Primary Author: James D. McCalley, Iowa State University

Transcription

1 Module T7: The ower Flow roblem 7 Module T7 The ower Flow roblem rimary Author: ames D. McCalley, Iowa State University Address: dm@iastate.edu Co-author: one rereuisite Cometencies:. Steady State analysis o circuits using hasors.. er-hase analysis o three-hase circuits, ound in module..one-line diagrams, ound in module. 4. er-unit analysis, ound in module enerator oeration, ound in module. 6. Transmission line characteristics and calculations or ower lowing across a line, ound in module T. Module Obectives:. Formulate the ower low roblem.. Aly the ewton-rahson algorithm to solve systems o nonlinear algebraic euations.. erorm a ower low solution. 4. Identiy basic inut and outut uantities associated with commercial ower low rograms. T7. Introduction The ower low roblem is a very well nown roblem in the ield o ower systems engineering, where voltage magnitudes and angles or one set o buses are desired, given that voltage magnitudes and ower levels or another set o buses are nown and that a model o the networ coniguration unit commitment and circuit toology is available. A ower low solution rocedure is a numerical method that is emloyed to solve the ower low roblem. A ower low rogram is a comuter code that imlements a ower low solution rocedure. The ower low solution contains the voltages and angles at all buses, and rom this inormation, we may comute the real and reactive generation and load levels at all buses and the real and reactive lows across all circuits. The above terminology is oten used with the word load substituted or ower, i.e., load low roblem, load low solution rocedure, load low rogram, and load low solution. However, the ormer terminology is reerred as one normally does not thin o load as something that lows. The ower low roblem was originally motivated within lanning environments where engineers considered dierent networ conigurations necessary to serve an eected uture load. Later, it became an oerational roblem as oerators and oerating engineers were reuired to monitor the real-time status o the networ in terms o voltage magnitudes and circuit lows. Today, the ower low roblem is widely recognized as a undamental roblem or ower system analysis, and there are many advanced, commercial ower low rograms to address it. Most o these rograms are caable o solving the ower low rogram or tens o thousands o interconnected buses. Engineers that understand the ower low roblem, its ormulation, and corresonding solution rocedures are in high demand, articularly i they also have eerience with commercial grade ower low rograms. The ower low roblem is undamentally a networ analysis roblem, and as such, the study o it rovides insight into solutions or similar roblems that occur in other areas o electrical engineering. For eamle, integrated circuit designers also encounter networ analysis roblems, although o signiicantly smaller All materials are under coyright o owerlearn. Coyright, all rights reserved.

2 Module T7: The ower Flow roblem 7 hysical size, are uite similar otherwise to the ower low roblem. For eamle, reerences [,] are well-nown networ analysis tets in LSI design that also rovide good insight into the numerical analysis needed by the ower low rogram designer. Similarly, there are numerous classical ower system engineering tets, [-] are a reresentative samle, that rovide advanced networ analysis methods alicable to LSI design and analysis roblems. Section T7. identiies an eature o ower generators imortant to the ower low roblem real and reactive ower limits. Section T7. deines some additional terminology necessary to understand the ower low roblem and its solution rocedure. Section T7. introduces the so-called networ -bus, otherwise nown more generally as the networ admittance matri. Section T7.4 develos the ower low euations, building rom module T where euations or real and reactive ower low across a transmission line were introduced. Section T7.5 rovides an analytical statement o the ower low roblem. Section T7.6 uses a simle eamle to introduce the ewton-rahson algorithm or solving systems o non-linear algebraic euations. Section T7.7 illustrates alication o the ewton-rahson algorithm to the ower low roblem. Section T7.8 rovides an overview o several interesting and advanced attributes o the roblem. Section T7.9 summarizes basic ower low inut and outut uantities and rovides an eamle associated with a commercial ower low rogram. T7. enerator Reactive Limits It is well nown that generators have maimum and minimum real ower caabilities. In addition, they also have maimum and minimum reactive ower caabilities. The maimum reactive ower caability corresonds to the maimum reactive ower that the generator may roduce when oerating with a lagging ower actor. The minimum reactive ower caability corresonds to the maimum reactive ower the generator may absorb when oerating with a leading ower actor. These limitations are a unction o the real ower outut o the generator, that is, as the real ower increases, the reactive ower limitations move closer to zero. The solid curve in Figure T7. is a tyical generator caability curve, which shows the lagging and leading reactive limitations the ordinate as real ower is varied the abscissa. Most ower low rograms model the generator reactive caabilities by assuming a somewhat conservative value or ma erhas 95% o the actual value, and then iing the reactive limits ma or the lagging limit and min or the leading limit according to the dotted lines shown in Fig. T.. lagging oeration ma ma min leading oeration Fig T7.: enerator Caability Curve and Aroimate Reactive Limits T7. Terminology ul high voltage transmission systems are always comrised o three hase circuits. However, under balanced conditions the currents in all three hases are eual in magnitude and hase searated by, we may analyze the three hase system using a er-hase euivalent circuit consisting o the a-hase and the neutral conductor. er-unitization o a er-hase euivalent o a three hase, balanced system results in All materials are under coyright o owerlearn. Coyright, all rights reserved.

3 Module T7: The ower Flow roblem 7 the er-unit circuit. It is the er-unitized, er-hase euivalent circuit o the ower system that we use to ormulate and solve the ower low roblem. For the remainder o this module, we will assume that all uantities are in er-unit. The reader unamiliar with er-hase euivalent circuits or the er-unit system should reer to modules and 4, resectively. It is convenient to reresent ower system networs using the so-called one-line diagram, which can be thought o as the circuit diagram o the er-hase euivalent, but without the neutral conductor module also rovides additional bacground on the one-line diagram. Figure T7. illustrates the one-line diagram o a small transmission system. Fig. T7. illustrates several imortant elements o the ower low roblem. First, one notices we may categorize each bus deending on whether generation and/or load is connected to it. Seciically, a bus may have generation only buses,, and, load only buses 5, 7, and 9, or neither generation or load buses 4, 6, and 8. In addition, a bus may have both generation and load, although none o the buses in Fig. all into this category. This categorization, which ocuses on the load and generation, leads us to deine the term bus inection or more simly, inection. We will use this term reuently, and the student is advised to careully note its meaning, given and discussed in the ollowing aragrah. An inection is the ower, either real or reactive, that is being inected into or withdrawn rom a bus by an element having its other terminal in the er-hase euivalent circuit connected to ground. Such an element would be either a generator or a load. We deine a ositive inection as one where ower is lowing rom the element into the bus i.e., into the networ; a negative inection is then when ower is lowing rom the bus i.e., rom the networ into the element. enerators normally have ositive real ower inections, although they may also be assigned negative real ower inections, in which case they are oerating as a motor. enerators may have either ositive or negative reactive ower inections: ositive i the generator is oerating lagging and delivering reactive ower to the bus, negative i the generator is oerating leading and absorbing reactive ower rom the bus, and zero i the generator is oerating at unity ower actor. Loads normally have negative real and reactive ower inections, although they may also be assigned ositive real ower inections in the case o very secial modeling needs. Figure T7. a and b illustrate the two most common ossibilities. Figure T.7. c illustrates that we must comute a net inection as the algebraic sum when a bus has both load and generation; in this case, the net inection or both real and reactive ower is ositive into the bus. Thus, the net real ower inection is = g- d, and the net reactive ower inection is = g- d. We may also reer to the net comle ower inection as S =S g-s d, where S = Figure T7.: Single Line Diagram or Simle ower System All materials are under coyright o owerlearn. Coyright, all rights reserved.

4 Module T7: The ower Flow roblem 74 = = = - 4 = - a b =+-4=6 =+-= c Fig T7.: Illustration o a ositive inection, b negative inection, and c net inection Although it is hysically aealing to categorize buses based on the generation/load mi connected to it, we need to be more recise in order to analytically ormulate the ower low roblem. For roer analytical ormulation, it is aroriate to categorize the buses according to what inormation is nown about them beore we solve the ower low roblem. For each bus, there are our ossible variables that characterize the buses electrical condition. Let us consider an arbitrary bus numbered. The our variables are real and reactive ower inection, and, resectively, and voltage magnitude and angle, and, resectively. From this ersective, there are three basic tyes o buses. We reer to the irst two tyes using terminology that remind us o the nown variables. uses: For tye buses, we now and but not or. These buses all under the category o voltage-controlled buses because o the ability to seciy and thereore to now the voltage magnitude o this bus. Most generator buses all into this category, indeendent o whether it also has load; ecetions are buses that have reactive ower inection at either the generator s uer limit ma or its lower limit min, and the system swing bus we describe the swing bus below. There are also secial cases where a non-generator bus i.e., either a bus with load or a bus with neither generation or load may be classiied as tye, and some eamles o these secial cases are buses having switched shunt caacitors or static var systems SCs. We will not address these secial cases in this module. In Fig. T7., buses and are tye. The real ower inections o the tye buses are chosen according to the system disatch corresonding to the modeled loading conditions. The voltage magnitudes o the tye buses are chosen according to the eected terminal voltage settings, sometimes called the generator set oints, o the units. uses: For tye buses, we now and but not or. All load buses all into this category, including buses that have not either load or generation. In Fig. T7., buses uses 4-9 are all tye. The real ower inections o the tye buses are chosen according to the loading conditions being modeled. The reactive ower inections o the tye buses are chosen according to the eected ower actor o the load. The third tye o bus is reerred to as the swing bus. Two other common terms or this bus are slac bus and reerence bus. There is only one swing bus, and it can be designated by the engineer to be any generator bus in the system. For the swing bus, we now and. The act that we now is the reason why it is sometimes called the reerence bus. hysically, there is nothing secial about the swing bus; in act, it is a mathematical artiact o the solution rocedure. At this oint in our treatment o the ower low roblem, it is most aroriate to understand this last statement in the ollowing way. The generation must suly both the load and the losses on the circuits. eore solving the ower low roblem, we will now all inections at buses, but we will not now what the losses will be as losses are a unction o the lows All materials are under coyright o owerlearn. Coyright, all rights reserved.

5 Module T7: The ower Flow roblem 75 on the circuits which are yet to be comuted. So we may set the real ower inections or, at most, all but one o the generators. The one generator or which we do not set the real ower inection is the one modeled at the swing bus. Thus, this generator swings to comensate or the networ losses, or, one may say that it taes u the slac. Thereore, rather than call this generator a bus as the above naming convention would have it, we choose the terminology swing or slac as it hels us to better remember its unction. The voltage magnitude o the swing bus is chosen to corresond to the tyical voltage setting o this generator. The voltage angle may be designated to be any angle, but normally it is designated as o. A word o caution about the swing bus is in order. ecause the real ower inection o the swing bus is not set by the engineer but rather is an outut o the ower low solution, it can tae on mathematically tractable but hysically imossible values. Thereore, the engineer must always chec the swing bus generation level ollowing a solution to ensure that it is within the hysical limitations o the generator. T7. The Admittance Matri Current inections at a bus are analogous to ower inections. The student may have already been introduced to them in the orm o current sources at a node. Current inections may be either ositive into the bus or negative out o the bus. Unlie current lowing through a branch and thus is a branch uantity, a current inection is a nodal uantity. The admittance matri, a undamental networ analysis tool that we shall use heavily, relates current inections at a bus to the bus voltages. Thus, the admittance matri relates nodal uantities. We motivate these ideas by introducing a simle eamle. Figure T7.4 shows a networ reresented in a hybrid ashion using one-line diagram reresentation or the nodes buses -4 and circuit reresentation or the branches connecting the nodes and the branches to ground. The branches connecting the nodes reresent lines. The branches to ground reresent any shunt elements at the buses, including the charging caacitance at either end o the line. All branches are denoted with their admittance values y i or a branch connecting bus i to bus and y i or a shunt element at bus i. The current inections at each bus i are denoted by I i. y 4 I y 4 I 4 y y y I y y I y 4 Fig. T7.4: etwor or Motivating Admittance Matri Kircho s Current Law KCL reuires that each o the current inections be eual to the sum o the currents lowing out o the bus and into the lines connecting the bus to other buses, or to the ground. Thereore, recalling Ohm s Law, I=/z=y, the current inected into bus may be written as: I = - y + - y + y T7. To be comlete, we may also consider that bus is connected to bus 4 through an ininite imedance, which imlies that the corresonding admittance y 4 is zero. The advantage to doing this is that it allows us to consider that bus could be connected to any bus in the networ. Then, we have: I = - y + - y + - 4y 4 + y T7. All materials are under coyright o owerlearn. Coyright, all rights reserved.

6 Module T7: The ower Flow roblem 76 ote that the current contribution o the term containing y 4 is zero since y 4 is zero. Rearranging e. T7., we have: I = y + y + y + y 4 + -y + -y + 4-y 4 T7. Similarly, we may develo the current inections at buses,, and 4 as: I = -y + y + y + y + y 4 + -y + 4-y 4 I = -y + -y + y + y + y + y y 4 I 4= -y 4+ -y 4 + -y 4+ 4 y 4 + y 4 + y 4 + y 4 T7.4 where we recognize that the admittance o the circuit rom bus to bus i is the same as the admittance rom bus i to bus, i.e., y i=y i From es. T7. and T7.4, we see that the current inections are linear unctions o the nodal voltages. Thereore, we may write these euations in a more comact orm using matrices according to: I y y I I I 4 y y y y 4 y 4 y y y y y y 4 y 4 y y y y y y 4 y 4 y 4 y y y y y 4 y 4 4 T7.5 The matri containing the networ admittances in e. T7.5 is the admittance matri, also nown as the - bus, and denoted as: y y y y4 y y y4 y y y y y4 y y4 T7.6 y y y y y y4 y4 y4 y4 y4 y4 y4 y4 y 4 Denoting the element in row i, column, as i, we rewrite e. T7.6 as: 4 4 T where the terms i are not admittances but rather elements o the admittance matri. Thereore, e. T7.5 becomes: I I I I T7.8 y using e. T7.7 and T7.8, and deining the vectors and I, we may write e. T7.8 in comact orm according to: All materials are under coyright o owerlearn. Coyright, all rights reserved.

7 Module T7: The ower Flow roblem 77, 4 I I I I I4 I T7.9 We mae several observations about the admittance matri given in es. T7.6 and T7.7. These observations hold true or any linear networ o any size.. The matri is symmetric, i.e., i= i.. A diagonal element ii is obtained as the sum o admittances or all branches connected to bus i, including the shunt branch, i.e., ii y i y i, i, where we emhasize once again that y i is nonzero only when there eists a hysical connection between buses i and.. The o-diagonal elements are the negative o the admittances connecting buses i and, i.e., i=-y i. These observations enable us to ormulate the admittance matri very uicly rom the networ based on visual insection. The ollowing eamle will clariy. Eamle T7. Consider the networ given in Fig. T7.5, where the numbers indicate admittances I I -5.. I.4 I 4 The admittance matri is given by insection as: Fig. T7.5: Circuit or Eamle T T7.4 The ower low euations We have deined the net comle ower inection into a bus, in Section T7., as S =S g-s d. In this section, we desire to derive an eression or this uantity in terms o networ voltages and admittances. We begin by reminding the reader that all uantities are assumed to be in er unit, so we may utilize singlehase ower relations. Drawing on the amiliar relation or comle ower, we may eress S as: S = I * T.7. From e. T7.8, we see that the current inection into any bus may be eressed as All materials are under coyright o owerlearn. Coyright, all rights reserved.

8 Module T7: The ower Flow roblem All materials are under coyright o owerlearn. Coyright, all rights reserved. 78 I T7. where, again, we emhasize that the terms are admittance matri elements and not admittances. Substitution o e. T7. into e. T7. yields: * * * S T7. Recall that is a hasor, having magnitude and angle, so that =. Also,, being a unction o admittances, is thereore generally comle, and we deine and as the real and imaginary arts o the admittance matri element, resectively, so that = +. Then we may rewrite e. T7. as S * * * * T7. Recall, rom the Euler relation, that a hasor may be eressed as comle unction o sinusoids, i.e., =={cos+sin}, we may rewrite e. T7. as S sin cos T7.4 I we now erorm the algebraic multilication o the two terms inside the arentheses o e. T7.4, and then collect real and imaginary arts, and recall that S = +, we can eress e. T7.4 as two euations, one or the real art,, and one or the imaginary art,, according to: cos sin sin cos T7.5 The two euations o T7.5 are called the ower low euations, and they orm the undamental building bloc rom which we attac the ower low roblem. It is interesting to consider the case o es. T7.5 i bus, relabeled as bus, is only connected to one other bus, let s say bus. Then the bus inection is the same as the low into the line. The situation is illustrated in Fig. T7.6.

9 Module T7: The ower Flow roblem All materials are under coyright o owerlearn. Coyright, all rights reserved. 79 us us Series admittance - us inection and Line low and Fig. T7.6: us Connected to Only us For the situation illustrated in Fig. T7.6, es. T7.5 become: cos sin sin cos T7.6 I the line admittance is y=-, as shown in Fig. T7.6, then =- and = see e. T7.6. I there is no bus shunt reactance or line charging, then = and = -. Under these conditions, es. T7.6 become: cos sin sin cos T7.7 I we simly rearrange the order o the terms in the reactive euation, then we have: sin cos sin cos T7.8 It is helul to comare es. T7.5 to the euations derived in Module T or ower lowing across the series imedance o a transmission circuit see es. T.7 and T.8 in Module T. For convenience, these euations rom Module T are reeated here: sin cos sin cos In comaring the module T euations to e. T7.8, we see that they are eactly the same, imlying that the module T euations or ower lowing across a series imedance are simly a secial case o the ower low euations. T7.5 Analytic statement o the ower low roblem Consider a ower system networ having buses, o which are voltage-regulating generators. One o these must be the swing bus. Thus there are - tye buses, and - tye buses. We assume that We have deined the line admittance to be y=-, instead o y=+, to remain consistent with Module T. The reason or deining in this way is because, since the line admittance y always reresents inductive suscetance, the imaginary art o y must be negative; thereore the deinition used here reuires that to be a ositive number.

10 Module T7: The ower Flow roblem 8 the swing bus is numbered bus, the tye buses are numbered,,, and the tye buses are numbered +,, this assumtion on numbering is not necessary, but it maes the ollowing develoment notationally convenient. It is tyical that we now, in advance, the ollowing inormation about the networ imlying that it is inut data to the roblem:. The admittances o all series and shunt elements imlying that we can obtain the -bus,. The voltage magnitudes, =,,, at all generator buses,. The real ower inection o all buses ecet the swing bus,, =,, 4. The reactive ower inection o all tye buses,, = +,, Statements and 4 indicate ower low euations or which we now the inections, i.e., the values o the let-hand side o es. T7.5. These euations are very valuable because they have one less unnown than euations or which we do not now the let-hand-side. The number o these euations or which we now the let-hand-side can be determined by adding the number o buses or which we now the real ower inection statement above to the number o buses or which we now the reactive ower inection statement 4 above. This is -+- =--. We reeat the ower low euations here, but this time, we denote the aroriate number to the right. cos sin cos,,..., sin,,..., T7.9 We are trying to ind the ollowing inormation about the networ: a. The angles or the voltage hasors at all buses ecet the swing bus it is at the swing bus, i.e.,, =,, b. The magnitudes or the voltage hasors at all tye buses, i.e.,, = +,, Statements a and b imly that we have - angle unnowns and - voltage magnitude unnowns, or a total number o unnowns o -+- =--. Reerring to the ower low euations, e. T7.5, we see that there are no other unnowns on the right-hand side besides voltage magnitudes and angles the real and imaginary arts o the admittance values, and, are nown, based on statement above. Thus we see that the number o euations having nown let-hand side inections is the same as the number o unnown voltage magnitudes and angles. Thereore it is ossible to solve the system o - - euations or the - - unnowns. However, we note rom e. T7.9 that these euations are not linear, i.e., they are nonlinear euations. This nonlinearity comes rom the act that we have terms containing roducts o some o the unnowns and also terms containing trigonometric unctions o some o the unnowns. ecause o these nonlinearities, we are not able to ut them directly into the amiliar matri orm o A=b where A is a matri, is the vector o unnowns, and b is a vector o constants to obtain their solution. We must thereore resort to some other methods that are alicable or solving nonlinear euations. We describe such a method in Section T7.6. eore doing that, however, it may be helul to more crisly ormulate the eact roblem that we want to solve. Let s irst deine the vector o unnown variables. This we do in two stes. First, deine the vector o unnown angles an underline beneath the variable means it is a vector or a matri and the vector o unnown voltage magnitudes., T7. Second, deine the vector as the comosite vector o unnown angles and voltage magnitudes. All materials are under coyright o owerlearn. Coyright, all rights reserved.

11 Module T7: The ower Flow roblem All materials are under coyright o owerlearn. Coyright, all rights reserved. 8 θ θ θ θ T7. With this notation, we see that the right-hand sides o es. T7.9 deend on the elements o the unnown vector. Eressing this deendence more elicitly, we rewrite es. T7.9 as,...,,...,,, T7. In es. T7., and are the seciied inections nown constants while the right-hand sides are unctions o the elements in the unnown vector. ringing the let-hand side over to the right-hand side, we have that,...,,...,,, T7. We now deine a vector-valued unction as: T7.4 Euation T7.4 is in the orm o =, where is a vector-valued unction and is a vector o zeros; both and are o dimension --, which is also the dimension o the vector o unnowns,. We have also introduced nomenclature reresenting the mismatch vector in e. T7.4, as the vector o s and s. This vector is used during the solution algorithm, which is iterative, to identiy how good the solution is corresonding to any articular iteration. In the net section, we introduce this solution algorithm, which can be used to solve this ind o system o euations. The method is called the ewton- Rahson method. T7.6 The ewton-rahson Solution rocedure There are two basic methods or solving the ower low roblem: auss-siedel S and ewton- Rahson R. oth o these methods are iterative root inding schemes.

12 Module T7: The ower Flow roblem 8 The S and R methods are oten classiied as root inding schemes because they are geared towards solving euations lie = or =. The solution to such an euation, call it * or *, is clearly a root o the unction or. The methods are called iterative because they reuire a series o successive aroimations to the solutions. The rocedure is generally as ollows. First, guess a solution. Unless we are very ortunate, the guess will be, o course, wrong. So we determine an udate to the old solution that moves to a new solution with the intention that the new solution is closer to the correct solution than was the old solution. A ey asect to this tye o rocedure is the way we obtain the udate. I we can guarantee that the udate is always imroving the solution, such that the new solution is in act always closer to the correct solution than the old solution, then such a rocedure can be guaranteed to wor i only we are willing to comute enough udates, i.e., i only we are willing to iterate enough times. Commercial grade ower low rograms may mae several dierent solutions rocedures available, but almost all such rograms will have available, minimally, the R method. It is air to say that the R method has become the de-acto industry standard. The main reason or this is that the convergence roerties o the R scheme are very desirable when the initial, guessed solution is uite good, i.e., when it is chosen close to the correct solution. In the ower low roblem, it is usually ossible to mae a good initial guess regarding the solution. One reason or this is that oten, we may actually now the solution o a articular set o conditions because we have already gone through the solution rocedure, and we want to resolve or a set o conditions that are almost the same as the revious ones, e.g., maybe remove one circuit or change the load level a little. In this case, we may utilize the revious solution as the initial guessed solution or the new conditions. This is sometimes reerred to as a hot start. ut even i we do not have a revious solution, we still may do very well with our guess. The reason or this is that the ower low roblem is always solved with all uantities in er-unit. ecause o the way we choose er-unit voltage bases, the er-unit voltages or all buses, under any reasonably normal condition, will be close to. erunit. O course, this tells us nothing about the angles, but it is something, and oten it is enough to simly guess that all voltages are. er-unit and all angles are degrees. This is sometimes called a lat start. ut what are convergence roerties o a root inding method? There are basically two o them. One is whether the method will converge. The second one is how ast the method will converge. For R, whether the method will converge deends on two things: how close the guessed solution is to the correct solution and the nature o the unction close to the correct solution. I the guessed solution is close, and i the unction is reasonably smooth close to the correct solution, then the R will converge. ot only that, but it will converge uadratically. uadratic convergence means that each iteration increases the accuracy o the solution by two decimal laces. For eamle, i the correct solution or a articular roblem is , and we guess., then the irst iteration will yield., the second iteration will yield.45, the third iteration will yield.4567, and the correct solution will be obtained eactly on the ourth iteration. In this module, we will not discuss the S method, but the interested reader may ind inormation about it in many tets on ower systems analysis or in boos on numerical methods. We will introduce the R method with a simle illustration, obtained rom []. Eamle T7. Consider the scalar unction = This unction may be easily actored to ind the roots as *=4,. Let us now illustrate how the R method inds one o these roots. We irst need the derivative: =-5. Assume we are bad guessers, and try an initial guess o =6. The ollowing rovides the irst two iterations:. =6=6-56+4=. = 6=6-5=7. = - / = -/7= = + =6+-.49=4.57 All materials are under coyright o owerlearn. Coyright, all rights reserved.

13 Module T7: The ower Flow roblem 8. =4.57=.94. = 4.57=4.4. = - / = -.94/4.4= = + = =4.787 One more iteration yields =4.. ote that by the third iteration, as it is getting very close to the correct solution, the algorithm has almost obtained uadratic convergence. Fig. T7.7 illustrates how the irst solution is ound rom the initial guessed solution during the irst iteration o this algorithm. The R algorithm is not smart enough to now which root you want, rather, it generally inds the closest root. This is another reason or maing a good initial guess in regards to the solution. Fortunately, in the case o the ower low roblem, alternative solutions are tyically ar away rom initial guesses that have near-unity bus voltage magnitudes. On the other hand, it is ossible or the solution to diverge, i.e., not to converge at all. This may occur i there is simly no solution, which is a case that engineers encounter reuently when studying highly stressed loading conditions with served by wea transmission systems. It also might occur i the initial guessed solution is too ar away rom the correct solution. For this reason, lat starts encounter solution divergence more reuently than hot starts. Fig. T7.7: Illustration o the irst iteration o the ewton-rahson algorithm et, we develo the R udate ormula. We begin with the scalar case, where the udate ormula may be easily inerred rom Eamle T7.. ewton Rahson or the Scalar Case: Assume that we have guessed a solution to the roblem =. Then because is ust a guess. ut there must be some which will mae + =. One way to study this roblem is to eand the unction in a Taylor series, as ollows: ' ''... T7.5 All materials are under coyright o owerlearn. Coyright, all rights reserved.

14 Module T7: The ower Flow roblem 84 I the guess is a good one, then will be small, and i this is true, then will be very small, and any higher order terms h.o.t. in e. T7.5, which will contain raised to even higher owers, will be ininitesimal. As a result, it is reasonable to aroimate e. T7.5 as Taing to the right hand side, we have ' T7.6 ' We may easily solve e. T7.7 or according to: T7.7 ' T7.8 ecause in e. T7.8 is scalar, it s inverse is very easily evaluated using simle division so that: T.9 ' Euation T7.8 rovides the basis or the udate ormula to be used in the irst iteration o the scalar R method. This udate ormula is: T7. ' and rom e. T7.8, we may iner the udate ormula or any articular iteration as: ' T7. et we develo the udate ormula or the case where we have n euations and n unnowns. We call this the multidimensional case. ewton Rahson or the Multidimensional Case: Assume we have n nonlinear algebraic euations and n unnowns characterized by =, and that we have guessed a solution. Then because is ust a guess. ut there must be some which will mae + =. Again, we eand the unction in a Taylor series, as ollows: ' ''... ' ''... T7. n n n ' '' n... All materials are under coyright o owerlearn. Coyright, all rights reserved.

15 Module T7: The ower Flow roblem 85 Euations T7. may be written more comactly as ' ''... T7. Assuming the guess is a good one such that is small, then the higher order terms are also small and we can write ' T7.4 One reasonable uestion to as at this oint is: ust what is? That is, what is the derivative o a vector-valued unction o a vector? Since we have n unctions and n variables, we could comute a derivative or each individual unction with resect to each individual unnown, lie /, which gives the derivative o the th unction with resect to the th unnown. Thus, there will be a number o such derivatives eual to the roduct o the number o unctions by the number o unnowns, in this case, nn. Thus, it is convenient to store all o these derivatives in a matri. This matri has become uite well-nown as the acobian matri, and it is oten denoted using the letter. ut how should the nn derivatives be stored in this matri? The rows o should be ordered in the same order as the unctions, that is, the th row should contain the derivatives o the th unctions. In e. T7.4, since the roduct must rovide a correction to the unction +, i.e., since =, it must be the case that elements o any row o the matri must be ordered so that the term in the th column contains a derivative with resect to the th unnown o the vector. The reasoning in the last aragrah suggests that we write the acobian matri as: n n n n n n T7.5 In e. T7.4, taing to the right hand side, we have ' or, in terms o the acobian matri, we have: Solving e. T7.7 or, we have: T7.6 T7.7 ' T7.8 All materials are under coyright o owerlearn. Coyright, all rights reserved.

16 Module T7: The ower Flow roblem 86 Euation T7.8 rovides the basis or the udate ormula to be used in the irst iteration o the multidimensional case. This udate ormula is: T7.9 and rom e. T7.9, we may iner the udate ormula or any articular iteration as: i i i i i T7.4 For roblems o relatively small dimension, where the inverse o the acobian is easily obtainable, e. T7.4 is an aroriate udate ormula. In general, however, it is a good rule, in rogramming, to always avoid matri inversion i at all ossible, because or high-dimension roblems, as is usually the case or large scale ower networs, matri inversion is very time consuming. Usually, it is ossible to avoid matri inversion through a techniue nown as matri actorization, sometimes nown as LU decomosition. We will not discuss this techniue in this module, as it can be readily ound in most ower system analysis tets or boos on numerical methods. However, we do desire to state the udate ormula a little dierently, so that one is reared to utilize it or the general case using matri actorization. To do this, we write e. T7.4 as i i i T7.4 where i is ound rom i i T7.4 using the matri actorization techniue. In this module, we will treat roblems o low dimension only so that matri inversion is reasonable, and the udate ormula T7.4 is aroriate. Eamle T7. Solve the ollowing two euations algebraically and using R: + - -=, - = The stes or the algebraic solution are to irst solve both euations or, resulting in =- + +/ and =. Euating these two eressions or, and maniulating, results in a cubic + - -=. This eression may be actored as: - + +=, and we see that the solutions to the cubic in are, - and. lugging these values or bac into either eression or yields, resectively,,, and 4, and thereore there are three solutions to the original roblem; they are:, =,, -,, -,4. ow let s solve this same roblem using R. Deine the unctions, = and, = -. Then the acobian matri is:,, 4,, Let s act lie we do not now the solution and guess at, =.9,.. Then the acobian, evaluated at this guessed solution, is All materials are under coyright o owerlearn. Coyright, all rights reserved.

17 Module T7: The ower Flow roblem 87, Inverting the acobian results in: We also need to evaluate: We can now udate the solution using e. T7.4, as , We see that the irst udate results in a solution that is very close to the actual solution o,. This good erormance is due to the act that we made a good initial guess. The student should reeat the above rocedure, but try starting rom other oints, e.g., -.9,., -.9,4., and,., using two iterations each time. Writing a simle rogram will greatly reduce the eort. In general, o course, we usually need to iterate several times in order to obtain a satisactory solution. How many times is enough? The R algorithm must emloy a stoing criterion in order to determine when the solution is satisactory. There are two ways to do this. Tye stoing criterion: Test the maimum change in the solution elements rom one iteration to the net, and i this maimum change is smaller than a certain redeined tolerance, then sto. This means to comare the maimum absolute value o elements in against a small number, call it. In eamle T7., = [-.6,.6] T, so the maimum absolute value o elements in is.6. I we had =.5, we could sto. ut i we had =.5, we would need to continue to the net iteration. Tye stoing criterion: Test the maimum absolute value in the unction elements o the most current iteration, and i this maimum value o elements in is smaller than a certain redeined tolerance, then sto. This means to comare the maimum absolute value o elements in against a small number, call it. In eamle T7., =[-.9, -.9] T, so the maimum absolute value o elements in is.9. I we had =., we could sto. ut i we had =., we would need to continue to the net iteration. This is the most common stoing criterion or ower low solutions, and the value o each element in the unction is reerred to as the ower mismatch or the bus corresonding to the unction. For tye buses, we test both real and reactive ower mismatches. For tye buses, we test only real ower mismatches. T7.7 Alication o R to ower Flow Solution Let s revisit the ower low roblem outlined in Section T7.5, in light o the R solution rocedure described in Section T7.6. We desire to solve e. T7.4, given the vector o unnowns are given by e. T7. and the unctions are in the orm o e. T7.9. These euations are reeated here or convenience: All materials are under coyright o owerlearn. Coyright, all rights reserved.

18 Module T7: The ower Flow roblem All materials are under coyright o owerlearn. Coyright, all rights reserved. 88 T7.4 θ θ θ θ T7.,...,, cos sin,...,, sin cos T7.9 The solution udate ormula is given by e. T7.4, reeated here or convenience: i i i i i T7.4 Clearly, an essential ste in alying R to the ower low roblem is to enable calculation o the acobian elements, given or the general case by e. T7.5 as n n n n n n T7.5 Evaluation o these elements is acilitated by the recognition, rom e. T7.4, that there are only two inds o euations real ower euations and reactive ower euations, and rom e. T7., that there are only two inds o unnowns voltage angle unnowns and voltage magnitude unnowns. Thereore, there are only our basic tyes o derivatives in the acobian. We denote our sub-matrices corresonding to these our basic tyes o derivatives as,,,, where the irst suerscrit indicates the tye o euation

19 Module T7: The ower Flow roblem All materials are under coyright o owerlearn. Coyright, all rights reserved. 89 we dierentiate, and the second suerscrit indicates the unnown with resect to which we dierentiate. Thereore, T7.4 The numbers above each sub-matri in e. T7.4 indicate its dimensions, which can be inerred by identiying the number o euations o that tye the number o rows o the sub-matri and the number o unnowns o that tye the number o columns o the sub-matri. We may then identiy an individual element o each sub-matri as: T7.44 ote that the element is not the element in row, column o the submatri, rather it is the derivative o the real ower inection euation or bus with resect to the angle o bus. Since the swing bus is numbered, the acobian matri will have as the element in row, column. The situation is similar or the other submatrices. To get these derivatives, it is helul to more elicitly write out the unctions o e. T7.4. They are: sin sin sin sin sin cos sin cos,,,, T7.45 So each o the our sub-matrices o e. T7.4 has elements given by the eressions o e. T7.44, resectively. These eressions are evaluated by taing the aroriate derivatives o the unctions in e. T7.45. One might thin that this reresents a ormidable roblem, since, based on e. T7.4, we have elements in the acobian and thereore the same number o derivatives to evaluate. A tyical ower low model or a US control area might have 5 nodes =5 and generators =, resulting in a acobian matri containing 8,98, elements, with each element reuiring a dierentiation o a unction lie those reresented in e. T7.45! Fortunately, all o the derivatives can be eressed by one o ust a ew dierentiations. At irst glance, one might thin that there would be our dierentiations, one or each sub-matri. However, or each submatri, the o-diagonal terms, with, are eressed dierently than the diagonal terms, with =. Thereore, there are eight dierentiations to erorm. The student should attemt to obtain a ew o these eressions. In doing so, the ollowing tis are helul. eore dierentiating, it is helul to ull out the term rom the summation that corresonds to the bus inection being comuted.

20 Module T7: The ower Flow roblem All materials are under coyright o owerlearn. Coyright, all rights reserved. 9 When dierentiating a sum o terms with resect to a articular unnown, the resulting derivative will be non-zero only or those terms in which the unnown aears. When dierentiating with resect to the angles, the chain rule must be roerly alied to account or the derivatives o the trigonometric unctions and the arguments o those trigonometric unctions. Each o the unctions aear in the orm o =g-a. ecause A is a constant reresented by,, and g+,, in e. T7.45, it has no eect on the resulting derivatives. The resulting eressions are given below. cos sin T7.47 T7.48 sin cos T7.49 T7.5 sin cos T7.5 T7.5 cos sin T7.5 T7.54 We are now in a osition to rovide the algorithm or using R to solve the ower low roblem. eore doing so, it is helul to more elicitly deine the mismatch vector, rom e. T7.4 or T7.45 as: T7.5 The R algorithm, or alication to the ower low roblem, is:. Seciy: All admittance data d and d or all buses g and or all buses

21 Module T7: The ower Flow roblem 9 or swing bus, with =. Set the iteration counter =. Use one o the ollowing to guess the initial solution. Flat Start: =. or all buses. Hot Start: Use the solution to a reviously solved case or this networ.. Comute the mismatch vector or, denoted as in e. T7.4 and e. T7.45. In what ollows, we denote elements o the mismatch vector as and corresonding to the real and reactive ower mismatch, resectively, or the th bus which would not be the th element o the mismatch vector or two reasons: one reason ertains to the swing bus and the other reason to the act that or tye buses, there are two euations er bus and not one. This comutation will also result in all necessary calculated real and reactive ower inections. erorm the ollowing stoing criterion tests: I < or all tye and buses and I < or all tye buses, Then go to ste 5 Otherwise, go to ste Find an imroved solution as ollows: Evaluate the acobian at. Denote this acobian as Solve or rom: or - where we must use actorization with the let euation i the system is large, but i the system is not large, we may use the right hand euation. Comute the udated solution vector as + = +. Return to ste with =+. 5. Sto. The above algorithm is alicable as long as all buses remain within their reactive limits. To account or generator reactive limits, we must modiy the algorithm so that, at each iteration, we chec to ensure bus reactive generation is within its limits see Section T7. regarding modeling o reactive limits. In this case, stes -4 remain eactly as given above, but we need a new ste 5 and 6, as ollows: 5. Chec reactive limits or all generator buses as ollows: a. For all tye buses, erorm the ollowing test: I g> g,ma, then g= g,ma and CHAE bus to a tye bus see ste 6a I g< g,min, then g= g,min and CHAE bus to a tye bus see ste 6b b. For all tye generator buses, erorm the ollowing test: I g= g,ma and >,set or i g= g,min and <,set, then CHAE this bus bac to a tye bus see ste 6b 6. I there were no CHAES in Ste 5, then sto. I there were one or more CHAES in ste 5, then modiy the solution vector and the mismatch vector as ollows: a. For each CHAE made in ste 5-a changing a bus to a bus: = - Include the variable to the vector and the variable to the vector. Include the reactive euation corresonding to bus to the vector. Modiy the acobian by including a column to and including a row to and. b. For each CHAE made in Ste 5-b changing a gen bus bac to a bus: = + Remove the variable to the vector and the variable rom the vector. Remove the reactive euation corresonding to bus rom the vector. All materials are under coyright o owerlearn. Coyright, all rights reserved.

22 Module T7: The ower Flow roblem 9 Modiy the acobian by removing a column to and removing a row rom and. Ater modiications have been made or all CHAES, go bac to Ste 4. When the algorithm stos, then all line lows may be comuted using S I * Eamle T7.4 [5] used with ermission o. ittal Find, * [ ],, S!, and or the system shown in Fig. T7.8. In the transmission system all the shunt elements are caacitors with an admittance y c =., while all the series elements are inductors with an imedance o z L =.. S =.666 = =.5 S D = Fig. T7.8: Three us System or Eamle T7.4 Solution: The admittance matri or the system shown in Fig. E.6 is given by us is the swing bus. us is a bus. us is a bus. We use the R method in the solution. The unnown variables are,, and. Thus, we will need three euations, and the acobian is a matri. We irst write e. T7.45 or the case at hand, utting in the nown values o,,, and the i s. ote that since we have neglected line resistance in the roblem statement, all i s are zero. sin sin =.5sin.5 sin T7.54a All materials are under coyright o owerlearn. Coyright, all rights reserved.

23 Module T7: The ower Flow roblem 9 sin sin =. sin.5 sin T7.54b The euation or will not hel since we do not now the reactive inection or bus, and its inclusion would bring in the reactive inection on the let-hand side as an additional unnown. ut this loss o an euation is comensated by the act that we now and this will always be the case or a tye bus. So we do not need to write the euation or. et, because bus is a tye bus, we do now its reactive inection, and so we will now the let hand side o the reactive ower low euation. This is ortunate, since we do not now and this will always be the situation or a tye bus. cos cos = - cos.5 cos 9.98 The udate vector and acobian matri is: T7.54c We obtain the various artial derivatives or the acobian rom es. T7.54a,b,c: cos cos cos =.5cos.5 cos cos sin.5sin = -.5 =.5 cos. cos.5 cos sin.5sin All materials are under coyright o owerlearn. Coyright, all rights reserved.

24 Module T7: The ower Flow roblem 94.5 sin sin = sin.5 sin sin sin cos cos = cos.5cos 9.96 We are ready to start iterating using T7.4. We note that the inections, to be used on the let hand side o es. T7.54a,b,c are = =.666, = - D = -.865, and = - D = -.44; these uantities remain constant through the entire iterative rocess. We use a lat start; thereore our initial guess is = = and =..Using es. T7.5 and T7.54a,b,c we get: As eected or a lat start, the mismatch is large. et we calculate the acobian matri: T ote that the acobian sub-matrices and are both illed with zeros. This is because these derivatives deend on sin terms, and because this is the irst iteration o a lat start, all angles are zero and thereore the sin terms are all zero. As mentioned, commercial ower low rograms normally use LU actorization to obtain the udate. In this case, however, because o the low dimensionality, we may invert the acobian. Taing advantage o the bloc diagonal structure, we have: ow we comute: All materials are under coyright o owerlearn. Coyright, all rights reserved.

25 Module T7: The ower Flow roblem 95 The elements o the udate vector corresonding to angles are in radians. We can easily convert them to degrees: o.5 rad.996 o -.66 rad We now ind as ollows: We note that the eact solution is o o o. o o o, so this is retty good rogress or one iteration! We roceed to the net iteration using the new values =-.996, =-9.59, and =.968. Substituting into.996 o, 9.59 o, and.968. Substituting in e. T7.54a, we get =.6, and thus = = Similarly, using es. T7.54b and T7.54c, we get the udated mismatch vector: ote that, in ust one iteration, the mismatch vector has been reduced by a actor o about. Calculating K using the udated values o the variables, we ind that The matri should be comared with rom the revious iteration. It has not changed much. The elements in the o-diagonal matrices and are no longer zero, but their elements are small comared to the elements in the diagonal matrices and. The diagonal matrices themselves have not changed much. It is also imortant to note that the uer let-hand bloc o the acobian matri is symmetric. This act allows or a signiicant savings in storage when dealing with large systems. The udated inverse is All materials are under coyright o owerlearn. Coyright, all rights reserved.