3- BASICS. YTransformation. for balanced load. \V ab 120 = \V bc. \V ab 240 = \V ca \I a 120 = \I b \I a 240 = \I c V ab I a

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1 3- BASICS YTransformation for balanced load Z =3Z Y Balanced 3- Systems \V ab 10 = \V bc \V ab 40 = \V ca \I a 10 = \I b \I a 40 = \I c V ab I a = p 3 V an = p 3 I ab \V ab 30 = \V an \I ab 30 = \I a S 3 =3V ani a S 3 =3V ab I ab S 3 = p 3V ab I a\30 PER UNIT SYSTEM S B = V BI B Z B = 1 Y B S 3 B V line line B = V B S B =3SB = p 3V B line line Z 3 VB B = ZB = S 3 B MAGNETIC CIRCUIT RELATIONS µ 0 = R = l µa L = N R = N µa l F = NI = l µa = R Faraday s Law V (t) = d (t) dt 1 = N d (t) dt

2 TRANSFORMERS Ideal transformer voltage and current relations E 1 E = N1 N = a = 1 n I 1 = N = 1 I N 1 a = n Transformer modelling - impedance relations secondary winding impedance reflected to primary Z 0 = N1 Z = a Z N primary winding impedance reflected to secondary Z1 0 = N Z 1 = n Z 1 N 1 equivalent impedance seen at primary Z in = Z 1 + a Z + a Z load Impedance Diagram Representation Z 1,B = V 1,B S B and Z,B = V,B S B = n V 1,B S B Z 1,p.u. = Z 1 Z 1,B and Z,p.u. = Z Z,B Autotransformer - impedance relations equivalent impedance seen at primary - step down, neglecting core loss Z in = Z 1 +(a 1) Z + a Z load equivalent impedance seen at primary - step up, neglecting core loss Z in =(a 1) Z 1 + a Z + a Z load conducted and transformed power S capacity = S transformed + S conducted capacity increase - step up (a 1) S capacity S rating = a a 1 capacity increase - step down (a apple 1) S capacity S rating = 1 1 a

3 VR. = E. = Pout P in 100% = ciency 1 P losses P in 100% Voltage Regulation V1,serving rated load at rated voltage 1 V 1,rated 100% POWER AND ENERGY FUNDAMENTALS P m =! mt m = e ai a Energy - power relationship P = dw dt Energy stored in inductor W = 1 LI Mechanical kinetic energy W = 1 mv Mechanical energy stored in rotating machine W m = 1 J! m Magnetic field energy stored in machine inductances W f = 1 Lssi s + 1 Lrri r + L sri si r FARADAY S LAW V = d dt CYLINDRICAL ROTOR AC MACHINE Some notation for rotating machines p. = # of poles n s.! s = sychronous frequency. = synchronous mechanical speed in rpm.! m = rotor mechanical frequency. = rotor mechanical speed in rpm n r N r. = # of rotor winding turns 3

4 L sr N s L rr L ss. = rotor self inductance. = # of stator winding turns. =statorselfinductance. = rotor-stator mutual inductance Self and Mutual Inductances L sr = 4µ0NsNrlr e s(t) = d(lsrir) dt L ss = 4µ0N s lr cos m = M cos m L rr = 4µ0N r lr Induced armature voltage = 4µ0NsNrlr Induced armature voltage d (i r(t)cos (t)) dt assuming constant speed and DC rotor current e s(t) = 4µ0NsNrlr!m I r cos (! mt +90 ) Electrical torque T e f (i s,i r, m assuming constant speed T e = Mi si r cos(! mt +90 ) and also T e = esis! m DC MACHINES Induced armature voltage E a = K a p! m = K 0 ai f! m Electrical torque T e = EaIa! m Shunt connected motor - p or I f constant E a = V t I ar a T e = K 0 ai f I a 4

5 ! m = Vt K 0 ai f T er a (K 0 ai f ) = R f K 0 a Series connected motor (fields additive)- I a = I f T e = K 0 ai a T er ar f (K 0 av t) p varies linearly with I a,i.e,! m = Vt R a + R f = Vt R a + R f p KaI 0 a Ka 0 K 0 a T e Ka 0 CYLINDRICAL ROTOR SYNCHRONOUS MACHINES Speed and frequency relationships! s = p!m n r = n s = 10 p fs Circuit relation E a = V t + R ai a + X si a Electrical power output - lossless, per phase P out = EaVt X s sin EaVt cos Vt Q out = X s INDUCTION MACHINES Speed and frequency relationships s = ns n s nr. =slip! r =! s! m = s! s. = rotor current frequency f r = sf s = s pns 10 Input impedance per phase Z in = R 1 + X 1 + R c//x c//( R s + X) Copper loss per phase P loss = R 1I 1 + R I Mechanical power developed P dev = 1 s R I s 5

6 LINE PARAMETERS Line inductance per phase L = µo GMD ln GMR b if transposed GMD = 3p D ab D bc D ac if bundled GMR b = bp r 0 d 1d 13...d 1b µ o r 0 = re µr/4 7 = 10 henrys/meter Line capacitance per phase C = o ln GMD GMR c b GMR c b = bp rd 1d 13...d 1b o = farads/meter CIRCUIT MODELS Line Characteristics ABCD parameters - medium line apple " V1 = I 1 Z = zl and Y = yl 1+ YZ Z Y (1 + YZ 4 ) 1+ YZ # apple V ABCD parameters - short line Z = zl apple apple V1 1 Z = I apple V POWER CAPABILITY OF LINES Power flow on a line - assuming A = D with A = A\, B = B\, V 1 = V 1\ and V = V \0 I I P 1 = A B V 1 cos( ) V 1V B cos( + ) 6

7 Q 1 = A B V 1 sin( ) V 1V B sin( + ) P = V1V B cos( ) A B V cos( ) Q = V1V B sin( ) A B V sin( ) Power flow on lossless line for medium line and short line B = X so P = V1V X sin P max = V1V X POWER FLOW Admittance Matrix Y bus = G + B = Y i = Yi\ i Y ii = X y i. =Sumofadmittancesconnectedtobusi. Y i = y i =( )Admittance between bus i and Power flow equations X n S i = S Gi S Di = V i V Y i\ ( i i) S i = V i or X n Y iv or P i = P Gi X n P Di = V i V [G i cos( i )+B i sin( i )] = f i(v, ) X n Q i = Q Gi Q Di = V i V [G i sin( i ) B i cos( i )] = g i(v, ) Numerical iteration General form update = old value + iteration matrix*error x i+1 = x i + D 1 [y f(x i )] Gauss D = diag{d i} Gauss-Seidel iteration calculations for power flow 7

8 at slack bus at PV (generator) buses (using latest voltage updates in summation for Gauss-Seidel form, k is the iteration count) "!# Q (k+1) i (k+1) i = \ = Im ( Y 1 ii V (k+1) i = Y 1 ii V (k) i S i V (k) i Xi 1 Y iv (k+1) + Xi 1 Y iv (k+1) at PQ (load) buses S i V (k) i Xi 1 Y iv (k+1) nx =i Y iv (k) nx =i+1 nx =i+1 Y iv (k) Y iv (k)!)! 8

Synchronous Machines

Synchronous Machines Synchronous generators or alternators are used to convert mechanical power derived from steam, gas, or hydraulic-turbine to ac electric power Synchronous generators are the primary