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Session 1b.

Session 1b. Electric Power Systems and How They Work. What really makes power flow ?. The amount of power flowing over the bulk power system is proportional to the difference between the power angles of the machines Actually, proportional to the sine of the angle. Power Swing Equation.

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Session 1b.

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  1. Session 1b. Electric Power Systems and How They Work

  2. What really makes power flow ? • The amount of power flowing over the bulk power system is proportional to the difference between the power angles of the machines • Actually, proportional to the sine of the angle

  3. Power Swing Equation E1 E2 sine δ12 P12 = _____________ X12 Note: Above equation can be derived from Kirchhoff’s Laws Note: Said derivation intended only for professional drivers on a closed course – do not attempt this at home!

  4. Stability The property of a system by virtue of which it will attain a new steady state condition following any one of a list of specified contingencies. Not unlike an Alexander Calder mobile.

  5. Stability (cont.) • Simulation of the dynamic response of the system, particularly the generators, to a sudden contingency. • Normally judged by examining plots of the power angles of the various generators vs. time. • In a stable case, the generator power angles will oscillate, but settle at a new equilibrium. In an unstable case, they will diverge, usually quite dramatically.

  6. Electric Systems of Pangaea

  7. VARs (Volt Amperes Reactive) • VARs are based on “Imaginary Numbers” • What’s an “Imaginary Number” ? • The square root of 9 is 3 • The square root of 4 is 2 • The square root of 1 is 1 • What’s the square root of minus 1 ? A number whose square is a minus

  8. Blame it on the Italians • 16th Century – Italian mathematicians • Cubic & quadratic equations • Solutions to some didn’t make sense • Square root of minus 1 • Raffael Bombelli • Developed first consistent theory of imaginary and complex numbers c1550 • Published Algebra in 1572

  9. Bombelli’s Great Insight “Imaginary” numbers include the square root of minus 1 – “real” numbers do not Cannot combine “real” and “imaginary” numbers • Ergo, must express them as “complex numbers” with 2 distinct parts • Form: a + bi (or, a + jb) • where a is the “real” part and b is the “imaginary” part • and i (or j) is the square root of minus 1

  10. Bombelli’s Algebra • What Newton’s Principia is to Physics … • What the Thomas Aquinas’ Summa is to Theology … • Bombelli’s Algebra is to Mathematics

  11. VARs (cont.) WATTs / VARs Real / Imaginary Active / Reactive VARs = VOLTAGE CONTROL • Too few VARs, voltage goes down • Too many VARs, voltage goes up • Can’t live without ’em, can’t shoot ’em ! (apologies to Stephen Wright)

  12. VARs (cont.) • Need VARs all along the way • Can’t move WATTs without VARs • VARs don’t travel well • Generally, they need to be produced locally • That’s because X >> R • “You want VARs with that?”

  13. Transmission LossesExamples New York “Central East” • 1 – 765 kV line operated at 345 kV • 1 – 345 kV line • 2 – 230 kV lines, a number of 115 kV lines At interface loading of about 3000 MW, losses are: • 87.7 MW • 861 MVAR (717 net after line charging) Typical ECAR (Ohio Valley) to NPCC • Incr. MW losses about 10% of incr. transfer

  14. Where do VARs Come From?(And Where do they Go to Die?) VARs are produced by : • generators & synchronous condensers • capacitors (or capacitor banks) • transmission lines (capacitive effect – “line charging”) VARs are consumed by : • the customers’ electrical load • series reactors • transmission lines (inductance – creates magnetic field) • generators & synchronous condensers French blackout – 12/7/78 – insufficient VARs

  15. VAR Analogies VARs are like your brother-in-law; they do no work. But, unlike your brother-in-law, they’re absolutely essential. • Steam locomotive • Dyeing fabric • Carrier wave • Telephone poles • Head on the beer (?) • Pushing car across inclined plane

  16. VAR Analogies (cont.) • You can think of power flow as consisting of two separate and distinct kinds of STUFF -- which can’t be mixed together, so you always have to identify them separately. • If you were ordering 6 hamburgers and 4 orders of fries, you wouldn’t try to combine them and say, “I want 10 (somethings)” !! • NO! You’d say, “I want 6 hamburgers and 4 fries.”

  17. Loehr’s First Law • VARs roll down the voltage hill • They flow in accordance with differences in voltage magnitude • WATTs roll down the angle hill • They flow in accordance with differences in voltage angle

  18. Useful Approximation Difference in voltage magnitude equals MVAR flow times inductive reactance (in per unit) E2 - E1 = Q x X12

  19. Prof. Vladimir Karapetoff (Cornell Univ.) upon death of Steinmetz: “He was allowed to try to generate electricity out of the square root of minus one.”

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