 7–9 February 2019, pp. 1–6. New York: IEEE.

31.

Eksin

Erol

OK.

A fuzzy identification method for nonlinear systems. Turk J Electr Eng Comp Sci 2000; 8(2): 125–137.

32.

Mills

AF.

Basic heat and mass transfer, 2nd ed. Sydney, NSW, Australia: Prentice Hall, 1998.

33.

Khan

Finkelstein

Mathematical modelling in measurement and instrumentation. Meas Control 2011; 44(9): 277–282.

34.

Neacă

AM.

Mathematical model for resistive tubular heater. In: International Conference on Applied and Theoretical Electricity (ICATE), Craiova, 6–8 October 2016, pp. 1–6. New York: IEEE.

35.

Arora

Singh

Brar

GS.

Thermal and structural modelling of arc welding processes: a literature review. Meas Control 2019; 52(7–8): 955–969.

36.

https://docs-emea.rs-online.com/webdocs/001b/0900766b8001b623.pdf (accessed 15 April 2019).

37.

Kalaiarassan

Krishnamurthy

Digital hydraulic single-link trajectory tracking control through flow-based control. Meas Control 2019; 52(7–8): 775–787.

38.

Neacă

MI.

Simulink model for resistive tubular heater. In: International conference on applied and theoretical electricity (ICATE), Craiova, 4–6 October 2018, pp. 1–6. New York: IEEE.

39.

Franklin

Powel

Naeini

AE.

Feedback control of dynamic systems, 6th ed. Boston, MA: Pearson, 2010.

40.

Jun

Shouyong

Chong

, et al. A spintronic memristor crossbar array for fuzzy control with application in the water valves control system. Meas Control 2019; 52(5–6): 418–431.

41.

Kosko

Neural networks and fuzzy systems. Upper Saddle River, NJ: Prentice Hall, 1992.

42.

Nguyen

Prasad

Walker

, et al. A first course in fuzzy and neural control. Boca Raton, FL: Chapman & Hall/CRC, 2003.

43.

Yilmaz

Dincer

Development of a fuzzy control program to reduce uncertainty in the measurement of dc HV resistive dividers. In: Proceedings of the third international conference on electrical and electronics engineering, 3–7 December 2003, Bursa, pp. 310–314.

44.

Calver

Delphi unleashed. Istanbul: Sams Publishing, 1997.

45.

Yilmaz

New approaches to reduce measurement uncertainty in HVDC resistive dividers. PhD Thesis, Kocaeli University Institute of Science and Engineering, Kocaeli, 2003.

46.

Ross

Fuzzy logic with engineering applications. London: John Wiley, 2004.

47.

Uysal

Gokay

Soylu

, et al. Fuzzy proportional-integral speed control of switched reluctance motor with MATLAB/Simulink and programmable logic controller communication. Meas Control 2019; 52(7–8): 1137–1144.

Modeling and simulation of a fuzzy heat distribution controlled high-voltage DC resistive divider

Abstract

Keywords

Introduction

Heat distribution controlled HVDC-ResDiv platform

Heat conduction and convention model of the 5-kV HVDC-ResDiv

Heat conduction and energy equations

Heat distribution in the HPCyT

Heat distribution in the CyT

Heat distribution in the CoT

Control methods

PID (three terms) control method

Fuzzy control method

Design of the controllers

Design of TD controller

Fuzzy TD controller design

Fuzzy VTG controller design

Programs developed for the system

Simulation model of the control process

Results and discussion

Conclusion

Footnotes

Appendix 1

Appendix 2

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References