A Cycle-Based Solvability Condition for Real Power Flow Equations

A Cycle-Based Solvability Condition f or Real Po wer Flow Equations Puskar Neupane and Bai Cui Abstract — The solvability condition o f the power flow equa- tion is important in operational planning and contro l as it guarantees the existence and un iqueness of a solution f or a giv en set of po wer injections. As renewable generation becomes more p re valent, th e steady-state operating point of the system changes more frequently , making it increasingly challenging to verify power flow solvability by ru nning the A C power flow solver after each change in p ower i n jections. This process can be comput ation all y intensive, and numerical solv ers d o not always con ver ge reliably t o an operational solution. In thi s paper , we p ropose a su fficient condit i on f or the solv ability of the lossless real p ower flow equation b ased on the cycle space of a meshed network. The proposed cond ition yields a less conserv ativ e solvability certifi cate th an existing suffi cient conditions on the tested systems and can serv e as a useful fo undation f or dev eloping solvability conditions f or th e fully coupled power flow equations. I . I N T RO D U C T I O N The nonline a r nature of the A C power flow equations makes the com putation of power flow so lu tions challengin g. One co mmon app roach to obtaining ap proximate solutio ns is throug h iterative numer ical methods. Am ong these, Newton- Raphson (NR) metho d is the most widely used techn iq ue in p ractice. Howe ver , the c on vergence of the NR m ethod depend s stro ngly on the c hoice o f the initial guess and is not gu a r anteed in general. Moreover, ev e n when convergence occurs, the method may fail to conv erge to the de sire d high-voltage oper a tin g so lution. When the system is h ighly loaded, multiple power flo w solution s are very clo se, an d there is a chance tha t NR co nverges to an unstable low- voltage solution. These limitations h av e motiv ated extensive research in to alternative iterative meth ods with improved con vergence proper ties an d robustness guaran tees. In parallel, analytical approa c h es have be en developed to ch aracterize the ex- istence of feasible power flow solutions indep e n dently of any particular nu merical solver . Such analytical con ditions provide valuable insight in to solvability limits and ser ve as a found ation for the d esign o f reliable solution metho d s. The problem of determinin g solvability limits has bee n explored in various form s in th e literature and ha s gain ed atten tio n in recent d ecades. As e arly as 19 82, the author in [1] proposed a sufficient condition fo r the existence of a solution to p ower flow equations. Then in 198 6, the work in [2] in vestigated the existence of solution s to the reactive power balance problem and proposed condition s for assessing solvability following The authors are with the Department of Electri cal and Computer Engineeri ng, Iowa State Univ ersity , Am es, USA. Emails: { puskar, baicui } @iastate.e du system d isturbances. Later in 199 0, an analytical result on the existence and uniqu eness of th e power flow solution was propo sed [3]. Th ere has bee n some work on th ese analytical fields [4], [5], which are consider ed to be starting research in th is dom ain. The deriv ation of sufficient co nditions has also be e n studied in different ways in the literature. The p aper in [6] used an energy f unction as a tool in order to propose necessary and sufficient con ditions for the existence of a power flow solutio n . T he to pology o f the power network has also been leveraged in p r ior work to derive solvability condition s for the A C power flow equatio n s [7]. In [8], a novel iterative algo rithm is propo sed fo r solving power flow equations in both meshed an d radial n e tworks, while [9] derives a sufficient condition for the existence of a high - voltage solution in r adial n e tworks. In a related line of work, power system networks are m odeled as systems of coupled oscillators, and solvability conditions are obtained by drawing analo gies with syn chronizatio n phenomen a in Kuramoto oscillator models. In particular, the work in [10] studies the lossless power flow problem by expressing the dynamics in an edg e - balance fixed-po int form via a cutset projection operato r and establishes existence and uniqu eness of solutio ns using the Brouwer Fixed Poin t The o rem. The work in [11] prop osed a su fficient conditio n for the coupled f u ll-power m odel to verify th e existence of p ower flow so lu tions an d provide d an app roximate solution to the nonlinear power flow eq uations. Having sufficient con ditions for the solvability of the co upled full power flo w equation re- duces computatio nal burden [12] and offers valuable in sights for security mon itoring and the design o f effective control strategies. These sufficient conditio n s can be incorpo rated as constraints in the optim a l p ower flow pr oblem [13] a nd u sed to e n sure voltage stability of the system [14], [1 5]. Existing resear c h has suc cessfully d eveloped sufficient condition s for the existence of power flow solutio ns. How- ev e r , these condition s are o f ten conservati ve an d rely on re- stricti ve assumption s. Given the inh erently mesh ed natu re of real-world transmission networks, it is impo rtant to develop solvability con d itions th at explicitly exploit such structure. Motiv ated b y this, we leverage the to pological structu re of meshed networks, par ticularly th eir cyclic ch aracteristics, to derive a sharper sufficient c ondition for the existence of a solution to the lossless r eal power flow equation . Since solvability con d itions for the deco upled power flow model can h elp p av e the way toward sharper and more robust condition s fo r the full co u pled A C power flo w equatio ns, we f o cus in th is pap e r o n the lossless real power flow case. The remaind e r of the p a per is organized a s follows. Sec- tion II presents backgr ound material and prelimin aries. Sec- tion I II in troduces th e pro blem for mulation and assumptio ns. Section IV derives th e sufficient condition f or the existence of a power flow so lu tion. Section V d escribes the method- ology and imp lementation deta ils. Section VI pre sen ts the numerical results on IEEE test cases and com p arison with existing work. Finally , Sectio n VII conclude s the pap er . I I . B A C K G RO U N D A N D P R E L I M I N A R I E S A. P ower S y stem Model A power network is represented by a co nnected, undi- rected, weighted graph G ( N , E ) wh ere N is th e set of nod es (buses) and E ⊆ N × N is the set of ed ges (lines). An arbitrary or ientation is assigned to each ed ge so that th e edg e is oriented from its source node to the sink nod e. W e d enote the numbe r o f buses and the n umber of line s by n and m , respectively . For e a ch line { i, j } ∈ E , let g ij and b ij denote the series cond uctance an d susceptan ce, respe cti vely , so th at the line admittance is given b y y ij = g ij + j b ij . I n this work, we consider a lossless power system without shun t elements; ther efore, g ij = 0 for all ( i, j ) ∈ E , and each line admittance is purely imagin ary , i.e ., y ij = j b ij . The bus admittance matrix Y ∈ C n × n captures both the ne twork ’ s topolog y and its weights. The d ia g onal element Y ii is equ al to j B ii , where B ii is the sum of the suscepta n ces b ij of all lines incident to bus i . Th e off-diagon al elemen t Y ij is j B ij , wh ere B ij equals the negative of the suscep ta n ce of the line connectin g buses i an d j . For all i ∈ N , V i is the voltage m agnitude and θ i is the phase ang le . I n this work, we study the lossless re a l power flow equatio n s, so all th e voltage magnitud es ar e fixed and assumed k nown. B. Algebraic Graph Theory The no de-edge inciden ce matrix of the gr aph G ( N , E ) is giv en by A ∈ {− 1 , 1 , 0 } n × m . The elem ent A ke will be + 1 if k is th e sour ce node of th e edg e e , − 1 if k is the sink n ode of the edge e , and zero othe r wise. T he diagon al susceptanc e matrix D ∈ R m × m encodes the e d ge weights. For an ed ge e connectin g buses i and j , the correspo n ding diago nal entry of D is given by D ee = V i V j B ij . Th e Laplacian matrix of the grap h G is defined as L = AD A ⊤ ∈ R n × n , which h as a one - dimensiona l n ullspace ker( L ) = spa n { 1 } , wh ere 1 is the vector of all 1 ’ s. The (Moore- Penrose) pseudoinverse of L is den o ted by L † . A walk in a gra ph is a sequ ence of no des su ch th at each consecutive pair of n odes is connec ted by an edge. If th e walk starts and end s at the same no d e, it is called a closed walk. A cycle is a closed walk in which no node is repea te d e xcept the first and last. A cycle in a gr a p h can b e represen ted by a cycle vector c k ∈ R m , whose entries indicate whether each edge belongs to the cycle and, if so , whether its or ie n tation agrees with the ch osen traversal direction of the cycle. Sp ecifically , ( c k ) e =      +1 , if e ’ s orien tation alig n s with the trav ersal , − 1 , if e ’ s orien tation is o p posite to the traversal , 0 , otherwise . Any set of linear ly independ ent cycle vecto rs th a t spans the cycle space f orms a cycle basis. The cycle space of the g raph G is k er( A ) , and its dimension is q = m − n + 1 [16, Theorem 9 . 5 ]. The cutset space o f the grap h G is the colum n space o f A ⊤ . Mor eover , the cycle space and cutset space ar e orthog onal co mplements in R m , so that R m = ker( A ) ⊕ im ( A ⊤ ) . Giv en a cycle basis Σ = { c 1 , . . . , c m − n +1 } , C Σ denotes the cycle basis m atrix whose columns are th e corr esponding cycle vectors. I I I . P O W E R F L O W E Q UAT I O N For a lo ssless network , th e real power flow eq uations are P i = V i n X j =1 V j B ij sin( θ i − θ j ) , i ∈ N , (1) where the unkn own variables are the p h ase angles θ , while the voltage mag nitudes V are fixed and known. Collecting the bus power injections into th e vecto r P ∈ R n , (1) can b e written comp a c tly as P = AD sin( A ⊤ θ ) . (2) For a real-value vector x , the fun ctions sin( x ) and arcsin( x ) are under stood compo n entwise. In this pap e r , we are inter- ested in d eriving a sufficient con dition on the nod a l injection vector P ∈ 1 ⊥ : = { x ∈ R n : 1 ⊤ x = 0 } und er which (2) admits a solution. W e see f rom (1) and (2 ) that the no dal power injectio n P i is expressed in term s of th e an gle differences θ i − θ j for lines ( i, j ) inc id ent to bus i . For each line ( i, j ) ∈ E , d efine ˜ b ij := V i V j B ij . Th en the real power flow on lin e ( i, j ) is f ij = ˜ b ij sin( θ i − θ j ) . (3) In vector fo rm, this becom es f : = D sin( A ⊤ θ ) ∈ R m . (4) Using this definition, the compact power flow eq uation can be rewritten as P = Af . (5) W e refer to ( 5) as the nod al b alance equation , and any vector f ∈ R m satisfying it is called a line flo w solution . The no dal b alance equ ation (5 ) is solvable if and only if P ∈ im( A ) . Since the graph is con nected, im( A ) = 1 ⊥ , and theref ore ( 5) is solvable f or every nodal injection vecto r P ∈ 1 ⊥ . Moreover, bec ause L = AD A ⊤ and im( L ) = 1 ⊥ , a p articular solution is given b y ˆ f = D A ⊤ L † P. (6) Indeed , A ˆ f = AD A ⊤ L † P = LL † P = P, (7) where the last eq uality follows fr om the fact that LL † is the orthog onal p r ojector onto im( L ) = 1 ⊥ and P ∈ 1 ⊥ . A standard result in linea r algebra [1 7] states that, if ˆ f is a p articular solution of (5), then the set of all line flow solutions to (5 ) is g i ven b y f = ˆ f + s, s ∈ ker( A ) . (8) Since ker( A ) is the cycle space of the g r aph, any s ∈ ker( A ) is referred to as a cycle flow . It follows from (8) th at any two line flow so lutions differ by a cycle flow . While every power flow solution θ induces a line flow solution throug h f ij = ˜ b ij sin( θ i − θ j ) , the converse is no t true in gener a l: a line flow solution f does not necessarily correspo n d to a power flow solutio n. T he n ext pro position characterizes exactly w h en this corresp ondence h olds. Pr oposition 1 : Gi ven a line flow solution f ∗ satisfying (5) and a cycle basis matrix C Σ of G , there exists a power flow solution θ ∗ such that f ∗ ij = ˜ b ij sin( θ ∗ i − θ ∗ j ) , | θ ∗ i − θ ∗ j | ≤ π 2 , ∀ ( i, j ) ∈ E , ( 9) if an d only if the fo llowing two cond itions hold: 1) Edge - wise fe a sibility: | f ∗ ij | ≤ ˜ b ij , ∀ ( i, j ) ∈ E , (10) 2) Cycle consistency: C ⊤ Σ arcsin( D − 1 f ∗ ) = 0 . (11) Pr oof: W e first prove th e “only if ” part. Given a power flow solution θ ∗ for f ∗ as given by e quation (9), we h a ve f ∗ ij / ˜ b ij = sin( θ ∗ i − θ ∗ j ) ∈ [ − 1 , 1] . (12) So, con dition 1 is satisfied. Since | θ ∗ i − θ ∗ j | is bound ed b y π / 2 , we kn ow arcsin  f ∗ ij / ˜ b ij  = θ ∗ i − θ ∗ j . (13) The ab ove equatio n can be written in vector form as: arcsin( D − 1 f ) = A ⊤ θ ∗ . (14) Premultiplyin g by C Σ giv es C ⊤ Σ arcsin( D − 1 f ) = C ⊤ Σ A ⊤ θ ∗ = ( AC Σ ) ⊤ θ ∗ = 0 , (15) where the last equality holds since each colum n of C Σ lies in ker( A ) . Ther efore, condition 2 is also satisfied. For the other direction , we show that there exists a θ ∗ correspo n ding to f ∗ whenever cond itions 1 and 2 hold. By condition 1 , the vector δ : = a rcsin( D − 1 f ∗ ) (16) is well defined, with each compo nent ly in g in [ − π / 2 , π/ 2 ] . In add ition, condition 2 imp lies C ⊤ Σ δ = 0 . Sin ce the cycle space and cutset space are o rthogo nal com plements to each other, we h av e ker( C ⊤ Σ ) = img( A ⊤ ) . It follows that there exists θ ∗ ∈ R n such that δ = A ⊤ θ ∗ . Substituting the definition of δ and applyin g sin( · ) com ponen twise yields D − 1 f ∗ = sin( A ⊤ θ ∗ ) . (17) Equiv alently , f ∗ ij = ˜ b ij sin( θ ∗ i − θ ∗ j ) f o r ev ery line ( i, j ) ∈ E . Since each δ ij ∈ [ − π / 2 , π / 2] , the co rrespond ing angle difference satisfies | θ i − θ j | ≤ π / 2 fo r e a ch ( i, j ) ∈ E . Thus, θ ∗ is a power flow solution satisfying ( 9). Finally , θ ∗ is un ique u p to an additive constant, since A ⊤ 1 = 0 and the graph is con nected [18, Theorem 5.1] . I V . S U FFI C I E N T C O N D I T I O N F O R P O W E R F L O W E Q UA T I O N S O LV A B I L I T Y T o de riv e a su fficient co ndition for power flow solvability , we seek a family of line flow solutions that preserves the nodal balance equation wh ile allowing adjustmen ts in the cycle flow . By (8), every line flow so lu tion can be written as f ( λ ) = ˆ f + C Σ λ, λ ∈ R q , (18) where q = m − n + 1 and C Σ ∈ R m × q is the cycle basis matrix. T hat is, f ( λ ) is a lin e flow so lu tion par ameterized by cycle flow . W e next d efine the line flow vector n ormalized by D as z ( λ ) : = D − 1 f ( λ ) = D − 1 ˆ f + D − 1 C Σ λ = z 0 + H λ, (1 9) where z 0 : = D − 1 ˆ f and H : = D − 1 C Σ . W e note the edge- wise feasib ility cond itio n in Pro position 1 r equires | z ij ( λ ) | ≤ 1 , ( i, j ) ∈ E . (20 ) Now we can d e n ote the cycle con sistency expression in (11) par ametrized by the cycle flow ind icator λ as g ( λ ) : = C ⊤ Σ arcsin( z ( λ ) ) = C ⊤ Σ arcsin( z 0 + H λ ) , (21) where g ( λ ) is a q -d imensional vector whose comp onents represent the sum of pha se ang le differences around the correspo n ding b asis cycles. Th erefore, by Proposition 1, a sufficient cond ition for the existence of a power flow solutio n is the existence o f λ th at satisfies both (20) an d g ( λ ) = 0 . T o obtain a verifiable existence co ndition, co nsider the bo x Λ : = { λ ∈ R q : ¯ λ ≤ λ ≤ ¯ λ } wh ere ¯ λ, ¯ λ ∈ R q satisfy ¯ λ ≤ ¯ λ compon entwise. W e will explo it a monoto nicity pr operty of g , wh ose verificatio n will be discussed later . Theor em 1 (Sufficient Condition fo r Solva bility): Given a connected power system with nodal injection P ∈ 1 ⊥ ⊆ R n described by the lossless power flow equations ( 2). Let the line flow f ( λ ) , norm a lized line flow z ( λ ) , and cycle consistency residu al g ( λ ) be as defined in (18), ( 19), and (21), resp ecti vely . Let Λ : = { λ ∈ R q : ¯ λ ≤ λ ≤ ¯ λ } (22) for som e ¯ λ, ¯ λ ∈ R q with ¯ λ ≤ ¯ λ . Suppose the following cond itions hold: 1) The n ormalized line flows ar e boun d ed by 1 for λ ∈ Λ : | z ij ( λ ) | ≤ 1 , ( i, j ) ∈ E . (23) 2) Each compo nent g i ( λ ) is non-d e creasing in every coordin ate λ i in Λ . 3) For every i = 1 , . . . , q , we h a ve ¯ g i ( ¯ λ, ¯ λ ) : = g i ( ¯ λ 1 , . . . , ¯ λ i − 1 , ¯ λ i , ¯ λ i +1 , . . . , ¯ λ q ) ≤ 0 , ( 24a) ¯ g i ( ¯ λ, ¯ λ ) : = g i ( ¯ λ 1 , . . . , ¯ λ i − 1 , ¯ λ i , ¯ λ i +1 , . . . , ¯ λ q ) ≥ 0 . (24b) Then the re exists λ ⋆ ∈ Λ such that g ( λ ∗ ) = 0 . Conseq uently , the lossless real power flow eq uations (2) admit a solution . Pr oof: Since th e nor malized line flow z ( λ ) = z 0 + H λ is affine in λ , it is continu ous on Λ . By (23), each co m ponen t of z ( λ ) lies in [ − 1 , 1] fo r all λ ∈ Λ . It follows that arcsin( z ( λ ) ) is well defined an d contin uous on Λ , g ( λ ) is in tu rn continu ous on Λ . T ake any i ∈ { 1 , . . . , q } . Sinc e g i ( λ ) is no n -decreasing on Λ f or eac h compo n ent, its max imum value on th e “ lower face” { λ ∈ Λ : λ i = ¯ λ i } of the box Λ is attain ed wh en all other co m ponents are m a ximized, i.e., λ j = ¯ λ i for every j 6 = i . I n o ther words, g i ( λ ) ≤ ¯ g i ( ¯ λ, ¯ λ ) for any λ ∈ Λ such that λ i = ¯ λ i . The r efore, ( 24a) implies th a t g i ( λ ) ≤ 0 for any λ ∈ Λ such that λ i = ¯ λ i . In the similar vein, (2 4b) implies that g i ( λ ) ≥ 0 fo r any λ ∈ Λ such that λ i = ¯ λ i . In su mmary , we h a ve shown that cond itions 2 and 3 imply that the com ponent function g i ( λ ) is n on-po siti ve on the lower face { λ ∈ Λ : λ i = ¯ λ i } an d no nnegative on the upper face { λ ∈ Λ : λ i = ¯ λ i } . fo r each index i ∈ { 1 , . . . , q } . By the Poincar ´ e-Miranda theor e m [19], there exists λ ∗ ∈ Λ such that g ( λ ∗ ) = 0 . Condition 1, together with g ( λ ∗ ) = 0 , ensure s that the line flow f ( λ ∗ ) = ˆ f + C Σ λ ∗ satisfies the two con ditions of Propo sition 1. It then follows from the prop o sition that f ( λ ∗ ) correspond s to a power flow solution. Therefor e, the lossless r eal power flow eq uation (2) admits a so lution. V . C Y C L E B A S I S C H A R AC T E R I Z A T I O N A N D S O LV A B I L I T Y V E R I FI C AT I O N In this section, we describe how the cycle basis is con- structed and how the sufficient co ndition f or solvability de- riv ed in Sectio n IV can b e verified. The p roposed pro cedure is car ried o u t entirely in the line flow doma in . W e start from the particu lar line flow solu tio n ˆ f = D A ⊤ L † P , co nstruct a cycle basis matr ix C Σ , fo rm the cycle flow param etrization by λ ∈ R q as f ( λ ) = ˆ f + C Σ λ , and verify the theor e m o n a b ox Λ := { λ ∈ R q : ¯ λ ≤ λ ≤ ¯ λ } . A key in gredient is th e constructio n of a cycle basis wh o se columns have a consistent edg e o rientation. That is, for an edge traversed b y multiple cycles, it is traversed in the same direction by all the se cycles. This yields a cycle basis matrix C Σ with non n egati ve entries, so that H : = D − 1 C Σ inherits the same sign structure. As a result, the nor malized line flow z ( f ) = D − 1 ˆ f + D − 1 C Σ λ = z 0 + H λ varies mon o tonically with respect to each coord inate of λ . T his mono tonicity r esult is the fou ndation for verif ying the h ypotheses of Theorem 1. A. Robbin s’ The o r em Giv en a connected undirected g raph G , a classical result due to Robbins [20] states that G adm its a stro n gly conn ected orientation if and only if it is bridgeless, or equivalently , 2 -edge - connected . A strongly conn ected orie n tation of an undirected gra p h is an assignme nt o f directio ns to all edges such that, in the resu lting directed grap h , e very node is reachable fr om every other n ode v ia d ir ected paths. Since ev e ry stron gly conne cted or ien tation co ntains a direc te d cycle throug h ea c h edge, Robb ins’ theorem ensures that a cycle 1 2 3 4 e 2 e 1 e 4 e 3 e 5 Fig. 1. Undi rected graph 1 2 3 4 e 2 e 1 e 4 e 3 e 5 Fig. 2. DFS Spanning tree 1 2 3 4 e 2 e 1 e 4 e 3 e 5 C 1 Fig. 3. Directe d graph with C 1 1 2 3 4 e 2 e 1 e 4 e 3 e 5 C 2 Fig. 4. Directe d graph with C 2 basis with consistent orientatio n s can be constructed for a connected b ridgeless graph . The graph in Fig. 1 re p resents an undir ected bridgeless graph. The n, by Rob b in’ s theorem, it adm its a stro n gly connected orien tation, and hen ce an o rientation compa tib le with a directed cycle structure . T his observation motivates the cycle-basis co nstruction discu ssed next. B. Cycle Basis Construction via De p th-F irst Sear ch-Ba sed Orientation Practical power systems, inclu ding many IE EE test cases, may contain brid ges an d radial sections. Sin ce bridge s do no t belong to any cycle, they do no t contribute to the cycle sp ace ker( A ) a n d do not affect the cycle consistency con dition in Proposition 1. Therefo re, suc h edges ca n be re m oved befor e constructing the cycle basis. After removing all bridg e s, the remaining g raph decomposes into one o r mor e bridg eless connected compo nents, and the cycle basis can b e con - structed in d ependen tly on eac h co nnected compo nent. In our implementatio n, th e cycle basis is ob ta in ed th rough a Depth-First Search (DFS) traversal using the Schmid t chain d ecomposition p rocedu re in [21]. In th is algo r ithm, an arb itrary node is first selected as the ro o t of the DFS tree. When ever DFS exp lores an edge fro m a visited n ode to a n unvisited node, that e dge is classified as a tre e edg e and orien ted f r om the child toward the parent. This pr oduces a directed spa nning tree who se edges poin t tow ard the root. In Fig. 2 , n ode 1 is ch osen as the root node, and the green edges ind icate the resulting dir ected tree edges. Any non - tree edge enco untered durin g DFS con nects a node to one of its an c estors in the DFS tr ee and is, therefor e, a so-called back edge. Each b ack edg e is or iented fro m th e ancestor to the d escendant. T ogether with the uniqu e d irected tree path fro m the descen dant b ack to the ancestor, the back edge forms a d irected cycle. Each such cycle contr ibutes o ne column to the cycle basis m atrix C Σ . For the examp le in Fig. 3 , the b ack edge e 4 : (1 → 4) forms th e directed cycle c 1 =  1 0 1 1 1  ⊤ . (25) 1 2 3 4 5 6 7 8 12 13 9 10 14 11 Fig. 5. IEEE 14 bus system repre- sented by an undirected graph 1 2 3 4 5 6 7 8 12 13 9 10 14 11 Fig. 6. Directed 14 bus system: green edge represents the tree edge and red edge represents the back edge Similarly , the back ed ge e 2 : (2 → 4) f orms c 2 =  0 1 1 0 1  ⊤ . (26) Hence the cycle b asis matrix is C Σ = [ c 1 , c 2 ] =  1 0 1 1 1 0 1 1 0 1  ⊤ . (27) By constru ction, th e cycle vectors are rep resented under a consistent glo b al orientatio n, and their en tries are nonnega- ti ve on th e edg es that belong to the correspo nding cycles. The same procedu re is app lied to the IEEE 1 4-bus network shown in Fig. 5. A f ter removin g the b r idge between buses 7 and 8 , Schmidt ch ain decomposition y ields th e directed graph shown in Fig. 6 , from which the cycle basis matrix C Σ can be con structed. C. V erification P r ocedure After d iscussing the cycle b asis con struction, we describe how to apply the sufficient so lvability con dition in Section IV to certify the existence o f a power flow solu tion. W ith the cycle basis matrix C Σ , we co mpute the particular line flow ˆ f = D A ⊤ L † P , norm alized line flow z 0 = D − 1 ˆ f , and the normalized cycle basis matrix H = D − 1 C Σ . The parametrize d line flows and th e cycle consistency residual can subsequently be d efined as f ( λ ) = ˆ f + C Σ λ , z ( λ ) = z 0 + H λ , and g ( λ ) = C ⊤ Σ arcsin( z ( λ ) ) . After th e definition s above, the verification pr o cedure can be broken down into th e f o llowing three steps. First, we construct a box Λ = [ ¯ λ, ¯ λ ] such that | z ( λ ) | ≤ 1 holds fo r all λ ∈ Λ . Since z ( λ ) is affine in λ , this amoun ts to enf orcing − 1 ≤ z 0 ,ℓ + H ℓ, : λ ≤ 1 , ℓ ∈ E (28) for all λ ∈ Λ . Hence, [ ¯ λ, ¯ λ ] is a n inne r app roximation of the edge-wise f easible region. Second, we verify th e mono to nicity con dition in The- orem 1. Because the cycle basis is constru cted with a consistent o rientation, the matrices C Σ and H are bo th nonnegative. This makes g i ( λ ) a fu n ction that is compo nent- wise mo notone for ev er y i . Third, we verify the face cond itions (24 ) in Theo rem 1 . For each k = 1 , . . . , q , we ev alu ate ¯ g k ( ¯ λ, ¯ λ ) and ¯ g k ( ¯ λ, ¯ λ ) . I f they satisfy (2 4) f or every k , th en th e hy p otheses of Theorem 1 hold. I t follows that there exists λ ∗ ∈ Λ such tha t g ( λ ∗ ) = 0 . Hence the line flo w f ( λ ∗ ) = ˆ f + C Σ λ ∗ satisfies both Algorithm 1 Sufficient Solvability Condition V erification 1: Remove a ll bridge edges and decom pose the remaining graph into bridg eless compo n ents 2: Construct a con sistently o riented cycle basis matrix C Σ using DFS/Schm idt chain decomp osition 3: Compute ˆ f = D A ⊤ L † P 4: Compute z 0 = D − 1 ˆ f and H = D − 1 C Σ 5: Construct a box Λ = [ ¯ λ, ¯ λ ] such that − 1 ≤ z 0 + H λ ≤ 1 , ∀ λ ∈ Λ 6: for k = 1 , . . . , q do 7: Evaluate ¯ g k ( ¯ λ, ¯ λ ) 8: Evaluate ¯ g k ( ¯ λ, ¯ λ ) 9: end for 10: if all ¯ g k ( ¯ λ, ¯ λ ) ≤ 0 and ¯ g k ( ¯ λ, ¯ λ ) ≥ 0 for all k = 1 , . . . , q then 11: Solvability certified by Theor em 1 12: else 13: Inconclu si ve 14: end if edge-wise feasibility and cycle consistency con ditions. By Proposition 1 , it therefor e correspon ds to a solution of the lossless r eal power flow eq uation. Algorithm 1 summ arizes the verificatio n procedu re. V I . R E S U LT S T o assess the acc uracy of the pr oposed sufficient condition , we evaluated its perf ormance on the I E EE 9-bus an d IEEE 14-bus systems an d compa r ed the resu lts with those obtain ed using the Newton-Raphson (NR) m ethod. T h e compar ison is based on determin ing the max imum level of system stress fo r which a feasible power flow solu tio n exists. System stress is introd uced b y un iformly scaling th e p ower injections b y a scalar par a meter y , i.e., P ( y ) = y P 0 , wh ere P 0 denotes the nominal injection vector . Th e co rrespond ing particular line flow solution is ˆ f ( y ) = D A ⊤ L † P ( y ) = y D A ⊤ L † P 0 . (29) For each value of y , we apply the verificatio n proced ure described in Section V. Let y cert denote the m aximum loading le vel for which the proposed sufficient condition certifies th e existence of a p ower flo w solu tion. Likewise, let y NR denote the largest loading le vel for which th e NR method con verges to a solu tion of th e lossless power flow equations when initialized in th e stand ard man ner, an d th e phase angle differences are b o unded b y π / 2 . The q uantity y cert therefor e represents the certified solvability margin giv en by th e proposed method, whereas y NR serves as a referenc e for the actu al solvability b o undary . T o quan tify th e tightness of the certificate, we rep ort η : = y cert y NR (30) as the tightness measur e. A value of η closer to 1 indicates a less con servati ve certificate. The NR simulations were carried T ABLE I C E RT I FIE D S T R E S S R A T I O y cert /y NR F O R D I FF E R E N T T E S T C A S E S . T est Case λ 2 test [23 ] ∞ -norm test [10] Proposed test IEEE 9 16.54% 73.74% 100% IEEE 14 8.33% 59.42% 100% out using M A T P O W E R [2 2]. T able I com pares the prop osed test with two existing su fficient conditio ns f r om the litera tu re, namely the λ 2 test [2 3] and th e ∞ -n orm test [ 10]. It can be seen fr om the table that f o r bo th the IEEE 9-bus and 14-bus systems, the prop osed meth o d yields essentially exact certificate, with the certified loa ding level m atching the NR-based r e f erence. The se results show that th e pro - posed cycle co nstraint app roach can p rovide a rem arkably tight sufficient so lvability co ndition, at least f o r small and moderately sized meshed networks. At the same time, the numerical studies indicate that the tightness of the certificate depends heavily on the ch osen cycle ba sis. Different cycle bases of the same network lead to different m atrices C Σ and H = D − 1 C Σ , which in turn affect the m onoton icity structure of g ( λ ) , the size o f the adm issible box Λ , and the conditio n (24 ) in Theo rem 1 . In particular, cycle bases with weaker overlap among cycles tend to y ie ld less conservati ve certificates, which is clear in the IEEE 14-bus case. These o bservations suggest that cycle basis selection is an imp ortant prac tica l compo nent of the p roposed method. Developing systematic proc edures fo r constructing fa vorable cycle bases and extending the ap proach to larger networks are amon g the imp ortant direction s for f uture work . V I I . C O N C L U S I O N In this p aper, we developed a n ew sufficient condition for th e solvability of the lossless real power flow equ ation based on the cycle c o nstraint in th e line flow domain. By combinin g a prec ise character iz a tio n of realizable line flows with an inv arian t set-based so lvability certificate for the cycle consistency constraint, we o btained a mathem atically rigoro u s an d p ractically accu rate test for power flow so lvabil- ity . Numerical results on the IEEE 9-bus and IEEE 1 4-bus systems show that the propo sed con dition can be significan tly less c o nservati ve than existing sufficient condition s and can provide a re m arkably tigh t certificate of solvability . At th e sam e time, the nu merical perfo rmance of the propo sed m e thod depend s on the chosen cycle basis. T his suggests that cycle b a sis construc tio n is an impor tant comp o- nent of the overall ap proach . Po tential futur e work s includ e developing sy stematic proc e d ures fo r constru cting fav orable cycle bases and extend ing the meth od to larger networks. The proposed fr a mew o rk provides a cycle-aware alterna- ti ve to conv e n tional DC-based a pproxim ations in settings where a rigoro us sufficient con dition for AC power flow solvability is d esired. 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