On Improving Distortion Power Quality Index in Distributed Power Grids

ABSTRACT

This paper presents the Euclidean norm based new power quality index (PQI), which is directly

related to the distortion power generated from nonlinear loads, to apply for a practical

distribution power network by improving the performance of the previous PQI proposed by the

authors. The proposed PQI is formed as a combination of two factors, which are the electrical

load composition rate (LCR) and the Euclidean norm of total harmonic distortions (THDs) in

measured voltage and current waveforms. The reduced multivariate polynomial (RMP) model

with the one-shot training property is applied to estimate the LCR. Based on the proposed PQI,

the harmonic pollution ranking, which indicates how much negative effect each nonlinear load

has on the point of common coupling (PCC) with respect to distortion power, is determined. Its

effectiveness and validity are verified by the experimental results from its prototype’s

implementation in a laboratory with a single-phase 3 kW photovoltaic (PV) grid-connected

inverter, which contributes to a small distortion in voltage at the PCC, and practical nonlinear

loads. Then, the harmonic current injection model based time-domain simulations are carried out

to prove the effectiveness of the proposed PQI under the other conditions with different

nonlinear loads.

Index Terms—Distortion power, distribution power system, Euclidean norm, harmonic pollution ranking, power quality index, reduced multivariate polynomial (RMP) model.

INTRODUCTION . AS THE increased utilization of power electronic devices and nonlinear loads aggravates the

distortion in voltage and current waveforms, the power quality (PQ) in modern power systems

has become a significant issue for both power suppliers and consumers. Moreover, there has

been an increasing trend towards electric deregulation and independent power producers (IPPs)

based on renewable energies such as fuel cell, photovoltaic, wind, and gas-fuelled micro-

turbines, etc. In addition, the distributed generation (DG) [1] by the IPP with poorly controlled

synchronization will make it more difficult to handle the PQ problems related with system

reliability and stability at both power generation and distribution levels. In other words,

electricity has been generally sold from on supplier to one consumer with ownership changing

hands at only one physical point: the revenue meter. In contrast, after deregulation accompanied

with the DGs, it is expected that the ownership of electric power will be exchanged at several

points along the generation-transmission-distribution chains. Then, the proper PQ solutions will

be necessary at each physical location where ownership is transferred [2]–[4]. Therefore, it is

important to develop the appropriate power quality index (PQI) as well as identity the sources

and disturbances deteriorating the PQ. The limits on the amount of harmonic current and

voltages generated by customers and/or utilities have been stablished in the IEEE standards 519

[5] and 1547 [6], and in the IEC-61000-3 standard [7]. Recently, some techniques to achieve the

specified levels of PQ while enhancing its performance have been reported [8]–[11]. In addition,

several power quality indices through the analysis of measured voltage and current waveforms

[12]–[14] and analytical tools to evaluate the harmonic contributions on a point of common

coupling (PCC) [15]–[17] have been developed. In particular, the distortion

SYSTEM CONFIGURATION

The system configuration of the compound cascaded , where the CMI means the cascaded

multilevel inverter and its circuit is shown in Fig. 2. The compound cascaded also uses the

carrier phase shifted SPWM technology (CPS-SPWM) [3] to modulate the CMI, which could

improve the equivalent switching frequency, reducing the harmonics, and minimizing the size of

passive filter components. usx, isx and ilx (x=a, b, c) are the grid voltages, grid currents and load

currents, respectively. iab, ibc and ica are the phase currents. ix are the line currents. R1, R2, L1

and L2 are the equivalent series resistance and inductance. The new topology inherits the Δ

structure and for that the output phase current of the compound cascaded can be controlled

equally as far as possible in order to make full use of the capacity of the CMI, when the

compensating line current is determined. Because the typical Δ-connected cascaded is parallel-

connected to the point of common coupling (PCC), so the CMI have to bear the line voltage of

the PCC directly [6]–[7]. Differently with the pure Δ-connected , the Δ structure in the

compound cascaded , as the green zone shows in Fig.1, is separated from the direct line voltage

of the PCC (Point U, V, W) by the three series CMIs which can be viewed as the Y structure of

the compound . So the Y structure as the yellow zone shows in Fig.1, in the compound , has born

the main phase voltage of the PCC while the Δ structure only bear the line voltage which has

been decreased low enough by the Y structure. So, for the Δ structure, Y structure plays a step-

down transformer role. Based on the duality principle, the conclusion that the essence of the

compound is that the low-voltage Δ-connected cascaded is series connected with the Y-

connected can be draw easily.

The compound cascaded multilevel .

MATHEMATICAL MODEL

In order to reveal the internal relations and distinctions among the Y-connected, the Δ-connected

and the compound cascaded , their equivalent circuit diagrams and mathematical models are

presented firstly. Fig. 3 shows the equivalent circuits of the Δ-connected, Yconnected and the

compound , respectively. Based on these circuits, their KVL equations can be written as follows.

For the Δ-connected cascaded ,

When the system is three-phase balanced, the following equations hold, The technological and

economic characteristics, including economy, reliability, fault-tolerant ability and three-phase

unbalance compensating ability, are analyzed comparatively in order to highlight the superiority

of the new topology.

For the Y-connected structure, the outputs of the CMIs, IA,IB and IC, are also determined,

thereupon. In this ondition, the 3-phase CMIs will output different capacity and their rated

capacity must be determined by the CMI which outputs the maximum capacity while other

phases cannot output the rated value. If the Y-connected structure is designed based on the rated

capacity of the balanced working condition in order to make full use of all the power units, it

may have not the ability to compensate the serious unbalanced systems and have to quit

operating. For the Δ-connected structure, nevertheless, the 3-phase CMIs can output equal

capacity in order to make full use of all the power units and can satisfy the compensating targets

simultaneously, which can be verified as follows.

Obviously, the relationship among (1), (2) and (3) can be expressed as follows, The fact that the

essence of the compound cascaded is that the low-voltage Δ-connected cascaded is series

connected with the Y-connected cascaded also can be illustrated from the mathematical models.

Based on the current direction labeled in Fig. 1, the KCL equations can be written as follows. the

mathematical model of the compound in the 3-phase static coordinate can be expressed as

follows,

Viewed from the structures of the state-space equations, we know that (11) and (12) are quite

similar. The only difference is the voltage uY which caused by the CMI of the Y structure viewed

as a step-up transformer. Based on the equation (12), the single-phase equivalent circuit model

and the voltage vector diagram can be derived, shown in From (11) and (12), the mathematical

model of the Y structure’s external characteristics can be expressed as follows,

The expressions (1) and (10) can be written as the state space equation form,

bears the line voltage of the PCC while the Y-connected bears the phase voltage. In the

compound , the Y structure which plays a step-down transformer role has born the main phase

voltage of the PCC while the Δ structure only bear the line voltage which has been decreased low

enough by the Y structure. So, the quantitative relation can be expressed as follows,

Where NY, NY/Δ and NΔ represent the number of power units in the Y-connected, the Δ-

connected and the compound cascaded , respectively. Viewed from the structures of the state-

space equations, we know that (11) and (12) are quite similar. The only difference is the voltage

uY which caused by the CMI of the Y structure viewed as a step-up transformer. Based on the

equation (12), the single-phase equivalent circuit model and the voltage vector diagram can be

derived, From (11) and (12), the mathematical model of the Y structure’s external characteristics

can be expressed as follows,

This equation can be used as the design basis for the CMI of the Y structure. The technological

and economic characteristics, including economy, reliability, fault-tolerant ability and three-

phase unbalance compensating ability, are analyzed comparatively in order to highlight the

superiority of the new topology.

The economic advantage is embodied in the number of the power units constituting the

cascaded . The more the power units consumed, the higher the cost. So, the number of the power

units is the most important economic indicators. The quantity can be calculated based on the

followed expression

Figure : Network

a coordination operation of generator excitation from the synchronous generatorand

employed to support electrical power networks that have poor voltage and power stability (both

small-signal and large-signal (transient)),(Song & John, 1999; Hingorani & Gyugyi, 1999)and

references therein.For simplicity, the dynamic behavior the generator excitation is based on a

third-order generator model while the is regarded as a first-order differential equation; thus, the

generator excitation/ dynamic model is expressed as follows:

(2)

with

where . is the power angle of the generator,

denotes a transient voltage source behind a direct axis transient reactance. denotes the relative

speed of the generator, is a damping constant, is the mechanical input power,

is the electrical power, without , delivered by the generator to the

voltage at the infinite bus is the synchronous machine speed, , represents the per

unit inertial constant, is the system frequency and .

is the reactance consisting of the direct axis transient reactance of SG, the reactance of the

transformer, and the reactance of the transmission line.Also, denotes the reactance of

the transmission line. Similarly, is identical to except that denotes

the direct axis reactance of SG. is the direct axis transient short-circuit time constant. is the

field voltage control input to be designed. denotes the injected or absorbed currents as a

controllable current source, is an equilibrium point of currents, is the control input to be

designed, and is a time constant of models.

4. It is well-known that the generator transient voltages and v current are often physically

not measurable while in practical and are always monitored, so and in (4) can be

considered as new state variables. Differentiating the electrical powers and terminal voltage

, respectively, in (2) and then defining the variables , the

dynamic model of the power system including excitation and can be expressed as the general

form (1) below

The Harmonic Oscillator

The harmonic oscillator is often used as a model for absorption of infrared radiation by

covalently bonded molecules. This motion is as simple as the oscillations of a mass on a spring.

The force required for this type of motion obeys Hooke's Law (F = kx) where x is the

displacement away from equilibrium, k is the proportionality constant (called the force constant),

and F is the force, usually expressed in Newtons. The restoring force to bring the mass back to

equilibrium would be equal and opposite to this force.

The period for one oscillation is given by Galileo' s equation

.

Since the

frequency (n) is the reciprocal of the period then . The m actually includes the

mass of the object and the mass of the spring itself(mm + ms). The equation can be rearranged

by inverting both sides and squaring to get the following form that is also useful for determining

the constant k from frequency data. This model should be applicable to vibrational harmonic

motion of molecules, such as chloroform (CHCl3). If we assume that the CCl3 group remains

motionless during vibration of the C-H bond, then a force constant for the stretching and bending

modes can be obtained from spectra and the change in the frequencies for isotopic substitutions

such as deuterium for hydrogen can be predicted from the modeled force constants. The

equation used to find the force constant k is analogous to the mass on a spring, namely If

deuterium is substituted for H in chloroform one can calculate the new effective mass (~the mass

of the deuterium atom) and use it to find the new frequency. It is also helpful to remember

that IR spectra is usually reported in units of cm-1 (1/l) and that c =ln. The purpose of this lab

is to investigate the harmonic oscillator model using a simple mass and spring and to apply the

model to predict the frequencies of deuterated chloroform.

The Harmonic Oscillator Model

Using a simple spring system, take several masses (at least 5) and find the displacement away

from the equilibrium position. Time 25 oscillations or more to find the average period. · Find

the best value of the spring constant k for the simple mass on spring system by graphing force

versus extension. · Determine the constant k using Galileo's equation. Graph (1/n) 2 versus the

mass on the spring. Compare to the Hooke's Law method. Calculate the % deviation.

Application to IR spectra · Take the IR spectra of chloroform and find the C-H stretch and C-H

bend. These bands should be around 3000 cm-1 and 1200 cm-1 . Determine them as precisely

as possible. · Determine the force constant for each motion by using the Galileo equation. You

will have to change wave number to frequency. · Predict using your model (and the force

constant above) the frequencies of the C-D stretch and C-D bend for deuterated chloroform. ·

Take the IR spectra of deuterated chloroform and compare to your predicted values for the C- D

stretch and bend. Calculate the % deviation.

Synergetic Control

The synergetic control theory has beenfirst introduced by the Russian researcher

(Kolesnikov, 2000). This design procedure follows the analytical design of Aggregated

Regulator (ADAR) method (Kolesnikov, Veselov, Kolesnikov, Monti, Ponci, Santi, & Dougal,

2002). Thetechniquehas successfully been applied and employed in its wide range of

applications, such as the area of power electronic controls. In this paper, we are interested, in

particular, in the area of power system control and operation (Jiang, 2007; Ademoye, & Feliachi,

2011; Ademoye, Feliachi,& Karimi, 2011; Ademoye,& Feliachi,2012), this method used to

design a stabilizing control lawproviding better performance than traditional power system

stabilizers.

Let us consider the nonlinear dynamic equation1 in the state space form as follows:

(4)

wherestate variable and control input , and an assignable equilibrium point

to be stabilized. Basically, synergetic control consists ofthe followingsteps.

(1) In order to construct a manifold for the nonlinear systems, a macro-variable is defined as

where is a function of the system states. The synergetic synthesis provides a

method to find a stabilizing control law as a function of some specified

macro-variable to force and restrict the system trajectories to operate on the manifold

defined by . The behavior of the macro-variable can be selected by designersin

accordance with the desired control specifications. Basically, a linear combination of the

state variables is a simple case which can be chosen so as to achieve the control objective,

the settling time, limitations in control output, and so on.Also, any variable constraints can

be included to form the macro-variable.

(2) Design or synthesizea stabilizing state feedback controller in order to drive the system states

to exponentially converge to, and then remain on the specified manifold . The selected

macro-variableis evolved in a desired manner by introducing a constraint expressed in the

following equation:

(5)

1

where is a controller parameter indicating the speed of convergence of the macro-variable

toward the manifolds specified by .

(3) Differentiate , by taking into account the chain rule of differentiation, and substitute (1)

or (4) into (5), we can obtain

(6)

Then solvealgebraic equation (6) to obtain the control law . Thus, the resulting control law

can be expressed as

(7)

The following result is used for our nonlinear controller design of generator excitation and

of power systems.

Theorem 1: Consider a class of nonlinear systems (1) or (3). The system states and their rate

will converge exponentially to zero with the speed of convergence depending upon the selected

parameter , if the control law is exerted as (7).

Synergetic Control Design

In this section, a synergetic controller is designed to accomplish the expected requirements

mentioned in Section 2. The aim is to define a stable and invariant manifold and to design a

control law capable of driving the system trajectories and force them to remain on the manifold.

Given a nonlinear power system with generator excitation and described by (1) or (3), the

synergetic synthesis of power systems considered starts with defining two macro-variables as

follows.

(8)

(9)

where , and are the referencesof the rotational angular speed

(frequency), active electrical power, and terminal voltage, respectively. The aim of the proposed

controller design are to steer the system trajectories and force them to remain on the manifolds

The dynamic of the evolution of each macro-variable is provided as:

(10)

where are a pre-specified controller parameter indicating the converging speed of the closed-

loop system to the manifolds . Therefore, denoting and by

and , respectively and by substituting (8)-(9) and their derivative into (10), which is

(11)

(12)

The expressions above gives

(13)

(14)

Directly substituting into (13)-(14), the result is

After rearranging the expressions above, the following control laws are obtained as:

(17)

and

(18)

Based on selecting some suitable choices of the controller gains ,

the proposed controller can not only power angle stability but also frequency and voltage

regulations following a large disturbance. From the synergetic control approach mentioned

previously, it is obvious that regardless of the steady-state operating point of the system, the

synergetic controller performs well on the full nonlinear system and in contrast to the traditional

control theory,do not need any linearization or simplification on the system model.

Simulation Results

In this section, simulation results of coordination of generatorexcitation and control in

SMIB power systems considered in previous sections are shown. Power angle stability as well as

voltage and frequency regulations are used to point out the transient stability enhancement and

dynamic properties.

Considering the single line diagram as shown in with SG connected through parallel

transmission line to an infinite-bus,such generators deliver 1.0 per unit, power while the terminal

voltage is 0.9897pu., and an infinite-bus voltage is 1.0 per unit. However, when a three-phase

fault (a large perturbation) occurs at the point , the midpoint of one of the transmission lines.

This leads to rotor acceleration, voltage sag, and large transient induced electromechanical

oscillations.

The interesting question is that after the fault is cleared from the network, will the system

return to a post-fault equilibrium state?

In this paper, the fault of interest is the following two fault sequences, namely temporary

and permanent faults. Usually, there are four basic stages associated with transient stability of a

power system:

Stage 1: The system is in a pre-fault steady state.

Stage 2: A fault occurs at .

Stage 3: The faults is isolated by opening the breakers at .

Stage 4: The transmission line is recovered without the fault at sec. Eventually, the system

is in a post-fault state at sec.

In this paper, two cases with different faults sequences are investigated as follows:

Temporary fault

The system is in a pre-fault steady state, a fault occurs at sec., the fault is

isolated by opening the breaker of the faulted line at sec., the transmission line is

recovered without the fault at sec. Afterward the system is in a post-fault state.

Permanent fault

The system is in a pre-fault steady state, a fault occurs at sec., the fault is

isolated by permanently opening the breaker of the faulted line at sec. The system is

eventually in a post-fault state.

In this section, the effectiveness of the combination of the coordinated (generator

excitation/) controller to improve transient stability applied of a power system through power

angle stability, as well as voltage, frequency, and power regulations, is investigated and

compared with the FeedBack Linearization controller: FBL (Gu, &Wang, 2007) and

conventional linear controller (PSS/AVR) (Kundur, 1994).

ADD YOYR SIMULATION OUTPUT

Figure 2:

The physical parameters (pu.) and initial conditions ( for this proposed

power system model are given as follows:

The tuning parameters of the coordinated controller selected to test in this paper are as

follows: . From our simulation results,

the following can be seen.

The transient stability of a power system with both generator excitation and can be

effectively improved by using the nonlinear coordinated controller proposed as seen in Figures 3

and 4. Although there is a large sudden fault (temporary or permanent) on the network, the

system is able to keep transiently stable.

Time trajectories of a power angle , SG relative speed (frequency), the transient

voltage of the proposed controller, the FBL controller, and PSS/AVRrespectively, are shown in

Figures 3(a)-(b). After the fault is cleared from the network, from temporary fault cases above

the power angles , and the SG relative speeds, , the transient voltage

, and current settle to the pre-fault steady state as expected. Note also that,

due to the presence of the permanent fault on the network, power angle, transient voltage and

current of the synergetic controller, the coordinated controller, and PSS/AVR cannot go to the

pre-fault state except for SG relative speed. In comparison with the FBL controller and

PSS/AVR, time histories of the proposed controller, particularly power angles, SG relative

speeds, and current have obviously smaller overshoot along with faster reduction of oscillation

excluding transient voltage. Regarding power and voltage regulation as shown in Figure 3(b) and

4(b), the synergetic controller not only provides clearly better transient responses thanboth the

FBLcontroller and PSS/AVR,but also quickly settles to their pre-fault steady state of active

power. In particular, the voltage sag of the proposed controller is quickly stabilized but has

higher overshoots when compared with the FBL controllerand PSS/AVR in terms of settling

time and rise time. From two different fault sequences, their active electrical power and voltage

responses of both the proposed and FBL controllers also can converge to the desired reference

values of active power and terminal voltage while terminal voltage responses of

PSS/AVR cannot.Those indicate that with the help of the synergetic control theory the

coordination of generator excitation and can obviously improve further transient stability along

with dynamic properties as compared with the FBL controller and linear PSS/AVR controller. In

the permanent fault, Figures 4(a)-(b) illustrate that and cannot return to the pre-fault state

whilethere are active electrical power and terminal voltage capable of settling to the desired

reference valves . Independent of the steady-state operating point of the

system and fault sequences above, the nonlinear coordinated controller can achieve the expected

requirements and accomplish better dynamic properties as seen in faster transient responses of

the closed-loop systems under a large sudden fault.

From the simulation results above, it can be concluded that not only transient stability is

enhanced but also power angle stability along with frequency, power, and voltage regulations are

simultaneously accomplished according to the two expected requirements for the proposed

controller.

SIMULATION WAVE FROMS 1

and superiorities of the proposed compound cascaded whose configuration is shown in Fig. 1.

The effective line voltage value of the power source is 10kV and the peak phase voltage value of

the PCC is about 8kV. The DC voltage of the power unit is 1kV and the switch frequency is

1kHz. In this model, CMIs of the Y-connected structure are designed to bear the PCC’s peak

phase voltage and contain 8 units in a phase, which are operating as voltage sources and not

controlled by any physical quantity.

SIMULATION WAVE FROMS 2

And the CMIs of the Δ-connected structure are designed to bear the instantaneous voltage of the

system’s equivalent series inductance and contain 3 units in a phase because the maximum

voltage is not over 1.5kV, which work as controlled current sources and whose control variable

is the sum of the reactive current and the negative-sequence current of the load

SIMULATION WAVE FROMS 3

In order to simulate the 3-phase unbalanced condition, this model sets a single phase load

between Phase A and Phase B. This load contains a 100Ω resistor and a 0.3H inductor. Since the

load is resistive-inductive, so the power grid must deliver the inductive reactive power which

will course much energy consumption in the power transmission lines. In addition, the three-

phase unbalanced power transmission results in power quality worsening seriously in the power

system. All of that can be seen After compensating, the PCC voltages, also the load voltages,

have restored to the three-phase balanced state, only using about 20ms for dynamic adjustment,

showing in Fig. 10. Since the negative-sequence currents have been compensated by the

compound cascaded , so the power grid do not deliver the negative-sequence currents to the load

and has restored to the three-phase balanced state. Meantime, the reactive power is also

compensated completely and the phase difference between the gird voltages and currents

thereupon maintain zero. The power grid only output the fundamental active currents, about 40A,

The simulation results have verified that the proposed compound cascaded can simultaneously

compensate the reactive power and the 3-phase unbalanced currents for the unbalanced load

perfectly and also can save power units significantly

Conclusions

This paper proposed the new distortion power quality index to replace the previously proposed

index . Its computation was carried out based on the load composition rate (LCR) and Euclidean

norm of total harmonic distortions (THDs) of the measured voltage and current waveforms at the

point of common coupling (PCC). The reduced multivariate polynomial (RMP) model with the

one-shot training property was successfully applied to estimate the LCR. Moreover, the use of

could avoid applying another RMP model, which is required in the implementation of to estimate

the nonlinear load harmonics. This advantage of allows for more effective and preferable use in

practice. Also, the experimental results showed that the can provide the relative harmonic

pollution ranking (HPR) of several nonlinear loads with good performance, which is directly

related to their distortion powers without the need for direct measurements. In contrast, the

results also verified that the has the serious drawback of obtaining wrong answers with an

incorrect HPR. This was the case when the load current was severely distorted with the high

THD and/or when it had a large phase difference with the PCC voltage with a low power factor.

Moreover, the good estimation performance of the proposed and its applicability in practice was

verified by the simulation results based on the harmonic current injection model.

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