EE369 POWER SYSTEM ANALYSIS Lecture 2 Complex Power, Reactive Compensation, Three Phase Tom Overbye and Ross Baldick

Reading and Homework Read Chapters 1 and 2 of the text. HW 1 is Problems 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 2.12, 2.14, 2.17, 2.19, 2.24, 2.25 and Case Study Questions A., B., C., D. from the text; due Thursday 9/5.

Review of Phasors Goal of phasor analysis is to simplify the analysis of constant frequency ac systems: v(t) = Vmax cos(wt + qv), i(t) = Imax cos(wt + qI), where: v(t) and i(t) are the instantaneous voltage and current as a function of time t, w is the angular frequency (2πf, with f the frequency in Hertz), Vmax and Imax are the magnitudes of voltage and current sinusoids, qv and qI are angular offsets of the peaks of sinusoids from a reference waveform. Root Mean Square (RMS) voltage of sinusoid:

Phasor Representation

Phasor Representation, cont’d (Note: Some texts use “boldface” type for complex numbers, or “bars on the top”.) Also note that the convention in power engineering is that the magnitude of the phasor is the RMS voltage of the waveform: contrasts with circuit analysis.

Advantages of Phasor Analysis (Note: Z is a complex number but not a phasor).

RL Circuit Example

Complex Power

Complex Power, cont’d

Complex Power, cont’d

Complex Power (Note: S is a complex number but not a phasor.)

Complex Power, cont’d

Conservation of Power At every node (bus) in the system: Sum of real power into node must equal zero, Sum of reactive power into node must equal zero. This is a direct consequence of Kirchhoff’s current law, which states that the total current into each node must equal zero. Conservation of real power and conservation of reactive power follows since S = VI*.

Conservation of Power Example Power flowing from source to load at bus Earlier we found I = 20-6.9 amps = 1600W + j1200VAr

Power Consumption in Devices

Example First solve basic circuit I

Example, cont’d Now add additional reactive power load and re-solve, assuming that load voltage is maintained at 40 kV. Need higher source voltage to maintain load voltage magnitude when reactive power load is added to circuit. Current is higher.

Power System Notation Power system components are usually shown as “one-line diagrams.” Previous circuit redrawn. Arrows are used to show loads Transmission lines are shown as a single line Generators are shown as circles

Reactive Compensation Key idea of reactive compensation is to supply reactive power locally. In the previous example this can be done by adding a 16 MVAr capacitor at the load. Compensated circuit is identical to first example with just real power load. Supply voltage magnitude and line current is lower with compensation.

Reactive Compensation, cont’d Reactive compensation decreased the line flow from 564 Amps to 400 Amps. This has advantages: Lines losses, which are equal to I2 R, decrease, Lower current allows use of smaller wires, or alternatively, supply more load over the same wires, Voltage drop on the line is less. Reactive compensation is used extensively throughout transmission and distribution systems. Capacitors can be used to “correct” a load’s power factor to an arbitrary value.

Power Factor Correction Example

Distribution System Capacitors

Balanced 3 Phase () Systems A balanced 3 phase () system has: three voltage sources with equal magnitude, but with an angle shift of 120, equal loads on each phase, equal impedance on the lines connecting the generators to the loads. Bulk power systems are almost exclusively 3. Single phase is used primarily only in low voltage, low power settings, such as residential and some commercial. Single phase transmission used for electric trains in Europe.

Balanced 3 -- Zero Neutral Current

Advantages of 3 Power Can transmit more power for same amount of wire (twice as much as single phase). Total torque produced by 3 machines is constant, so less vibration. Three phase machines use less material for same power rating. Three phase machines start more easily than single phase machines.

Three Phase - Wye Connection There are two ways to connect 3 systems: Wye (Y), and Delta ().

Wye Connection Line Voltages Van Vcn Vbn Vab Vca Vbc -Vbn (α = 0 in this case) Line to line voltages are also balanced.

Wye Connection, cont’d We call the voltage across each element of a wye connected device the “phase” voltage. We call the current through each element of a wye connected device the “phase” current. Call the voltage across lines the “line-to-line” or just the “line” voltage. Call the current through lines the “line” current.

Delta Connection Ica Ic Iab Ibc Ia Ib

Three Phase Example Assume a -connected load, with each leg Z = 10020W, is supplied from a 3 13.8 kV (L-L) source

Three Phase Example, cont’d

Delta-Wye Transformation

Delta-Wye Transformation Proof +

Delta-Wye Transformation, cont’d

Three Phase Transmission Line