Representation of parts of electric power systems by generalized static characteristics has been considered. The design procedure of steepness factors of generalized static characteristics depending on features of equivalent subsystems is discussed. The generalized static characteristics and their steepness factors give the equivalent information on power subsystems and can be used at estimation of static aperiodic stability of power supply systems

Gotman, V. I.

Markman, G. Z.

Electronic archive of Tomsk Polytechnic University

REFERENCES1. Rastrigin L.A. Statistical search technique. – Moscow: Nauka, 1968.– 375 p.2. Minakov I.A. Comparative analysis of some methods of random search and optimization // Bulletin of Samara scientific center of Russian academy of science. – 1999. – № 2. – P. 286–293.3. Lewis R., Virginia T. Pattern Search Algorithms for Bound Constrained Minimization // SIAM Journal on Optimization. – 1999. –№ 4. – P. 1082–1099.4. Zhezhelenko I.V. et al. Quality of electric energy at industrial enterprises. – Kiev: Tekhnika, 1981. – 160 p.5. Drekhsler R. Power and loss factor in network at asymmetrical andnonlinear consumer // Elektrichestvo. – 1982. – № 2. – P. 12–16.6. Avvakumov V.G. Statement and solution of electric power problemsof operation studying. – Kiev: Vischa Shkola, 1983. – 240 p.Received on 18.09.2007Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 4124Steepness factors and load and generator effects arewidely used in mode calculations and estimation of static stability of electric systems [1]. Static characteristicsand steepness factors not only of separate elements ofthe systems but of their complex combinations formingsubsystem are of practical interest. The generalized static characteristics and their steepness factors give theequivalent information on subsystem state. Let us singled out the subsystems in which there are no slack nodes by active power and subsystems containing slack nodes of active power depending on conditions accepted atequivalenting. In this paper the calculation expressionsfor determining steepness factors of static characteristics of complex electric power subsystems are introduced for the first time.Let us turn to subsystem without slack nodes by active power. Let us arrange to call a part of a system havingconnection with the main or adjacent part in a single node i, the limited part (Figure). Let us suppose that in acertain initial steadystate condition in the main part ofelectric system the stationary disturbance occurred. Itresulted in change of module ΔUi and phase Δδi of nodevoltage to which the limited subsystem abuts. Circuit layout, composition of equipment (including consumers),transformation ratios, settings of the first and secondaryregulators are stable in the given subsystem. The two latter conditions are adequate to constancy of source active power. Let us analyze the changes which occur in thesingled out subsystem at alternate and independent occurrence of disturbances ΔUi and Δδi.Under the action of applied disturbance ΔUi(δi=const) the subsystem operating conditions are deformed. Active power of power sources is constant andload node power (according to static characteristics)and loss in network elements change. Unbalance of active (ΔP) and reactive (ΔQ) powers which tends to thenode i is formed in subsystem.At stationary increment of phase Δδi (Ui=const) thesubsystem moves as a unit relative to synchronous axisof node i. Stress vectors of all nodes in subsystem change by the same angle as phase increment in common angle i. Reference angles of stress vectors in subsystem areconstant. Its mode is constant as well that indicates theabsence of subsystem reaction on the specified disturbance Δδi. As the limited subsystem has the only connection with the main part then functional dependences(1)are the full mode equivalent of subsystem.At known reset conditions of the system the generalized static characteristics are found out by calculationsof a number of steadystate conditions of equivalentingsubsystem at stress module variation of equivalentingnode. Equivalenting node is accepted as a balancing oneby active and reactive powers. Calculations allow introducing dependences (1) in Table, diagram or analytically at proper results processing.The subsystem diagram is given in the form of passive multiport (npole network). Twoterminal networkssubstituting generators, sources of reactive power andload are connected to its vertices. Let us suppose thatelements of twoterminal networks are introduced bytheir static characteristics by voltage.( );  ( )ic i ic iP P U Q Q U= =UDC 621.311.016.361GENERALIZED STATIC CHARACTERISTICS OF ELECTRIC POWER SUBSYSTEMS AND THEIR STEEPNESS FACTORS V.I. Gotman, G.Z. MarkmanTomsk Polytechnic UniversityEmail: mo@elti.tpu.ruRepresentation of parts of electric power systems by generalized static characteristics has been considered. The design procedure of steepness factors of generalized static characteristics depending on features of equivalent subsystems is discussed. The generalized staticcharacteristics and their steepness factors give the equivalent information on power subsystems and can be used at estimation of staticaperiodic stability of power supply systems.Let us consider the calculation of steepness factorsαic and βic of generalized static characteristics of theequivalenting subsystem (Figure)on the basis of the increment equations given below inwhich αi, βi are the steepness factors of static characteristics of twoterminal networks:(2)In order to calculate the coefficient βic we supposethat disturbance ΔQic at fixed phase of stress vector inequivalenting node i (δi=const) which is simultaneously balancing by active power that is reflected by the element ai,n+i=–1 in matrix Wβ is applied to node i. solvingthe system (2) relative to ΔUi at specified ΔQic the coefficient(3)is found out where Dβ is the determinant, Aβ(n+i,i) is the algebraic complement of matrix Wβ.Calculating the coefficient αic similarly to the previous one let us consider that external disturbance ΔPicat condition δi=const is applied to the node i. The nodei is balancing by the reactive power. In order o obtainmatrix Wα it is sufficient to consider an+i,n+i=–1 to be anonzero element in column n+i of matrix Wβ. Then:(4)where Dα is the determinant, Aα(i,i) is the algebraic complement of matrix Wα. In this case Aα(i,i)=–Aβ(n+i,i).Figure. The equivalenting subsystemLet us consider the substitution of the limited subsystem containing nodes balancing by active power by theequivalent twopole network with specified static characteristic. For the given subsystem it is appropriate touse the same form of introducing static characteristicsas for generator [2]:(5)We get(6)from (5).In order to differentiate singletype steepness factorsthe index of coefficient β determined at conditionPic=const for subsystems with nodes balancing by activepower is parenthesized.The value of coefficients β(iс) and ηi depends on distribution of «unbalance» between generator and loadnodes of subsystem. Depending on the solved problemthis «unbalance» may be distributed between several nodes of generation and load or fully referred to one node.The linearized system of equations (7) for calculation ofβ(iс) of subsystems (Figure) is obtained provided that disturbance ΔQic and δi=const (stress phase of any nodemay be taken as a basic one) is applied to the node i.(7)For community all generator and load nodes of subsystems except the equivalenting node are taken balancing by active power. A part of general «unbalance» ofactive power taken by an arbitrary node j is accepted inratio to increment of the power of node n (ΔPj=KinΔPn).Coefficients ηj in (7) are nonzero only for generatornodes involved in balancing. We have:(8)from (7) where D(β) is the determinant, Aβ(n+i,i) is the algebraic complement of matrix W(β).Coefficient ηi shows by how many units the reactivepower transferred to the node i of the equivalenting subsystem should be changed for Ui to stay constant atchanging active power of the node i by a unit stress. Being founded on this determination and retaining the same nodes of active power balancing that at β(ic) calculation the linearized system similar to (7) may be obtained. In the main system matrix W(β) is replaced by matrix Wη which differs from the first one by the fact thatelements of the column i equal zero except an+i,i=–1.Elements of this column are the coefficients at ΔQic. Theright part of the examined system contains disturbanceΔPic. At these conditions we have:where Dη is the determinant, Aη(i,i) is the algebraic complement of matrix Wη. Let us note that Dη=–Aβ(n+i,i).( , )/ / ,i ic ic i iQ P A Dη ηη = ∂ ∂ =( ) ( ) ( , )/ / ,ic ic i n i iQ U D Aβ ββ += ∂ ∂ =( )ic icic i ic ic i i ici icQ QQ U P U PU Pβ η∂ ∂Δ = Δ + Δ = Δ + Δ∂ ∂( , ).ic i icQ Q U P=2 1n j i ?Pic, ?Qic?ic, ?icUi( , )/ / ,ic ic i i idP dU D Aα αα = =( , )/ / ,ic ic i n i iQ U D Aβ ββ += ∂ ∂ =/ ;  /ic ic i ic ic iP U Q Uα β= ∂ ∂ = ∂ ∂Power engineering125Dividing expression (6) by ΔUi we obtain the interaction between the steepness factors of the examinedtypes of static characteristics:(9)Conditions of connection (9) may be obtained fromphysical considerations relying on notions putting in theseor those coefficients and conditions of their calculation.Applying disturbance ΔUi to the node i of power system (Figure) provided that the whole «unbalance» byactive power is received by twoterminal networks andΔPic=0, we obtain the reaction ΔQic. The ratio of this reaction to disturbance ΔUi gives the value β(ic). The sameresult may be obtained carrying out two experimentssuccessively and introducing respectively the resultantreaction ΔQic in the form of two components:(10)Really, applying external disturbance ΔUi and considering that only node i is balancing by active and reactive powers we obtain the reaction in the form of ΔQ 'icand ΔPic which are connected with disturbance byknown ratios:(11)Condition ΔPic=0 necessary at β(ic) calculation is notobserved here. In order to balance ΔPic reaction let usapply disturbance –ΔPic to the node i provided thatUi=const. Considering that «unbalance» of active powerin this experiment is received by twoterminal networkswe have the second component of the reaction by activepower in the node i:(12)Taking into account the (11) and (12) condition (10)is written as:This equality gives known interaction being dividedby ΔUi (9).The subsystem having several joint nodes with themain part may be artificially divided into some subsystems limited by a number of abutments. For this purpose it is necessary in the singled out subsystem to openbranches in the nodes where «hard» voltage control isfulfilled saving previous values of parameters on theends of open branches. The procedure of branch cuttingin nodes with U=const is very strict. Branch cutting inthe most remote nodes from the equivalenting one maybe assumed at violation of voltage stability condition.The circumstance that static characteristics and steepness factors are mainly determined by the elementsand sections of the system located closer to the equivalenting node may serve as the base for this.At slight changes of normal mode the steepness factors βic, αic change insignificantly and may be acceptedas constant. At deeper mode change they should be adjusted.The possibility of presenting the equivalenting subsystem in the form of fourterminal network containingthe branch of impedance Z, to a free end of which theconstant equivalent emf is applied, is considered in [3].The initial information for such equivalenting are theparameters of steadystate mode of the equivalentingsubsystem and steepness factor β, α.Conclusion Substitution of electric system parts by the equivalent twoterminal networks with specified static characteristics and steepness factors allows in some cases reducing considerably the dimension of electric system (anumber of required parameters). Such approach is appropriate at studying and modeling the modes of somestations, substations, calculations of ladder networkmodes limiting by static aperiodic stability (for example,longdistance power transmissions with intermediatesystems). It gives an opportunity to take into account ingeneralized form the reaction of separate system partsand localize directly the studied object.( ) .ic i i ic iicQ U Uβ ηαΔ = Δ − Δ.ic i icQ Pη′′Δ = − Δ;  .ic ic i ic ic iQ U P Uβ α′Δ = Δ Δ = Δ( ) .ic ic icQ Q Q′ ′′Δ = Δ + Δ( ) .ic ic i icβ β ηα= −Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 4126REFERENCES1. Levinshtein M.L., Scherbachev O.V. Design procedure of static stability of complex electric systems by the equivalent controlling effects of stations and loads // Izvetiya Vuzov. Energetika. – 1962. –№ 8. – P. 11–19.2. Tsukernik L.V., Korobchuk K.V. Design of static stability limit ofcomplex energy system by digital computer // Report at the II allUSSR scientific and technical conference on stability and reliabilityof USSR power systems. – Moscow: Energiya, 1969. – P. 56–62.3. Gotman V.I., Gotman O.V. Equivalenting of subsystems of powerpool on the basis of operating conditions // Izvetiya Vuzov. Elektromekhanika. – 2006. – № 3. – P. 111–114.Received on 20.09.2007

Generalized static characteristics of electric power subsystems and their steepness factors

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