EE 742 Chapter 3: Power System in the Steady State. Y. Baghzouz
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1 EE 742 Chapter 3: Power System in the Steady State Y. Baghzouz
2 Transmission Line Model Distributed Parameter Model: Terminal Voltage/Current Relations: Characteristic impedance: Propagation constant: π equivalent Circuit: π equivalent Circuit parameters: where is the total series impedance and is the total shunt admittance:
3 Line shunt and series elements Line shunt admittance: Line series reactance:
4 Transmission Line Model Simplified models for medium and short lines: Medium line ( 50 mi < l < 150 mi): Short line ( l < 50 mi): In a typical line, g can be neglected. If r is also neglected (since r << x), then the line becomes lossless and the characteristic impedance becomes a pure real number ( Zc = (L/C) 1/2 ), while propagation constant become pure imaginary number (γ = jβ = jω(lc) 1/2 ). The voltage/current equations become Note that the voltage and current vary sinusoidally along the line length. The wavelength of the full cycle is λ = 2π/β.
5 Surge Impedance Loading (SIL) Power delivered at rated voltage to a load whose impedance is equal to the surge impedance: Under such condition The voltage and current profiles are flat The reactive power loss in the line is zero. In reality, the loading is rarely equal to the natural load. See voltage variation with loading below
6 Real Power Transmission Complex power at receiving end (assuming lossless line): Real power at receiving end: Maximum power at receiving end For lines less than 150 miles,
7 Reactive Power Considerations Complex power at receiving end (assuming lossless line): Reactive power at receiving end: Approximate expression of lines less than 150 mi long: And for small transmission angles,
8 Reactive Power Considerations Change in Q R and in Q S (when the influence of real power flow is neglected) Reactive power loss under the presence of real power flow:
9 Homework # 4
10 Underground cables Modeled the same way as overhead lines The typical per-unit series reactance of a cable is about half of that of an overhead line of the same rating. The per-unit charging current of a cable is about 30 times larger that that of an overhead line of the same rating thus severely limiting transmission capacity.
11 Transformers Equivalent circuit (see above figure) Approximate equivalent π circuit (see figure below)
12 Transformers Equivalent circuit with off-nominal turn ratio (including phase shift) turn ratio is represented by a complex number.
13 HW # 5 Two transformers A and B (each with a series impedance of j0.1 pu) are connected in parallel and they supply a constant impedance load of 0.8+j0.6 pu. The secondary voltage if fixed at V 2 = o pu. Find the active and reactive powers transmitted through each transformer to the load. Repeat the above if the off-nominal turn ratio of transformer A is set to o. Repeat the above if he off-nominal turn ratio of transformer A is set to o.
14 Synchronous Generators: 2-pole round rotor For a 2-pole machine, ω m = ω e, and γ m = γ e Rotor flux produced by field current : Induced emfs (internal voltages) in stator windings:
15 Synchronous Generator under no-load Mutual inductance: Time variation of flux linkages and induced emfs (fig. a) Phasor representation of flux linkages and induced emfs (fig. b) RMS value of each emf:
16 Synchronous generator armature reaction The stator phase currents produce pulsating phase mmfs that are shifted in both space and time. The resultant stator mmf F a is constant (=1.5 N a I m ) and rotates with angular velocity ω (i.e., stationary w.r.t. the field mmf F f ). The new air-gap flux mmf F r is equal to the vector sum of F a and F f. (weaker than without load)
17 Total mmf in phase a : Equivalent Circuit Flux linkage in phase a : Induced emf in phase a : Phasor form: Terminal voltage (after taking stator winding resistance and leakage flux into account):
18 Equivalent circuit Armature reaction (or magnetizing) reactance: X a Leakage reactance: X l Synchronous reactance (or direct-axis synchronous reactance): X d = X a +X l Internal emf (or voltage behind the synchronous reactance: E f
19 Electromagnetic Torque For the rotor speed to be constant, the two opposing mechanical torque τ m and electromagnetic torque τ must be equal. Neglecting the mechanical losses, the air gap power must equal the mechanical power: τ m ω m = 3E f Icosβ (where β is the angle of I with respect to E f ) For a machine with p-poles, the electromagnetic torque can be written in a number of forms ( note: λ=π/2+β, and the torque angle δ fr is the angle between F r and F f.) for a 2-pole machine
20 Salient pole machines 2 poles Non-uniform air gap, hence variable reluctance. Decomposition of mmfs and current into d-component and q-component. Let X ad and X aq represent the direct-axis and quadrature-axis armature reaction reactances.
21 Resultant air gap emf: Salient pole machines 2 poles Resultant terminal voltage: Herein, and X d and X q are the direct- and quadrature-axis synchronous reactances. Note: X d > X q because the reluctance along the q-axis is highest. Note: for a round rotor, X d = X q, and E Q = E f.
22 Phasor Diagram and Equivalent Circuit
23 Torque Electromagnetic torque of a generator with 2-pole salient rotor (see derivation on pp. 88) Synchronous torque Reluctance torque
24 Generator-transformer unit The generator impedance is increased by the transformer impedance. For the round rotor machine; For the salient rotor machine
25 Real and reactive power of generatortransformer unit
26 Power-angle characteristic of generator with salient rotor When r is ignored and E q is held constant, then the power contains two terms. The latter (i.e., reluctance term) deforms the sinusoid. Hence peak power occurs at an angle that is less than 90 o.
27 HW # 6 Consider a 230 MVA generator with salient-pole rotor. The per-unit values of R, X d and X q are 0. 05, 0.95 and 0.65, respectively. The generator is delivering j0.25 pu to of complex power to the network and the terminal voltage is at 1 pu. Determine the d-axis and q-axis components of the generator current, terminal voltage, and internal emf. Calculate the active and reactive powers supplied by the generator using the expressions in slide # 23 and compare to the above values. Finally, determine the maximum power and corresponding angle δ if the E and V are held constant.
28 Real and reactive power capability curves Field current and voltage limit Turbine/boiler limit Stator current limit End region temperature limit Power angle stability limit
29 Voltage-reactive power capability curve Consider a round rotor generator connected to a power system through a transformer. Voltage is to be regulated at a fictitious point (V c ) within the transformer.
30 V(Q) Generator Curves Curve I: Field current less than its maximum value. Curve II: Maximum field current Curve III: Maximum power angle Curve IV: Maximum stator current P = 0 P 0
31 Power System Loads Only simple static composite load models are described (dynamic models will be seen in later chapters). The active and reactive power demand of a static composite load depends on the voltage and frequency. Voltage and frequency sensitivity: slope of load-voltage or loadfrequency characteristics (see fig. below)
32 Lighting and heating load characteristics Voltage characteristics of incandescent and fluorescent bulbs (see fig. below) Heating load equipped with thermostat is considered a constant power load. If not, its is considered a constant resistance load.
33 Induction Motors Equivalent circuit with stator impedance neglected. Voltage characteristic under constant load torque.
34 Influence of tap-changing transformer on composite load voltage characteristics
35 ZIP and Exponential and Frequency-Dependent Load Models ZIP model: Exponential model: Frequency dependent model:
36 Bus admittance matrix Network Equations Bus impedance matrix
37 Power flow equations
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