Newton s Power Flow algorithm

Transcription

1 Power Egeerg - Egll Beedt Hresso ewto s Power Flow algorthm

2 Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow 2 Calculatos. For the referece bus #, we set : V = p.u. ad δ = 0 For all other buses, =2,3,, guess a soluto, whch could be : V = p.u ad δ = 0 degrees ad set the terato couter m = tlt tlt Defe P ad Q whch s a specfed ow et power jecto at each bus of the power system except the referece bus. Substtute the guessed values of voltages to the power flow equatos ad calculate the followg quattes step 2:

3 Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow 3 Calculatos (2) re re 2. Calculate P ad Q for stace accordg to the equatos, Assumg a polar coordate decomposto all varables: re = cos = P V V y [ δ γ ] re = s = Q V V y [ δ γ ] for all buses except the referece bus,.e.: = 2, 3,...,. where j V = V δ y = y γ = y e γ

4 Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow 4 Calculatos (3) We get the followg dfferece vector: Δy tlt re P2 P2 tlt re P3 P3 tlt re ΔP P P = = tlt re Q2 Q ΔQ 2 tlt re Q3 Q3 tlt re Q Q

5 Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow 5 Calculatos (4) 3. ext calculate the Jacob-matrx : P P J J δ V 2 J = = J3 J4 Q Q δ V where the elemets of the matrx are show o the ext 2 sldes : J ad J 2 are the real power dervatve equatos J 3 ad J 4 are the reactve power dervatve equatos

6 Power Egeerg - Egll Beedt Hresso 6 The Jacob-Matrx P Wth the J sub-matrx s as follows: δ =, ( ) = V V y s δ δ γ ( ) P = V V y cos δ δ γ = P δ s ( ) = V V y δ δ γ The J 2 sub-matrx s as follows : P V = 2V y cos + V y cos ( γ ) ( δ δ γ ) j j j j j=, j P V = V y cos( δ δ γ )

7 Power Egeerg - Egll Beedt Hresso 7 The Jacob-Matrx (2) Wth ( ) Q = V V y s δ δ γ j j j j j= The J 3 sub-matrx s as follows : Q δ j=, j ( ) = V V y cos δ δ γ j j j j Q δ = V V y cos( δ δ γ ) The J 4 sub-matrx s as follows : Q V ( γ ) ( δ δ γ ) = 2V y s + V y s j j j j j=, j Q = V y s ( δ δ γ ) V

8 Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow 8 Calculatos (4) 4. Solve the lear set of equatos: ( m+ ) ( m) tlt ( m) re P P2 δ δ ( m+ ) ( m) tlt ( m) re P P δ δ 3 P P ( m ) ( m tlt ( m) re + ) P P δ V δ δ tlt ( m) re = Q2 Q2 Q Q ( m+ ) ( m) V2 V tlt ( m) re 2 Q3 Q δ V ( ) ( ) 3 m+ m V3 V 3 tlt ( m) re Q Q ( m+ ) ( m) V V óþetur vetor með leðréttgum Δy P P δ V = Δx Q Q δ V

9 Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow Calculatos (5) 9 The last matrx ca also be wrtte: P P ΔP δ V Δδ = ΔQ Q Q Δ V δ V ΔP J J2 Δδ = ΔQ J 3 J4 Δ V 5. Calculate a correcto to the varables, or: 6. ad ( m+ ) ( m) ( m) δ δ δ = +Δ. If the chages Calculate the load flow system braches accordg to: * S = P + jq = V I * * * 2 * s p S = V V V Y + V Y ( m) Δ V betwee teratos are sgfcat (>ε) go bac to step # 2, otherwse go to step # 6 Calculate the power of the referece bus ( = ) accordg to the equatos: I ( m+ ) ( m) ( m) = +Δ V V V ad Y p Δδ ( m) Y p [ δ γ ] P = V V y cos re = re = s = S Y s S Q V V y [ δ γ ] I

10 Power Egeerg - Egll Beedt Hresso 0 Gauss/ewto Comparso of Methods Both are soluto methods to solve a set of olear equatos wth the use of terato. The ewto s method s more dffcult programmg, but gves a faster soluto. The Gauss-method s smple to program but gves a more accurate soluto ad requres a loger soluto tme.

11 Power Egeerg - Egll Beedt Hresso umber of Iteratos for Load Flow Solutos (Gauss ad ewto) From: Stagg, El-Abad: Computer Methods of Power System Aalyss

12 Computato Tme for Each Power Egeerg Iterato - Egll Beedt Hresso for 2 Load Flow Methods From: Stagg, El-Abad: Computer Methods of Power System Aalyss

13 Power Egeerg - Egll Beedt Hresso 3 DC Load Flow (A lear approxmato for stuatos where may estmates of load flow are eeded a short tme such as dyamc models)

14 Power Egeerg - Egll Beedt Hresso A Lear Approxmato to Load Flow (DC Load Flow) 4 The electrcal power system s ductve ad resstaces are comparatvely small,. e.: g 0 Phase agles dffereces are geerally small betwee adjacet buses a electrcal power system,.e.: δ δ s cosδ Voltages the system uder ormal operatg crcumstaces are close to uty, or.: V

15 Power Egeerg - Egll Beedt Hresso A Lear Approxmato to Load Flow (DC Load Flow) (2) By relaxg the last assumptos (where voltages are assumed costat ad phase agles small), we get the reactve power quadratc equatos as follows: 5 [ cos( δ δ ) s( δ δ )] P = V V g + b = [ s( δ δ ) cos( δ δ )] Q = V V g b = Q = V V b = However, the reactve power quadratc equatos are ot as terestg as the equatos for real power (see ext slde):

16 Power Egeerg - Egll Beedt Hresso A Lear Approxmato to Load Flow (DC Load Flow) (3) 6 By assumg that all of the above assumptos are vald, we get the followg DC-load flow equatos, where the relatos of real power ad phase agle s a lear set of equatos: [ cos( δ δ ) s( δ δ )] P = V V g + b = [ s( δ δ ) cos( δ δ )] Q = V V g b = P = b δ = b = = [ δ δ ]

17 Power Egeerg - Egll Beedt Hresso A Lear Approxmato to Load Flow (DC Load Flow) (4) 7 These equatos are aalogous to equatos for a drect curret crcut, f : P s aalogous to the curret the curret source (Kow for load buses) b s aalogous to the resstace or coductace (Kow for load buses) δ s aalogous to the voltage at each ode. (These are the uow varables) DC power flow s approxmately 0-00 tmes faster tha covetoal" load flow (Also called full AC - load flow), but may have assocated errors.

18 Power Egeerg - Egll Beedt Hresso 8 Fast Decoupled Load Flow (FDLF) Ths s aother approxmato to the full load flow equatos to crease speed)

19 Power Egeerg - Egll Beedt Hresso The dervatves wth rectagular 9 represetato of admttaces The load flow equatos wth rectagular represetato of admttaces are repeated here: y = g + jb 2 = + + = ( δ δ ) s ( δ δ ) ( cos ) P V g V V g b 2 = + = ( δ δ ) cos( δ δ ) ( s ) Q V b V V g b [ cos( δ δ ) s( δ δ )] P = V V g + b = [ s( δ δ ) cos( δ δ )] Q = V V g b = or wrtte alteratvely :

20 Power Egeerg - Egll Beedt Hresso 20 Full AC Dervatves Real Power dervatve equatos (J ad J 2 ) are (wth dex replacg former dex ad dex m replacg former dex ) P δ m ( δ δ ) cos( δ δ ) = V V g s b m m m m m P V m ( δ δ ) cos( δ δ ) = V g s b m m m m P ( δ δ ) cos( δ δ ) = V Vm gm s m bm m δ + m= m P = 2V g + V g cos( δ δ ) + b s( δ δ ) V m m m m m m= m Reactve Power dervatve equatos (J 3 ad J 4 ) are: Q δ m ( δ δ ) s ( δ δ ) = V V g cos b m m m m m Q V m ( δ δ ) cos( δ δ ) = V g s b m m m m Q ( δ δ ) s ( δ δ ) = V Vm gm cos m bm m δ + m= m Q = 2V b + V g s( δ δ ) b cos( δ δ ) V m m m m m m= m

21 Power Egeerg - Egll Beedt Hresso 2 Decoupled Power Flow Equatos Mae the followg assumptos: δ δ m V cos( δ δ m ) s( δ δ m ) 0 The dervates smplfy to: 0 r m << x m 0 g m P δ = m= m b m P δ m = b m P V = 0 P V m = 0 Q Q = 2 b + ( bm ) = b Q m = 0 V V δ m= m m How do we terpret the above dervatve terms? δ Q m = 0

22 Power Egeerg - Egll Beedt Hresso To do that we remember the Y bus Matrx Composto Rule: The dagoal elemets of the Y bus matrx cosst of the sum of all the (seres ad shut) admttaces coected to the bus questo The off-dagoal elemets of the Y bus matrx cosst of the egatve value of the seres admttaces coectg the 2 buses questo. Therefore shut terms at bus # of : shut, m m= m are the sum of each row or colum of Y. The shut terms are show o the ext slde: Y bus bus y y y = +

23 V V 3 Power Egeerg - Egll Beedt Hresso A example of shut terms or shut elemets the Y bus matrx The shut terms at bus # are Y s,,3 y = Y + Y shut, p,,2 p,,3 23 I Y p,,3 Y p,3, I 3 Y p,,2 V 4 Y s,,2 Y s,2,4 Y p,4,2 Y p,2, V 2 The seres term betwee bus # ad bus #2 I 2 Y p,2,4 The shut terms at bus #2 are y = Y + Y shut,2 p,2, p,2,4

24 Power Egeerg - Egll Beedt Hresso 24 We chec out the Shut terms B bus g g2 g b b2 b g2 g22 g 2 b2 b22 b 2 Y j bus = + = Gbus + jb g g g b b b 2 2 Dagoal elemets the magary part of Y (that s B ) are composed of seres ad shut terms, or: b = b + b bus Shut part of b : b = b + b shut, m m= m shut. seres, bus bus Seres terms of b : b = ( b ) seres, m m= m

25 Power Egeerg - Egll Beedt Hresso 25 Defto of the B matrx: Therefore the shut terms at bus # of : b = b + b = b ( b ) shut, m m m= m= m m are the sum of all elemets each row or colum of B. Compare ths P to the above dervatves: = bm ad δ m m= m B P P = B = J. Sce = bm = ( b bshut, ) = bseres, we fd that: δ δ bus P δ = m= m b bus m ad defe the matrx B : B s the s the egatve of the magary part of the Y bus matrx (B bus ) oly wth seres terms or wth all the shut terms removed from B bus.

26 Power Egeerg - Egll Beedt Hresso 26 Defto of the B matrx: ow loo at the dervatves: Q V Q = 2 b + ( b ) ad = b ad defe the matrx B : m m m= Vm m Q = B = J. Sce b = b + b ad b = ( b ) ad sce V Q V 3 shut, m seres, m m= m= m m = 2 b + ( b We fd that: m= m ) = ( b + 2 b ) m seres, shut, B s the s the egatve of the magary part of the Y bus matrx (B bus ) wth all the shut terms couted twce (double couted) B bus.

27 Power Egeerg - Egll Beedt Hresso Comparso wth textboo verso (Saadat) 27 Wth o shut terms (all ls as serse mpedaces) B ad B are smply the egatve of B bus (page 242, Saadat). Ths s the case the textboo. Wth some voltages ot set equal to pu we get equatos (6.75) ad (6.76) Saadat. Wth Q = 0 (6.7) o shut elemets are preset B ad B ther textboo verso. It s a cruder approxmato tha the above where they are ether removed or double couted.

28 Power Egeerg - Egll Beedt Hresso 28 Soluto wth B ad B Matrces ow terate the decoupled equatos Δδ = ΔV = [ ] ' B ΔP [ B ] '' ΔQ

29 Power Egeerg - Egll Beedt Hresso 29 Fast Load Flow ΔP J J2 Δδ = ΔQ J2 J4 Δ V J2 0 J depedet sets of lear equatos. The dmesos for each of them s half the orgal set of equatos ΔP B 0 Δδ ΔQ 0 B Δ V ' = '' ΔP = J Δδ ΔQ = J Δ V 4

30 Power Egeerg - Egll Beedt Hresso Fast Decoupled Load Flow (FDLF) 30 FDLF s based o a approxmato ad s faster but more accurate tha covetoal power flow The approxmato taes to accout: That voltage a power system s tghtly coected to reactve power That phase agles a power system are tghtly coected to actve power

31 - securty rule Power Egeerg - Egll Beedt Hresso 3 Fally we meto a very well ow practcal securty rule of thumb: The system ca loose oe ut wthout losg customers or a blacout occurrg the system Ths ut ca be a trasmsso le, a trasformer or a geerator, for stace We specfy that durg a dsturbace customers wll ot be terrupted Ths rule s called the - rule

32 Hstorc refereces Power Egeerg - Egll Beedt Hresso 32 B. Stott ad O. Alsac, Fast decoupled load flow, IEEE Trasactos o Power Apparatus ad Systems, Vol. PAS-93, Jue 974, pp W. F. Tey ad C. E. Hart, Power Flow Soluto by ewto s Method, IEEE Trasactos o Power Apparatus ad Systems, Vol. PAS-86, o., ov. 967, pp B. Stott, Revew of Load-Flow Calculato Methods, Proceedgs of the IEEE, Vol. 62, o. 7, July 974, pp