an observer-based optimal voltage control scheme for three...

ISSN: 2278 – 7798

International Journal of Science, Engineering and Technology Research (IJSETR)

Volume 5, Issue 12, December 2016

3358 All Rights Reserved © 2016 IJSETR

An Observer-Based Optimal Voltage Control Scheme

for Three-Phase UPS Systems using Fuzzy Logic Prasanthi Morugu1 and Prof. G.V.Marutheswar2

1PG student,Department of Electrical and Electronics Engineering, S.V.University , Tirupathi, India 2professor, Department of Electrical and Electronics Engineering, S.V.University, Tirupathi, India

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Abstract – This paper proposes a simple optimal voltage

control method for three-phase uninterruptible-

power-supply systems. An UPS also known as

uninterruptable power supply, or battery/flywheel

backup is an electrical apparatus that provides

emergency power to a load when the input power

source or mains power fails. AC voltage controller

electronic module based on either thyristors,

TRIACs, SCRs or IGBTs, which converts a fixed

voltage, fixed frequency alternating current (AC)

electrical input supply to obtain variable voltage in

output delivered to a resistive load. The proposed

voltage controller is composed of a feedback control

term and a compensating control term. The former

term is designed to make the system errors converge

to zero, whereas the latter term is applied to

compensate for the system uncertainties. Moreover,

the optimal load current observer is used to optimize

system cost and reliability. Particularly, the closed-

loop stability of an observer-based optimal voltage

control law is mathematically proven by showing that

the whole states of the augmented observer-based

control system errors exponentially converge to zero.

Unlike previous algorithms, the proposed method can

make a tradeoff between control input magnitude and

tracking error by simply choosing proper

performance indexes. The effectiveness of the

proposed controller is validated through simulations

on MATLAB/Simulink. Finally, the comparative

results for the proposed scheme and the conventional

feedbacklinearization control scheme are presented to

demonstrate that the proposed algorithm achieves an

excellent performance such as fast transient response,

small steady-state error, and low total harmonic

distortion under load step change, unbalanced load,

and nonlinear load with the parameter variations. A

fuzzy logic controller to improve the voltage total

harmonic distortions and steady state and transient

state responses to get less harmonic distortions than

fuzzy less observer voltage controller.

Key Words: Optimal load current observer, optimal voltage control, three-phase inverter, total harmonic distortion

(THD), uninterruptible power supply (UPS), fuzzy logic

controller.

1. INTRODUCTION

A UPS differs from an auxiliary or emergency power

system or standby generator in that it will provide near-

instantaneous protection from input power interruptions,

by supplying energy stored in batteries, supercapacitors,

or flywheels. The onbattery runtime of most

uninterruptible power sources is relatively short (only a

few minutes) but sufficient to start a standby power

source or properly shut down the protected equipment. Recently, the importance of the UPS systems has been

intensified more and more due to the increase of sensitive

and critical applications such as communication systems,

medical equipment, semiconductor manufacturing

systems, and data processing systems [1]-[3]. Uninterruptible power supply (UPS) systems supply emergency power in case of utility power failures. These applications require clean power and high reliability regardless of the electric power failures and distorted utility supply voltage. Thus, the performance of the UPS systems is usually evaluated in terms of the total harmonic distortion (THD) of the output voltage and the transient/steady state responses regardless of the load conditions: load step change, linear load and nonlinear load.[4]-[7].

So far, the optimal control theory has been

researched in various fields such as aerospace,

economics, physics, and so on [8], since it has a

computable solution called a performance index that can

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quantitatively evaluate the system performance by

contrast with other control theories. In addition, the

optimal control design gives the optimality of the

controller according to a quadratic performance

criterion and enables the control system to have good

properties such as enough gain and phase margin,

robustness to uncertainties, good tolerance of

nonlinearities, etc. [9]. Hence, a linear optimal

controller has not only a simple structure in

comparison with other controllers but also a

remarkable control performance similar to other

nonlinear controllers [10]–[12]. Compared to

number of control algorithms that have been

proposed such as proportional–integral (PI) control,

H∞ loop-shaping control, model predictive control,

deadbeat control, sliding-mode control, repetitive

control, adaptive control, and feedback linearization

control (FLC) the proposed optimal voltage control

and fuzzy based control is showing better

performances in voltage correction and reduction of

total harmonic distortions(THD). This paper proposes

an observer-based optimal voltage control scheme for

three-phase UPS systems. This proposed voltage

controller encapsulates two main parts: a feedback

control term and a compensating control term. The

former term is designed to make the system errors

converge to zero, and the latter term is applied to

estimate the system uncertainties

Fig.1.Three-phase inverter with an LC alters for a

UPS system.

The Lyapunov theorem is used to analyze the

stability of the system. Specially, this paper proves

the closed loop stability of an observer based optimal

voltage control law by showing that the system errors

exponentially converge to zero. Moreover, the

proposed control law can be systematically designed

taking into consideration a tradeoff between control

input magnitude and tracking error unlike previous

algorithms.

In FLC, basic control action is determined by a set

of linguistic rules. These rules are determined by

the system. Since the numerical variables

areconverted into linguistic variables, mathematical

modeling of the system is not required in FC

2. SYSTEM DESCRIPTION AND PROBLEM FORMULATION:

Fig.2. Block diagram of the proposed observer-based optimal voltage

control system.

Fig. 2 illustrates the overall block, of the

proposed observer-based optimal voltage control system.

In practice, the inverter phase currents and line-to-neutral

load voltages are measured via the CTs and PTs to

implement the feedback control. In this paper, a space

vector PWM technique is used to generate the control

inputs (Viα and Viβ) in real time. The main objective of

static power converters is to produce an ac output waveform

from a dc power supply. According to the type of ac output

waveform, these topologies can be considered as voltage

source inverters (VSIs), where the independently

controlled ac output is a voltage waveform. These

structures are the most widely because they naturally

behave as voltage sources as required by many industrial

applications, such as adjustable speed drives(ASDs),

which are the most popular application of inverters. Here

we are using an LC filter to filter the harmonics. The

proposed algorithm is verified through two different

types of loads as explicitly depicted in Fig. 3(a). linear-

load circuit that consists of a resistor per phase, where as

3(b) nonlinear-load circuit that is comprised of a three-

phase full-bridge diode rectifier, an inductor (Lload), a

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capacitor (Cload), and a resistor (Rload). Using these

loads we depict the performance of the sytem. Along

with these an improved technique using fuzzy logic

has been implememted to get better results as optimal

voltage controller.

Fig. 3. Two types of load circuits. (a) Resistive linear

load. (b) Nonlinear load with a three-phase diode

rectifier.

Based on Fig. 1, the dynamic model of a three-phase

invertercan be derived in a d − q synchronous

reference frame as follows [13]:

2 2id iq id Ldi wi k v k v , 1 1Ld Lq id Ldv wv k i k i

2 2iq id iq Lqi wi k v k v , 1 1Lq Ld iq Lqv wv k i k i

where k1 = 1/Cf , and k2 = 1/Lf . In system model (1),

vLd,vLq, iid, and iiq are the state variables, and vid and

viq are the control inputs. In this scheme, the

assumption is made to construct the optimal voltage

controller and optimal load current observer as

follows:

1) The load currents (iLd and iLq) are unknown and

vary very slowly during the sampling period [11].

3. PROPOSED OPTIMAL VOLTAGE CONTROLLER DESIGN AND

STABILITY ANALYSIS:

A. Optimal Voltage Controller Design:

Here, a simple optimal voltage controller is

proposed for system (1). First, let us define the d − q-

axis inverter current references ( ,id iqi i ) as

1 1

1 1,id Ld Lq iq Lq Ldi i wv i i wv

k k

(2)

Then, the error values of the load voltages and

inverter currents are set as

,de qeLd Ld Lq Lqv v v v v v

,de qeid id iq idi i i i i i (3)

Therefore, system model (1) can be transformed into the

following error dynamics

( )dx Ax B u u (4)

Where x= [ dev vqe ide iqe]T, u=[vid viq]

T,

ud=[dd dq]T

A=

2

1

2

2

0 0

0 0

0 0 0

0 0 0

w k

w k

k

k

2

1

2

2

0 0

0 0

0 0 0

0 0 0

w k

w k

k

k

,

B=2

2

00

00

0

0

k

k

2

2

00

00

0

0

k

k

2

1* ( )q Ld Lqd v wik

and 2

1* ( )q Lq Ldd v wik

Note that ud is applied to compensate for the

system uncertainties as a compensating term.Consider the

following Riccati equation for the solution matrix P

[14]:

PA+ATP-PBR

-1B

TP+Q=0 (5)

where Q and R are the positive definite weighting matrices

with sufficient dimensions

Remark 1: Recall that Q and R are the weighting

matrices excessive large error or control input values

can be penalized by using properly chosen Q and R.

Generally, the large Q means a high control

performance, whereas the large R means a small input

magnitude. Consequently, there is a tradeoff between Q

and R in the control system. The Q and R parameters

generally need to be tuned until satisfactory control results

are obtained.

Let the diagonal matrices Q and R be defined as

Q=

1

2

0 0........... 0

0 0........... 0

....... .... ........ .....

0 ...... ........ m

Q

Q

Q

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R=

1

2

0 0........... 0

0 0........... 0

....... .... ........ .....

0 ...... ........ K

R

R

R

where Q and R have positive diagonal entries such that √Qi =1/ymax i , where i = 1, 2, . . .,m, and √Ri = 1/umax

i , where i = 1, 2, . . .,m. The number ymaxi is

the maximally acceptable deviation value for the ith component of output y. The other quantity

max

iu is the ith component of input u. With an

initial guessed value, the diagonal entries of Q and R can be adjusted through a trial-and-error method. Then, the optimal voltage controller can be designed by the following equation:

du u Kx (6)

where K = −R-1 BT P denotes the gain matrix, and ud

and Kx represent a feed forward control term and a feedback control term, respectively.

Fig. 4. Block diagram of the proposed optimal

voltage control scheme.

B. Stability Analysis of Voltage Controller Consider the following Lyapunov function:

(𝑥)=𝑥𝑇𝑃𝑥 (8)

From (4)–(6), and (7), the time derivative of V(x) is given by the following:

1

1

( ) 2 ( )

2 ( )

( 2 )T

T T

T T

T T T

dV x x Px x P A Bk x

dx

x P A BR B P x

x PA A P PBR B P x x Qx

implies that x exponentially converges to zero.

Remark 3: By considering the parameter variations, the state-dependent coefficient matrix A is rewritten as 𝐴′= A + ΔA, where ΔA means the value of system parameter variations. Thus, (4) can be transformed into the following error dynamics:

1( ) ( )T T TV x x P A A P Q PBR B P x (9)

If the following inequality holds for the given ΔA:

𝑃Δ𝐴+Δ𝐴T𝑃<𝑃𝐵𝑅-1+𝑄 (10)

then V < 0 for all nonzero x. Therefore, the proposed

optimal voltage control system can tolerate any parameter

variation satisfying(10)

4. OPTIMAL LOAD CURRENT OBSERVER DESIGN

AND STABILITY ANALYSIS

A. Optimal Load Current Observer Design

To avoid using current sensors, a linear optimal load current observer is introduced in this algorithm. As seen in (2) and (4), the inverter current references (i*Q and i*d) and feed forward control term (ud) need load current information as inputs. From (1) and the assumption, the following dynamic model is obtained to estimate the load current:

0 00 0 0

0 0

x A x B u

y C x

(11)

where xo = [iLd iLq vLd vLq] T, uo = [k1 iid k1 iiq] T,

A0= 1

1

0 0 0 0

0 0 0 0

0 0

0 0

k w

k w

, B0=

0 0

0 0

010

0 1

Then, the load current observer is expressed as

0 0 0 0 0 0 0ˆ ˆ ˆ ( ) x A x B u L y C x

(12)

where [ ]d d qu d d , x0 = [ iLd iLq vLd vLq]T , and ˆ

Ldi

ˆLdi and ˆ

Lqx ˆLqx are estimates of iLd and iLq, respectively.

In addition, L is an observer gain matrix calculated by

L=1

0 0 0

TPC R (13)

and Po is the solution of the following Riccati equation

1

0 0 0 0 0 0 0 0 0 0 0T TA P P A PC R C P Q (14)

where Qo and Ro are the positive definite weighting

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matrices with sufficient dimensions. The manner of

choosing Qo and Ro is the same as in Remark 1.

B. Stability Analysis of Load Current Observer

The error dynamics of the load current observer can

be

obtained as follows:

. ( ) e ex A LC x

(15)

Define the Lyapunov function as

Vo(xe) = xT

eXxe (16)

Where

1

0X P . Its time derivative along the error

dynamics (19) is represented by the following:

1

0 0 0 0 0 0

1

0 0 0 0 0 0 0 0 0

0

ˆ( ) 2 ( )

( 2 )

T T T

e e e e

T T T

e e

T

e e

dV x x Xx x XA XP C R C x

dt

x X A P P A P C R C P Xx

x XQ Xx

This implies that xe exponentially converges to zero. 4.5. OBSERVER-BASED CONTROL LAW AND

CLOSED-LOOP STABILITY ANALYSIS

4.5.1Observer-Based Control Law:

With the estimated load currents achieved

from the observer instead of the measured quantities,

the inverter current errors and feedforward control

term can be obtained as follows:

du Kx ,1

1ˆqe iq Lq Ldi i i wv

k

2

1d Ld Lqd v wv

k ,

2

1q Lq Ldd v wv

k (17)

Then, (16) can be rewritten as the following

equations:

ˆ 1 0 0 0de de ei i x

ˆ 0 1 0 0qe qe ei i x

2

0 1 0 0d d e

wd d x

k

2

1 0 0 0q q e

wd d x

k (18)

From (6) and (18), the proposed observer-based control

law can be achieved as

u = du Kx (19)

where

[ ]de qe de qex v v i i T and [ ]d d qu d d (20)

B. Closed-Loop Stability Analysis:

For the purpose of analyzing the stability, (20) is rewritten as the following: d eu u Kx Hx (21)

Where H=(w/k2)E +KF, As a result, the design procedure of optimal voltage control law can be summerised:

Step 1) Build system model (1) in the d − q coordinate

frame and then derive error dynamics (4) by using

system parameters.

Step 2) Set the optimal voltage controller (6) with the

feedforward control term (ud) and feedback control

term (Kx).

Step 3) Define the load current estimation model (13)

and build the load current observer (14) by using the

Kalman–Bucy optimal observer.

Step 4) Select the observer weighting matrices Qo and Ro

in Riccati equation by referring to Remark 1. Then,

choose the observer gain L in (12) using Qo and Ro

5. FUZZY LOGIC CONTROLLER:

In FLC, basic control action is determined by a set

of linguistic rules. These rules are determined by the

system. Since the numerical variables are converted into

linguistic variables, mathematical modeling of the system is

not required in FC.

Fig. 5. Fuzzy Logic Controller

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The FLC comprises of three parts: fuzzification, interference engine and defuzzification. The FC is characterized as (i). seven fuzzy sets for each input and output. (ii). Triangular membership functions for simplicity. (iii). Fuzzification using continuous universe of discourse. (iv). Implication using Mamdani’s, ‘min’ operator.(v). Defuzzification using the height method.

Fuzzification: Membership function values are assigned to the linguistic variables, using seven fuzzy subsets: NB, NM, NS, ZE, PS, PM, and PB. The partition of fuzzy subsets and the shape of membership CE(k) E(k) function adapt the shape up to appropriate system. The value of input error and change in error are normalized by an input scaling factor.

Table 1: Fuzzy Rules

In this system the input scaling factor has been

designed such that input values are between -1 and +1.

The triangular shape of the membership function of this

arrangement presumes that for any particular E(k) input

there is only one dominant fuzzy subset. The input error

for the FLC is given as

E(k) = Pph(k)−Pph(k−1)/ Vph(k)−Vph(k−1) (26)

CE(k) = E(k) – E(k-1) (27)

Inference Method: Several composition methods such

as Max–Min and Max-Dot have been proposed in the

literature. In this paper Min method is used. The output

membership function of each rule is given by the

minimum operator and maximum operator. Table 1

shows rule base of the FLC.

Defuzzification: As a plant usually requires a non-

fuzzy value of control, a defuzzification stage is

needed. To compute the output of the FLC, height

method is used and the FLC output modifies the control

output. Further, the output of FLC controls the switch in the

inverter. In UPQC, the active power, reactive power,

terminal voltage of the line and capacitor voltage are

required to be maintained. In order to control these

parameters, they are sensed and compared with the

reference values. To achieve this, the membership functions

of FC are: error, change in error and output.

The set of FC rules are derived from

u=-[αE + (1-α)*C]

5. PERFORMANCE VALIDATIONS

A. Testbed Description:

The proposed observer-based optimal voltage

controller has been performed through both simulations

with MATLAB/Simulink . Moreover, the conventional

FLC scheme [14] is also adopted to exhibit analysis of

the proposed control scheme since it reveals a reasonable

performance for nonlinear load and has the circuit model

of a three-phase inverter similar to our system. In the

testbed, the inverter phase currents and line-to-neutral

load voltages are measured via the CTs and PTs to

implement the feedback control. In this paper, a space

vector PWM technique is used to generate the control

inputs (Viα and Viβ) in real time. The proposed algorithm

is verified through two different types of loads as

explicitly depicted in Fig. 3.More specifically, Fig. 3(a)

shows a linear-load circuit that consists of a resistor per

phase, whereas Fig. 3(b) depicts a nonlinear-load circuit

that is comprised of a three-phase full-bridge diode

rectifier, an inductor (Lload), a capacitor (Cload), and a

resistor (Rload). Note that during simulation and the

experiment, observer gain L and controller gain K are

selected based on Remark 1 as

L= 105

1.0029 0.003

0.0003 1.0029

1.1877 0

0 1.1877

,

K=1011

0.3241 0 3.1623 0

0 0.3241 0 3.1623

B.Simulation and Experimental Results: The proposed voltage control algorithm is carried

out in various conditions (i.e., load step change,

unbalanced load, and nonlinear load) to impeccably

expose its merits. In order to instantly engage and

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disengage the load during a transient condition, the

on–off switch is employed as shown in Fig. 3.The

resistive load depicted in Fig. 4(a) is applied under

both the load step change condition (i.e., 0%–100%)

and the unbalanced load condition (i.e., phase B

opened) to test the robustness of the proposed

scheme when the load is suddenly disconnected. In

practical applications, the most common tolerance

variations of the filter inductance (Lf ) and filter

capacitance (Cf ), which are used as an output filter,

are within ±10%. To further justify the robustness

under parameter variations, a 30% reduction in both

Lf and Cf is assumed under all load conditions such as

load step change, unbalanced load, and nonlinear

load.

Time(s)

(a)

Time(s)

(b)

Fig. 6. Simulation and experiment results of proposed

observer based control using linear load (a)simulation

(b)experiment

This Fig. 6 shows the simulation and

experimental results of the proposed control method

during the load step change. Moreover, Specifically,

the figures display the load voltages (First waveform:

VL), load currents (Second waveformJ(IL).

(a)

Time(s)

(b)

Fig.7. simulation results of proposed optimal scheme

using nonlinear load

It is important to note that the load current error

waveform in the results of the conventional FLC method

is not included because the FLC scheme does not need

load current information. It can be observed in Fig. 6 that

when the load is suddenly changed, the load output

voltage presents little distortion. However, it quickly

returns to a steady-state condition in 1.0 ms, as shown in

the simulation results in Fig. 6(a). Moreover, it has

revealed a fast recovery time of 1.5 ms in a real

experimental setup as shown in Fig. 6(b).

It can be observed that the load root mean square

(RMS) voltage values in both schemes are appropriately

regulated at steady state. Moreover, small load current

error (ieLA) between the measured value (iLA) and the

estimated value (ˆiLA).

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Time(s)

(b)

Fig 8. Combined linear and nonlinear loads

(a)simulation (b) experiment without fuzzy

Next, the characteristic performances of the

transient and steady state under unbalanced load are

verified through Figs. 7. Precisely, this case is

implemented under a full-load condition by suddenly

opening phase B. It is shown that the load output

voltages are controlled well, although the rapid

change in the load current of phase B is observed as it

is opened, small steadystate voltage errors under

unbalanced load are observed because the load RMS

voltage values of both methods are almost 110 V. In

addition, the load current observer provides high-

quality information to the proposed controller . To

evaluate the steady-state performance under

nonlinear load, a three-phase diode rectifier shown in

Fig. 7(b) is used.

Fig 9. Combined linear and non linear load with fuzzy

In the case of the conventional FLC scheme, the

corresponding load voltage THD values are 4.4% . It can

be also observed that the proposed control strategy

provides a better load voltage regulation in steady state

similarly as the conventional FLC method. In Fig. 8, it

can be evidently seen that the load current observer

guarantees a good estimation performance because of a

small load current error (ieLA). Finally, all THD and load

RMS voltage values under the three load conditions. On

the other hand, the THD values of the load output voltage

at steady-state full-load operation are found as 4.4% for

proposed optimal voltage controller without using FLC

and 3.02% for optimal voltage controller using fuzzy

logic controller. However, the conventional FLC scheme

shows better performance as well as proposed optimal

voltage controller. Therefore, it is explicitly

demonstrated that the proposed algorithm attains lower

THD and also the optimal controller using fuzzy logic.

The proposed optimal voltage controller with fuzzy has a

little bit better performance than the optimal voltage

controller without fuzzy.

6.Conclusion: This paper has proposed a simple observer-based

optimal voltage control method of the three-phase UPS

systems The proposed controller is composed of a

feedback control term to stabilize the error dynamics of

the system and a compensating control term to estimate

the system uncertainties. Moreover, the optimal load

current observer was used to optimize system cost and

reliability. This paper proved the closed-loop stability of

an observer-based optimal voltage controller by using

the Lyapunov theory. Furthermore, the proposed voltage

ISSN: 2278 – 7798




control law can be methodically designed taking into

account a tradeoff between control input magnitude

and tracking error unlike previous algorithms. The

superior performance of the proposed control system

was demonstrated through simulations and

experiments. Under three load conditions (load step

change, unbalanced load, and nonlinear load), the

proposed control scheme revealed a better voltage

tracking performance such as lower THD, smaller

steady-state error, and faster transient response. The

optimal voltage controller that has fuzzy logic

controller has attained a better performance as

compared to optimal voltage controller without

having fuzzy logic circuit with less harmonic

distortions in voltage.

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