A Linear LMP Model for Active and Reactive Power with Power Loss

1 A Linear LMP Model f or Active an d R eactive Power with Power Loss Yanghao Yu, Qingch un Hou, Student Me mber, IEEE , Yi G e, Guojing Liu, and Ning Zhang, Senior Me mber, IEEE   Abstract — Pricing the reactive power is more n ecessary than ever b efore bec ause of the increasing challenge of renewabl e energy integration on reactive pow er balance and voltage control. However, react ive pow er price is ha rd to be efficientl y cal cul ated because of the non-line ar nature of optim al AC p ower f low equation. This paper proposes a li near model to calculate acti ve and reactive pow er LMP si multaneously consid ering pow er loss. Firstly, a l ineariz e d AC power flow equation is p roposed based on an augmented Gene r atio n Shift Distribut i on Factors (GSD F ) matrix. Second ly, a linearized LMP model is derived using GSDF and loss factors. The f ormulation of LMP is furt her d ecomposed into four components: energy, co ngestion, voltage limitation and power loss. Finally, an iterate algorithm is proposed for calculating LMP with the pro posed model. The performance of the proposed model is validated by the IEEE-118 bus system. Index Terms — Reactive power pricing, optimal pow er flow, Generation Shift Distribution Factor, LMP, power loss. N OMENCLATU RE P , Q Vectors of bus injected active and reactive power , i i P Q Injected active and reactive power at bus i j Y = G + B Admittance matrix of the power system j    Y = G + B Admittance matrix without shunt elements ij ij ij Y G jB   Element in the i th row and j th column of Y ij ij Y G jB      Element in the i th row and j th column of  Y V , θ Vectors of Ma g nitud es and phase an g les of bus voltages , i i V  Voltage magnitude and phase angle at bus i , M N The number of branches and buses ij ij ij y g jb   Admittance of branch ( , ) i j ii ii ii y g jb   Shunt admittance at bus i , , l l l m m m P Q I Power flow and current through branch m , G D i i P P The active power generation and active load at bu s i , G D i i Q Q The reactive power gen e ration and reactive lo ad at bus i This work w as supported by the Major Smart G rid Joint Project of National Natural Science Foundation of Chin a and State Grid (No. U17 66212) and Scientific & t echnical project of State Grid "Res earch and application of large-scale ene rgy storage pl anning for receiving- end power sy stem" (No . 5102- 20 19 18309A-0- 0-00). Y. Yu , Q . Hou, N . Zhan g are with the State Ke y Lab of P ower Systems, Departm ent of E lectrical Engineering, Tsinghua University , Beijing 100084, China  (e-m all: ningzhang@tsinghua.ed u.cn ). Y. Ge, G. Liu a re with E conomi cs a nd Te chnol ogical Res earch I nstitute, State G rid Jiangsu Electric Po wer Co., Ltd., Nanji ng, China. , P Q i i F F The fictional nodal demand at bus i , Loss Loss P Q Total power loss of the system , P Q i i LF LF The active and reactive power loss factor at bus i , P Q i i DF DF The active and reactive power d e livery factor at bus i I. I NTRODUC TION OCATIONAL Ma rginal Price ( LMP) is defined as the marginal expenses of both generat ors and transmissi on system to suppl y the in crement of active pow er at each b us. A pow er market usin g L MP can p rovide the right p r ice si g nal to relieve the congestion of p ower system. T herefore, LMP has been widel y ad opted i n Indepen dent S y stem Operators (ISOs), such as PJM int erconnection, the New York market , and the New England ISO [1]. The LMP can be di vided i nto active powe r LMP (ALMP) and reacti ve power LMP (RLMP). The l ater denot es marginal cost to supply t he incr ement of reactive p ower d emand at each bus. Currently the LMP ad opted by all of the po wer market i s ALMP onl y . Most re active power sources like stati c VAR com pensators (SVCs) and switched capacitors ( SCs) are n ot pri ced by RLMP but compensated as ancillar y servic e i n power market. Nowada y s, i ncreasing ren ewable energy penet ration brin gs g reat challenge to power system voltage securit y . The adequate reactive pow er supply ma y become crit ical. Sinc e the reactive power cannot be transmitt ed by long dist ance and i s usuall y locall y b alanced, the need of r eactive power for different node would be distinct. The val ue of providing reactive power would be different even there is no congestions in t he power system. T herefore, intro ducing RLMP in the market woul d provi de significant p rice signal fo r both shor t-term operation and l ong-term plannin g. The value of LMP can be decompose d into s everal com ponents. Referen ce [ 1] prop oses the DCOP F model with loss to d ecompose LMP into th e cost s of marginal energy, margin al loss and congestion. Reference [2] i ntroduc e s the concept of LMP to a distribution system. The distributi on LMP (DLMP) is decom p osed i nto margin al energy c ost, l oss cost and conge stion c omponents. Reference s [3 ], [4 ] further anal y zes both ac ti ve and reactive power D L MP w ith consi deration of congesti on and voltage suppo rt. The DLMP i s divid ed into five compon ents: marginal costs for active power, reactive po wer, congest ion, vol tage s upport, and powe r loss. The value of different LMP c omponents togeth er r eveal the econo mic and oper ation condition s of power system, whic h provi de price signal t o influence the beh avior of participants t o miti gate t ransmission congestion and s timulate reacti ve p ower L 2 supply for volt age support. LMP can be d erived f rom optimal power flow (OPF) [ 1]. In practi cal application of LMP, AC opt imal power flow (ACOPF) and DC optimal power flow (DCOPF) are two cl assic models. ACOPF can s imult a neousl y calc ulate ALMP and RLMP accuratel y . But it is a no n-convex prob lem which may be computatio nally int ractable even for s maller sy ste ms [5]. Addition a lly, LMP derived from ACOPF are not a nalytical so that it can no t be dec omposed into differ ent c ompone nts and is lack o f transpare ncy [ 6]. Comp aratively, DCOPF m odel is more conci se and efficient , wh ich m akes it wi dely utili zed by LMP-based mar kets. However, since it neglect s the voltage fluctuation and p ower l o ss, it has a relativel y low acc urac y i n LMP calculatio n. In a ddition, DCOPF cannot calc u late RLMP since it i gnores t he reactive power balan ce and reactive power flow. Recent researches have been condu cted to improve th e L MP calculatio n. Referen ce [7] uses en ergy ref erence b us independe n t model t o o btain a more accurate ca lculation on energy, conges tion and l oss components o f LMP. In Refer ence [8], an it erative DCOPF algorit hm is propos ed to address the nonlinear marginal loss. Reference [9] proposes a linear loss-em bedded mo del t o calcul ate LMP with l oss compon ents by deal ing wit h margi nal generator s under lossless DCOPF solutio n . Reference [10] pr oposes a line ar-constrained and convergence-guara nteed OPF mo del that explicitly con siders the c onstraints of react ive power b alance and bus voltage. Reference [ 11] proposes a lineari zed power flow model with decouplin g of volt age magnitu de and phase an gle to ac hieve accurate estimatio n of voltage magni tude. Base d on t he lineariz ed power flow mod e l, Refere nce [6] f urther proposes the d ecoupled OP F model using Generation Shift Distributio n Factors (GSDF) and furth er applies it to jointly cal culate ALMP and RLMP wi th r espect to volt age con straints, b ut the impact of network loss is neglecte d in this model. This paper proposes a linearized iterati ve mod e l t o calculat e both ALMP and RLMP considering power loss. A l inearized AC po wer fl ow e quation is proposed base d on the augmented GSDF matri x. T he active and reactive power loss are linearized by introd ucing l oss f actor (LF). An iterat e a lgorithm is propose d to calculate power l oss and th e LMP . Finally, the accurac y a nd perfor mance of the pro posed m odel is validated on an IEEE-118 bu s case. The rest o f thi s paper i s organized as follows: Secti on I I constructs the linear ized LMP mo del. Sectio n III validat es the accurac y of the A L MP and RLMP cal culation of the model through the IEEE 1 18-bus case on MATPOWER. Finally, Section I V concludes the pa per. II. L I NEARIZED LMP M ODEL W ITH P OWER L OS S This Section builds the LMP mo del b ased on GSDF. Th e modeling is di vided into five s ubproblems: 1) Introducing GSDF-based linear AC power flow model; 2 ) Lin earizing the power losses; 3) Buil ding GSDF - based LMP mod el; 4) Derivi ng ALMP and RLMP ba sed on the LMP model; 5) Proposing iterate al gorithm to solve L MP. This Section wil l present th e tec hnical details of each subproblem. A. Power Fl ow Equations w ith GSDF A lineari zed powe r flow e quation with respect to volta g e magnitu de and p hase a ngle is prop osed b y ref erence [11] as follow s. -                           P B -G θ θ = - C Q G B V V  (1) The ma trix C in Eq. ( 1) is a 2 2 N N  matrix det ermined solel y by the element of admitta nce matrix. Based on Eq. (1), the active p ower b alance e quation is der ived as follows: 1 1 1 1 1 1 1 1 1 1 1 0 N N N N N N N N N i ij j ij j ij j ij j i i j i j j i j i N N jj j jj j j P G V B G V B g V g                                              (2) The two appr ox imation ste ps in Eq. (2) are ba sed on t he ass umpt ions t hat 1 j V  and 0 jj g  . Similarl y , we have the reactive power balance equat ion as follo ws: 1 1 1 N N N i jj j jj i j j Q b V b           (3) Acco rding to Eq. (2 ), the eq uations Eq . ( 1) is not inde pendent. I n other wor ds, matrix C in (1) i s close to singul ar. So we remove the row a nd co lumn of t he re ference bus in C and invers e the ne w     2 1 2 1 N N    matri x. We then f ill the row and c olumn of th e reference bus i n the i nversed matri x wit h zeros. Hence, we have a new 2 2 N N  matrix X satisf ying:              θ P X V Q (4) Unfoldin g the matrix form of E q. (4), th e voltage equation is acqui red as follows: , , 1 1 N N i N i k k N i N k k k k V X P X Q          (5) Acco rding to Eq. ( 4), t he p ower flow o f b ranches can be linea rly express ed by t he bus power injection both of ac tive and reactive parts:                 , , , , 1 , , , , 1 = = l m ij i j ij ij i j N ij N i k N j k ij i k j k k k N ij N i N k N j N k ij i N k j N k k k P P V V g b g X X b X X P g X X b X X Q                          (6)                 , , , , 1 , , , , 1 = l m ij i j i j ij i j N ij i k j k i j N i k N j k k k N ij i N k j N k ij N i N k N j N k k k Q Q b V V g g X X b X X P g X X b X X Q                              (7) The coefficients of k P and k Q in Eqs. (6) -(7) forms t he GSDF m atrix between the b ranch po wer flow and power inj ec tio n: 3                 , , , , , , , , , , , , , , , , P P m k ij N i k N j k ij i k j k P Q m k ij N i N k N j N k ij i N k j N k Q P m k ij i k j k ij N i k N j k Q Q m k ij i N k j N k ij N i N k N j N k GSF g X X b X X GSF g X X b X X GSF g X X b X X GSF g X X b X X                                           (8) Finall y , t he active a nd reactive power flow equations (Eqs. (6)-(7)) can be expressed using GSDF:         1 1 1 1 1 1 1 1 , for = 1 , 2 , for = 1 , 2 N N l P P P Q m m i i m i i i i N N P P G D P Q G D m i i i m i i i i i N N l Q P Q Q m m i i m i i i i N N Q P G D Q Q G D m i i i m i i i i i P GSF P GSF Q GSF P P GS F Q Q m M Q GS F P GSF Q GSF P P GSF Q Q m M                                               (9) Diff erent from classi c DC power flo w model , t he p roposed model consid ers both active and reactive power, which provide foundations for deriving ALMP a nd RL MP. B. Conside ri ng Power Loss in GSDF-bas ed Power Flow Equati ons It should be noted that in the derivation of last subsection we do no t consider power l osses. In other words, th e t otal active power generation and reactive power generation i n E q. (2) equals t he active lo ad and reactive lo ad respectively ( plus the load on th e shunt branch ). Such loss le ss consumpti on does not coincide with practic al power s y stem. Therefore, we need to consid er the po wer losses in p ower bala nce equations a nd allocat e the power losses to e ach node as virtual lo ad. Acco rding to Joule’s Law, b oth acti ve and r eactive po wer losses, can be calculated using branch po wer flow (a ssuming that 1 i V  ):                 2 2 1 1 2 1 1 1 2 2 1 1 2 1 1 1 M M l l Loss m m m m m m M N N P P G D P Q G D m i i i m i i i m m i i M M l l Loss m m m m m m M N N P P G D P Q G D m i i i m i i i m m i m P I R P R GSF P P GSF Q Q R Q I X P X GSF P P GSF Q Q X                                                     (10) In o rder to consider power losses, t he concept of loss factor (LF), delivery factor (D F) an d fictio nal n odal demand (FND) [ 8] are i ntroduced in t his paper . LF indicates t he ma rg inal power losses with re spect t o an incremen tal power injection on a bus, while DF indi cate th e active and reactive p ower th at is equivalentl y inje c te d to t he gri d. LF and DF for act ive a nd reactive p ower can be stated a s Eq. (11). 1 1 = 2 , 1 for = 1 , 2 = 2 , 1 for = 1 , 2 M P l P P P P Loss i m m i m i i G m i M Q l P Q Q Q Loss i m m i m i i G m i P LF P GSF R DF LF i N P Q LF P GSF X D F L F i N Q                     (1 1 ) It shoul d be noted that P i LF and Q i LF ma y be negat ive for som e node, denoting t hat the power inj ection would reduce the pow er losses. We int r oduc e t he DF into the power balance Eqs. (2)-(3) so that t he power losses c an b e explicit ly formula ted i n the equation:     1 1 1 0 N P G D i i i Loss i N N Q G D i i i Loss jj i j DF P P P DF Q Q Q b              (12) FND is introduced to evenly allocate th e loss of each br anch to its t wo end buses, math ematically:     2 2 1 1 , , for = 1 , 2 2 2 P l Q l i m m i m m i m i m F P R F P X i N            ( 13) Each bus t akes th e P i F and Q i F on their connected branches as a v irtual injectio n. T herefore, the power injectio n a t each bus is re stated as: for = 1 , 2 for = 1 , 2 G D P i i i i G D Q i i i i P P P F i N Q Q Q F i N         (14) By i ntroducing the FND in each bus, the total active power generati on no lo nger equa ls to the active l oad since the generati on need to bal ance the extra virtual i njection. It suggests that bot h the acti ve and reactive power loss es are consi dered in the proposed GSDF -based power flow equations. C. LMP Model Based on GSDF Acco rding to the derivations above, especial ly wit h Eqs. (5), (9), (12), and (14), we draw the GSDF-based LMP model as show n in Eq. (1 5). , , i i i a b c represent t he cost coefficients of generat or at bus i , t he cost functions of active and re act ive pow er gen eration are both qu adratic while re act ive fun ction does not have primary term; , , , P Q k i u v   denote t he L agrange multi plier of t he con st raints of acti ve power b alanc e, reactive pow er b alance, power flow c ongesti o n and vol ta ge res pectively; , P Q i i DF DF denote t he DF at bus i ; , Lo ss Loss P Q are th e t otal system a ctive and rea ctive power loss; , P Q i i F F are ac tive and reactive FND at bus i ; lim k P is the active power limits of branc h k ; min max , i i V V denote the minim u m a nd maximum voltage at bus i respecti vely ; min max mi n max , , , G G G G i i i i P P Q Q denot e the active and react i ve power generation li mits at bus i resp ectively. 4               2 2 1 1 1 1 1 1 , , 1 . . 0 , for = 1 , 2 N G G G i i i i i i i N P G D P i i i Loss P i N N Q G D Q i i i Loss jj Q i j N N P P G D P P Q G D Q l m m i i i i m i i i i m m i i N V G D P i N i k k k k N i k Min a P b P c Q s t f DF P P P f DF Q Q Q b g GSF P P F GSF Q Q F P m M u h X P P F X                                                1 lim lim min max min max min max , for = 1 , 2 , for = 1 , 2 , for = 1 , 2 , for 1 , 2 , for 1 , 2 N G D Q N k k k k i i k l m m m i i i G G G i i i G G G i i i Q Q F V i N P P P m M V V V i N P P P i N Q Q Q i N                        (15) D. A LMP and RLMP Deco mposition When the optimal solu tio n of the model is reached, LMP can be calcula ted by the Lagrange Functi on:     2 2 1 1 1 N G G G i i i i i i P P Q Q i M N V m m i i m i a P b P c Q f f g h                         (16 ) ALMP is defined as the marginal cost o f an incr emental change of active power demand at a bus. AL MP at bus i can be calculat ed a s the f irst-ord er part ial derivati ve o f t he Lagrange Function w ith respect to the active po wer load at bus i : 1 1 , 1 1 , 1 1 = = = = i D i V M N j P m P m j D D D m j i i i M N P P P i P m m i j N j i m j M N P P P P m m i j N j i i P m j ALMP P h f g P P P DF GSF X GSF X LF                                              (1 7) Simil arly , RLMP i s d efined as marginal cost of a n incrementa l change o f r eactive powe r demand at a bus. T he calculatio n of RLMP at bus i is: 1 1 , 1 1 , 1 1 i D i V M N Q j m Q m j D D D m j i i i M N Q P Q i Q m m i j N j N i m j M N P Q Q Q m m i j N j N i i Q m j RLMP Q f h g Q Q Q DF GSF X GSF X LF                                                    (18) Eqs (17) an d (18) derives t he analy ti cal f orm d iff eren t component s of ALMP and R LMP. The first term of LMP is the energy compone nt derived from th e power balance constrai nts. The second term is the congestio n comp onent which is related to the power f low equations. The t hird term is the volt a ge constraint compon e nt. The la st term is l oss component. If loss is not consid ered in a model, LF is zero . When t he constraints on power flow or voltage magni tudes are active, t heir corre sponding components of L MP are non-zero. E. Solvin g Algorithm An i teration algorithm i s propose d to s olve the LMP model with power l oss, as shown in Fig. 1. First, w e sol ve t he LMP model without power loss. Then t he power loss is calculate d usin g E qs. (10)-(11) and is allocated using Eqs. (14). The LMP model is later upd ated with the calculated DF and FND. After one i teration, we solve the updated LMP m odel and re-cal culate the po wer losses. Wh en t he differ ence of po wer l o sses i n two iteratio ns is smaller than a certain threshold, iteration stops and a final LMP is obtained from the newest LMP model. Figure 1 Flow char t of the iteration algorithm III. C ASE S TUDY In this section, the GSDF- base d LMP mod el is appli ed to an IEEE 118-bus case on MATPOWER 6.0 to validate its 5 accurac y in cal culating A LMP a nd RLMP. The model i s impl emented in Ma tlab R2016a with the solver Gurobi 8.0. The simulation platfo rm is a per sonal compute r with a n Intel Core i7 CP U @2.50GHz and 8GB RAM. The IEEE 1 18-bus t est c ase is from MAT POWE R 6.0 toolbox of MATL AB [12] . Meanw hile, t he original case does not contain l imits on b ranch po wer flo w. The bran ch MV A rating dat a are t aken from I ndex of Data Ill inois I nst itute of Technolog y [ 13]. Moreover, th e cas e does n ot p r ovide cost for reactive power . We add the re active cost function of generators and let i c in Eq . (15) be the same as i b . The LMP res u lts of our model (denot e d as LMP-L) are compare d w ith t he LMP model without l oss (denoted as LMP-O), and t he L MP results derived from ACOPF and DCOPF solver in MATP OWER (denoted as LMP -AC and LMP-DC). The result obtai ned from ACOPF is s et as t he benchm ark re sult t o calculate the accuracy of other res ults. A. ALMP Accura cy Evaluation. Since the ALMP is mostly conc erned i n the market, we compare the accur acy of ALMP obtai ned from t he proposed method and that from result derived from LMP-O and DCOPF, taking ACOPF as t he st andard r esult. The index o f th e average error of ALMP (AEA) is introdu ced as:   1 / N AC AC i i i i ALMP ALMP ALMP AEA N     (19) Here AC i AL MP is the ALMP result at bus i from ACOPF. In the ACOPF model, when the volt age c onstraint is active, the voltage component influenc es the LMP. In o rder to demonstrat e t he effe ctiveness of our p roposed mo de l i n consid er ing t he voltage co nstraint, three scenarios with different voltage constraints are carried out in th e case study : 1) Loose voltage constraint: min max 0. 90 , 1. 10 V V   ; 2) Normal voltage constraint: m in m ax 0 .9 5 , 1.0 5 V V   ; 3) T ight voltage co nstraint: min ma x 0.9 7, 1 .03 V V   . Fig.2 shows th e A EA of the l oss-embedded L MP model (ALMP-L), LMP m odel without loss ( ALMP-O) and DCOP F model (ALMP-DC) whe n load level changes i n different voltage constr aint scenarios . ALMP-O per forms almost t he same as ALMP-DC i n lo ose and normal scena r ios and sl ightl y better than ALMP-DC i n tigh t sc enario. This indicat es t hat the lineariz ed model without powe r loss is com parable with DCOPF. In add ition, the ALMP-L res ults outpe rform both ALMP-O an d ALMP -DC i n all sc enarios and under all load levels. A t 0.95 p.u. load level, the AEA of ALMP-DC is ab ove 3% in all sc enarios, while the AEA of ALMP-L i s a round 1.5%. LOPF per forms 50% better than DCOPF. As loa d level rises, AEA o f A LMP-L has a t rend o f inc reasing, b ut it still outperforms DCOPF on AL MP. The results show th at th e propose d lineari ze LMP model with powe r l oss shows better accurac y i n calculatin g A LMP when comparing wi th D COPF, and power loss component pla y s a n important role in ALMP accurac y . Fig ure 2 AEA in t hree voltage scenarios B. ALMP and RLMP Analysi s The case in tight voltage constraint is further a nalyzed. Fig.3 pres ents the A LMP and RLMP at each b us from LMP-L, DCOPF and ACOP F respectivel y ( DCOPF do es not have RLMP) . LMP-L tra cks the fluct uation of ALMP deri v ed f ro m ACOPF at most buses. However, DCOPF gives a fixed ALMP value at all buses, th us it has m a ssive er ror compared with ACOPF. One of the reasons is that DCOPF do es not consider the limit o f bus voltage and reacti ve po wer. In the meanwhil e, the RLMP resul ts from L MP-L mo del is almost th e sa me a s that fro m ACOPF, which indicates that model also performs well in RLMP c alculation. ($/MW) ($/Mvar) Fig ure 3 ALMP and RL MP in tight volt age constraint Fig. 4 illu strates th at voltage magnit udes calculat ed from LMP-L are also accurate comparing to LMP-AC. Moreover, it indic ates wh ether the voltage magnit ude at cert ain bus reaches its boundary or not, which is clearly r eflected on the LMP value as voltage comp onent. For example, the voltage magnitudes at bus 10, 2 5, 66 and 69 reac h their upper limit, which means they alr eady have abundant reactive power. Consequentl y , RLMP in Fig. 3 are r elatively low at these b uses. On t he contrary, the volta ge ma gnitudes at bus 58 and 11 7 reac h their low er limit , makin g t heir RLMP r elatively h igher. I n fact, ALMP f ollows the sam e rul e as R LMP but it’s n ot a s apparent as RLMP s hown 6 in Fig. 3. A s a result, LMP especially R LMP can work as a n indicator of r eactive po wer de mand. It further pro v ides incenti v es for the reactive po wer compensat ors to join the market. Figure 4 Bus vol tage magnitude at each bus Mor eover, Fig. 5 s hows the active and reactive power generated by all generators. The ac tive power gener ation result of LMP-L is c loser to ACOPF than DCOPF. As for t he reac tive power gen erat io n, LMP-L is nearl y the same as tha t from ACOPF at t he margin al gene r ato rs such as gene rator NO. 23, 24, and 35. The marginal generators altogether d etermine t he energy ter m o f LMP which is t he majorit y of LMP. Therefore, locatin g the right marginal generat ors grea tl y improves the LMP result s. 5 10 15 20 25 30 35 40 45 50 Generator 0 200 400 600 800 (MW) Active power generation Pm ax LMP-DC LMP-AC LMP-L 5 10 15 20 25 30 35 40 45 50 Generator -50 0 50 70 (Mvar) Reactive power generation Qm ax Qm in LMP-AC L MP-L Figure 5 Generato rs output (MW) (Mvar) Figure 6 Activ e and reactive power flows Fig. 6 presents the act ive and r eactive power flow t hrough all 186 branches. It in dicates that none of the br anch li mits i s reache d. Thus, the con gest ion co m pon e nt do es n ot have cont ribution to LMP in thi s case. That’s another reason wh y the DCOPF gives a fixed ALMP value at all bus es. Above all, through th e case stud y on I EEE- 118 b us system, this mo del is proved to be more accurate than DCOPF when calc ulating ALMP. This model als o s hows high accurac y i n RLMP c alculation compa red with ACOP F mod el. IV. C ONCLU SION In this paper, a GSDF-based L MP model for both A L MP and RLMP calculat ion with pow er l oss i s proposed. The necessit y to price the reactive power is first discusse d. T hen a li nearized LMP m odel i s derived from an augment ed GSDF matrix. Los s equat ions and l oss-relat ed factors are added to enrich the mo del. Thro ugh derivation of Lagrange functio n, LMP can b e decompos ed to four components (energ y , congestion, voltage limita tion and po wer loss). A case study on t he IEEE-118 bus cas e validate s t he accu racy of our model in LMP calculation . The c a se demonstra te s that ACOPF co nstraints can be cle arly reflec ted o n LMP compo nents. Therefore, with t he congestion and voltage compon ents of LMP , th e o perator c an e ffectively man ag e the branch c ongestio n and v olta ge s upport respectivel y . 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