Further Rejection: Signal vs Background Optimisation

Computational Techniques for Distribution Systems

2.1 Introduction

Distribution system networks consisting of transmission lines, generators, loads, fuses, capacitors, and reclosers have been presented in the introduc-tory chapter. To fully analyze the system under steady-state or transient conditions requires some fundamental concepts of computational tech-niques. This chapter therefore presents a review of basic computational fundamentals, terminology, and notation used for the analysis of single-phase or multisingle-phase distribution systems.

The review presented here covers such topics as instantaneous and com-plex power, power factor, loss calculations, and management in distribution systems, as well as single- and three-phase load-flow analysis techniques used in distribution networks. This background will provide a basis for the computational tools needed for subsequent operational and planning studies in distribution systems. These tools are applicable in fault studies, distribu-tion reliability assessments, and distribudistribu-tion automadistribu-tion funcdistribu-tion computa-tions. In addition, the fundamental tools used give us a greater appreciation for the use of communication and software tools designed specifically for distribution planning, protection, and control.

2.2 Complex Power Concepts

The computation of power in a circuit is generally found using the instan-taneous current injection and the potential difference across the circuit ele-ments. Consider a simple load circuit given as a generalized load Zload

connected to a voltage source v(t), as shown in Figure 2.1. The sinusoidal representation of the source voltage is given as

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10 Electric Power Distribution, Automation, Protection, and Control

v(t) = Vm cos(ωt) (2.1) and the corresponding current is

i(t) = I_m cos(ωt – φ) (2.2) where

Vm = amplitude of the source voltage (in volts) ω = angular frequency (in rad⋅sec⁻¹)

φ = phase shift of the voltage waveform with respect to the current (in rads)

Vrms = root-mean-square (rms) value of voltage computed as

= current magnitude

Now, the complex power or apparent power is defined as the total orthog-onal components in a vector space given by

S = P + jQ = VI^*

= VI cos φ + jVI sin φ

= VI (cos φ + j sin φ) (2.3) Using Euler’s identity,

FIGURE 2.1 Load circuit.

+

v(t)

i(t)

Zload=Z ф

V V

rms= m

2

I V

m Zm

=

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Computational Techniques for Distribution Systems 11

S = VIe^jφ = VI∠φ (2.4)

and defining I^* = I∠ φ, we can write S = VI^* as an equivalent apparent power.

We also note that V = ZI and I = YV yield alternate forms of S = (VI)I^* or S = V(YV)^* = VV^*Y^*. These relationships for power using the phase relation-ship can be denoted in graphical form, as shown in Figure 2.2.

2.2.1 Power Equations

For a typical power network consisting of R (resistive), L (inductive), and C (capacitive) elements, the following subsections summarize power equations in terms of the voltages and currents for the power dissipated or developed across these elements.

2.2.1.1 Resistive Element

In the case of a purely resistive network, we develop the following power relations:

(2.5)

(2.6)

FIGURE 2.2

Phasor representation of complex power relationships.

Imaginary

S=VI*

Ф -Ф

I= I -Ф

0 V= V 0° P= V I cosΦ Real Axis

Q= V I sinΦ

P t T v t i t dt

T

R( )= ¹

∫

₀ R

( ) ( )

R

P V I t V I

average= m mcos² = m m ⎡⎣ +cos t⎤⎦

2 1 2

ω ω

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12 Electric Power Distribution, Automation, Protection, and Control

Using and ,

(2.7)

The real power loss or power dissipated by the resistor in watts is com-puted using

(2.8)

2.2.1.2 Inductive Element

Similarly, for a purely inductive network, we have the following formulation for the reactive power absorbed, P_L(t) = v_L(t)i_L(t), such that

(2.9)

(2.10) and the reactive power loss in VArs is V²/X_Lwhere X_L is the reactive inductance.

2.2.1.3 Capacitive Element

For a purely capacitive element, the power developed is given as

(2.11) and the power loss in VArs is V²/XCwhere XC is the capacitive reactance.

Finally, depending on the configurations of the RLC elements (e.g., series, parallel, or series-parallel arrangements), the total real and reactive powers are computed according to the voltage and current distribution.

V V

rms= m

2 I I

rms= m

2

P V I

average= m2 m2⎡⎣1+cos2ωt⎤⎦

P VI V

R I R

R = R = ² = R²

P t_L( ) = 1V I_{m m}cos( t+ )cos( t+ − / )

2 ω φ ω φ π 2

=V Im msin[ (2ω φt+ )]

P tC( )=v t i tC( ) ( )C =V Im mcos[ (2ω θ πt+ +) / ]2

= −VIsin[ (2ω θt+ )]

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Computational Techniques for Distribution Systems 13

2.2.2 Single-Phase Power Formulations

Consider a single power source supplying a load with or without feeder impedance, as shown in Figure 2.3. Using the nomenclature defined above, recall that the sinusoidal representation of the source voltage is given as

V = V_m cos(ωt) (2.12)

and the corresponding current is

ILine = ILoad = Im cos(ωt – φ) (2.13) The power delivered to the system consisting of the network and the load is given as

(2.14) Similarly, the received power or power developed across the load is

(2.15)

At the load, the real and reactive power losses are computed using

and , respectively. The voltage sag is computed as a function of the voltage drop across the line or feeder given by ΔV = V – V_Load.

FIGURE 2.3

Single-phase generator connected to a load.

V

ILinc ZLinc

VLoad +

ZLoad

n

~

S_S=VI_L^* =VI_m cos( )cos(ωt ω φt+ )

S_R =V_{Load L}I^* =V_{Load m}I cos( )cos(ωt ω φt+ )

V R

Load Load

2

V X

Load Load

2

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2.2.3 Balanced Three-Phase Power Formulations

Consider a three-phase (3φ) power source supplying a load. For a 3φ, if the system (loads/generator) connected as wye (Y) or delta (Δ) is to be balanced, the following conditions must be satisfied:

1. For loads: impedances in all three phases are equal.

2. For sources: voltage magnitude and current magnitude are equal but evenly and spatially distributed by 120° for a three-phase system.

Now, let Va, Vb, and Vc represent the voltages of phases a, b, and c, respec-tively, given as

Va = Vm cos(ωt) (as reference) (2.16) V_b = V_m cos(ωt – 120°) (2.17) Vc = Vm cos(ωt + 120°) (2.18) Similarly,

I_a = I_m cos(ωt + φ) (2.19) I_a = I_m cos(ωt + φ – 120°) (2.20) Ic = Im cos(ωt + φ + 120°) (2.21) Then

(2.22)

(2.23)

But , therefore

(2.24) S_3φ=(V I_{a a}^*+V I_{b b}^*+V I_{c c}^*)

P3φreal=V Ia a+V Ib b+V Ic c= ℜ [ ]e S3φ

=V Im m[cos²ωt+cos (² ωt−120° +) cos (² ωt+120°)

cos₂ 1 cos2 θ= + 2 θ

P V I

t t

3_φ = ^{m m}2 _⎡⎣1+^cos2ω + +

(

1 ^cos(2ω −240°⁾

)

^{+ +}

(

1 ^cos(2^{ωωt +240 )°}

)

_⎤⎦

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(2.25)

And the per-phase power is therefore

(2.26)

2.3 Balanced Voltage to Neutral-Connected System 2.3.1 Wye- or Y-Connected System

Consider Figure 2.4.

FIGURE 2.4

Wye or Y-connected system.

P V I

t t t

m m

3_φ = 2 ⎡⎡⎣3+cosω −cos(2ω −240° +) cos(ω +240°)⎤⎦

= 3 2 V Im m

P V I V I

φ= ^{m m} = ^{m m}

2 2

~ ~

~ a

c b

a’

n’

b ’

c’

In

I bn Icn

Ian

V a’n

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and for a Y-connected system,

(2.31)

(2.32)

(2.33) Overall, the line-to-line voltages are given as a phase-to-ground quan-tity, and I_L = I_P for a Y-connected system. In the Y-connected network, if the currents are balanced in magnitude and phase, then I_a + I_b + I_c = I_n = 0.

Otherwise, unbalanced line currents lead to I_a + I_b + I_c = I_n≠ 0.

2.3.2 Delta- or ΔΔΔΔ-Connected System

Consider Figure 2.5. For the 3φ -connection shown, the phase voltages are equivalent to the line voltages such that

VL = VP (2.34)

and the line current is

V V V

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(2.35) The phase currents are

(2.36)

(2.37)

(2.38) And, using Kirchoff’s current law (KCL), the line currents are

(2.39)

(2.40)

(2.41)

FIGURE 2.5

Delta-connected system.

~

~ ~

a

c

b

a’

b ’

c’

Iab

Ica

Ibc

I aa’

Ibb’

Icc’

IL= 3IP

I_ab = ∠ °IP 0

Ibc= ∠ −IP 120°

I_ca= ∠IP 120°

∴I_aa'=I_ca−I_ab

Iaa'= 3IP∠150°

I_bb'= 3I_P∠ °30

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(2.42)

In document plus Missing Transverse Energy Final States at the Tevatron using the CDF Detector (página 84-88)