J. Mod. Power Syst. Clean Energy https://do.org/10.1007/s40565-018-0404-6 Unfed optmal power flow model for AC/DC grds ntegrated wth natural gas systems consderng gas-supply uncertantes Jale FAN 1, Xaoyang TON 1, Junbo ZHAO Abstract A unfed optmal power flow (OPF) model for AC/DC grds ntegrated wth natural gas systems s proposed for the real-tme schedulng of power systems. Heren, the prmary physcal couplngs underlyng ths coordnated system are modeled and nvestgated. In addton, the uncertantes of gas loads are consdered when studyng the role of gas supply for gas-fred unts n power system operatons. The nonlnear gas system constrants are converted to the second-order cone forms that allow for the use of the Benders decomposton technques and the nteror-pont method to obtan the optmal soluton. The numercal results of the modfed IEEE 118-bus test system that ntegrates the Belgum 0-node natural gas system demonstrate the effectveness of the proposed model. The effects of gas demand uncertantes on the optmal schedule of thermal generators are nvestgated as well. CrossCheck date: 7 February 018 Receved: 13 October 017 / Accepted: 7 February 018 Ó The Author(s) 018 & Xaoyang TON xytong@swjtu.cn Jale FAN fjl@my.swjtu.edu.cn Junbo ZHAO zjunbo@vt.edu 1 School of Electrcal Engneerng, Southwest Jaotong Unversty, Chengdu 610036, Chna Bradley Department of Electrcal and Computer Engneerng, Vrgna Polytechnc Insttute and State Unversty, Blacksburg, VA 043, USA Keywords AC/DC optmal power flow (OPF), Natural gas system, Benders decomposton, Second-order cone programmng 1 Introducton Currently, owng to ther low prces and envronmentfrendly propertes, gas-fred unts (FUs) have been wdely used worldwde. For example, they accounted for 4% of the total nstalled capacty n the Unted States n 015 [1]. Meanwhle, the development of power electroncs and DC transmsson technology enhance the capablty of long-dstance energy transmsson []. Accordngly, nterconnectons and nteractons between power system, DC connectons, and natural gas systems are ncreasng rapdly. As a result, modern power systems have become an AC/DC and mult-energy coupled system, and any of them can have sgnfcant effects on the securty operatons of the system. Thus, the effcency mprovement of power utlzaton, accurate modelng, and stablty analyss of mult-energy couplng systems are ongong efforts. Ths paper ams to propose a unfed optmal power flow (OPF) framework to mprove the effcency of power usage and the relablty of ths mult-energy couplng system. To date, two types of OPF problems have been formulated and solved to handle ether AC grds coupled wth gas networks or AC/DC power systems. The relevant works on these formulatons are as follows: 1) Power/gas flow modelng for the power-natural gas ntegrated system The power flow and gas flow models for the power-gas coupled system have been addressed n [3 11]. The basc steady-state optmal power flow model for AC grds 13
Jale FAN et al. ntegrated wth natural gas nfrastructures was elaborated n [3, 7]. In [5], the temperature effect n natural gas constrants was consdered. To mprove the model accuracy for long tme-nterval applcatons, a multperod probablstc OPF model was desgned and the role of power-to-gas unts was studed n [4]. The transent gas flow model was combned wth the steady-state power flow model to study the operatonal characterstcs of the dynamc energy flow model of coordnated systems n [6]. ) Power flow modelng for AC/DC grds The unfed power flow model of the AC/DC grd ncorporatng multtermnal voltage source converter-hgh voltage drect current (VSC-HVDC) was proposed n [, 1, 13]. In [1], a quadratc loss model for converters n VSC statons was adopted, whle n [], a lnear loss model was utlzed. Snce the VSC-HVDC technologes are advantageous for wnd power ntegraton, a herarchcal securty-constraned OPF was proposed n [14] and the Benders decomposton (BD) was used to handle the ntermttent property of wnd power generaton. In [15], a multperod OPF for AC/DC grds wth wnd power ntegraton was presented and a scenaro-based (SB) approach was adopted to handle the uncertanty of wnd energy. 3) Soluton method for OPF problem The power flow equatons are nonlnear and nonconvex, whch pose challenges to the tradtonal determnstc OPF usng the nteror-pont method. Recently, the convexty of the OPF problem has pqued great research nterest [16 0]. The prmary dea s to transform the ntal nonlnear/nonconvex problem nto a convex one by usng convex relaxatons, such as the semdefnte relaxaton and second-order cone relaxaton. In [18], the OPF problem for AC/DC grds was converted to a second-order cone programmng (SOCP). However, the suffcent condtons that guarantee the exactness of ths convex relaxaton are not dscussed. Ths was solved by usng the semdefnte programmng (SDP) approach [19]. Although the SDP formulaton acheves a more accurate soluton than the SOCP, ts computaton effcency s too low to render the practcal applcaton capablty. In ths sense, the SOCP formulaton of the OPF appears to be promsng. Table A1 n Appendx A shows a statstcal study of dfferent OPF formulatons n the electrcty-gas couplng system, where Y and N denote whether the subject s beng consdered. Most related works focus on the nterdependency between AC grds and natural gas systems or the nterconnecton between the AC and DC power systems; none have studed the OPF problem for AC/DC grds ntegrated wth gas systems. Snce practcal modern power systems are coupled wth both DC grds and natural gas systems, neglectng any of them may yeld suboptmal solutons. More mportantly, accordng to a report [1], the gas supply of gas-fred generators could suffer shortages durng wnter peak hours n some Amercan regons. Ths s because the gas supply of FUs does not prortze the natural gas loads and once the FU generaton s curtaled, the thermal generators need to ncrease ther actve power generaton to compensate the decrease n the FU generaton. Thus, the gas-supply uncertanty plays an mportant role n the schedule of thermal generators. To solve the aforementoned problems, ths paper presents a unfed OPF model (master-sub-opf, denoted as MS-OPF n ths paper) for the AC/DC grd coordnated wth gas systems, consderng gas-supply uncertantes. Specfcally, a two-stage BD-based optmzaton problem s desgned to handle data uncertantes. In the frst stage, the nonlnear OPF model for AC/DC grds s reformulated as a master problem and solved by a large-scale nonlnear programmng solver,.e., the nteror-pont optmzer (IPOPT). Nevertheless, the SOCP formulaton for a natural gas system s consdered a subproblem and s solved by a convex optmzaton solver,.e., the urob. It s noteworthy that each subproblem represents a gas-supply scenaro; as a result, the effects of gas-supply uncertanty on FUs can be evaluated. In addton, owng to the SOCP formulaton of the subproblems, the orgnal nonconvex and nonlnear formulaton s transformed to a convex optmzaton problem and can be solved n polynomal tme. The remander of ths paper s organzed as follows. In Secton, the detaled nonlnear OPF model for the AC/DC grd s presented and an SOCP formulaton of the natural gas system constrants s proposed. The soluton method for the MS-OPF s elaborated n Secton 3. In Secton 4, the detals of the smulaton results performed on a modfed AC/DC grd-natural gas coupled system s presented. Fnally, Secton 5 draws the relevant conclusons. Problem formulaton In ths secton, the steady-state power (gas) flow models of AC/DC grds and the natural gas systems are ntroduced to determne the real-tme schedule of both thermal power plants and gas-fred generators. Meanwhle, the gas loads and power demands are satsfed durng the schedulng horzon. The followng assumptons are made n our proposed formulaton: 1) The steady-state model s adopted n both power systems and natural gas systems. 13
Unfed optmal power flow model for AC/DC grds ntegrated wth natural gas systems... ) To convert nonlnear gas system constrants nto SOCP forms, the natural gas systems are consdered to be lossless,.e., compressors are not consdered. Ths s prmarly because the gas consumpton of compressors only accounts for a very small fracton of the total gas loads. 3) The power demands are unchangeable durng the dspatch nterval, whch means only the uncertanty of natural gas systems was consdered. It s noteworthy that uncertantes exst n both power systems and natural gas systems. However, f the power system and gas system uncertantes are consdered smultaneously, the senstvty of the optmal soluton to gas system uncertantes may not be evaluated..1 OPF model of AC/DC grds.1.1 Objectve functon The OPF formulaton for AC/DC grds ams at fndng an optmzed operatonal pont by mnmzng the followng total generaton costs:! mn XN t ¼1 f P þ XNg ¼1 f R P R ð1þ where N t and N g are the number of thermal generators and gas-fred generators,respectvely; f s a quadratc functon that represents the fossl-fuel cost functon of thermal power plants; f R s a lnear functon that represents the natural-gas forced cost functon of FUs; P s the actve power generaton of thermal generator ; P R s the actve power generaton of gas-fred generator. It s noteworthy that the gas-fred generators are suppled wth low-prced natural gas. Q Q D Q S ¼ XN AC ð3þ B j e e j þ f f j þ j e f j e j f j¼1 P ;mn P P ;max ð4þ Q ;mn Q Q ;max ð5þ P R ;mn PR P R ;max ð6þ V ;mn e þ f V ;max ð7þ where P D and Q D are the actve/reactve power loads at bus ; P S and Q S are the actve/reactve power transformed from AC grd to the th VSC staton; Q s the reactve power generaton of thermal generator ; N AC s the number of AC buses; j and B j are the real/magnary part of the AC grd admttance matrx; e and f s the real/magnary part of the AC voltage. If no generator s connected to bus, the values of P and Q are 0 n () and (3). Rectangular coordnates are chosen because the Hessan matrx of the equalty or nequalty constrants wll be constant, whch s convenent for the nteror-pont-method-based solver. To yeld a more accurate power flow formulaton, a precse steady-state VSC model s adopted. A VSC staton ncludes a couplng transformer, an AC flter, a phase reactor, and a converter, as shown n Fg. 1 [1]. The converters are based on nsulated gate bpolar transstor values that are controlled wth pulse-wdth modulatons. For smplcty, the VSC statons are assumed to have unfed parameters of the transformer, the AC flter, and the phase reactor. Fnally, a Y D transformaton s used to obtan the power flow formulatons [1]. Thus, we can obtan a modfed VSC staton model as shown n Fg...1. Power flow models of AC/DC grds Two couplngs are present n the AC/DC grds: couplng between the AC grd and voltage source converter (VSC) statons, and couplng between the DC grd and VSC statons. Hence, the AC/DC grd power flow models can be dvded nto three nterconnected parts: AC grd constrants, VSC staton constrants, and DC grd constrants. The AC grd constrants can be expressed n the followng rectangular form: P þ P R P D P S ¼ XN AC ðþ j e e j þ f f j Bj e f j e j f j¼1 Bus s AC grd Z tf B f AC flter Bus f Fg. 1 VSC staton model S P+ e s + jf s jq S R pc Z pc X pc Phase reactor Y 1c Y c VSC staton e c + jf c Y 3c P Bus c Converter Converter P DC U DC Bus dc DC grd Fg. Equvalent crcut of VSC staton 13
Jale FAN et al. The buses that connect the AC grd and VSC statons are called the ponts of common connectons (PCCs). Usng the notatons n Fg., an AC grd s assumed to nject actve and reactve powers, P S and Q S, respectvely, nto the VSC staton. The converter can ether absorb or nject actve (reactve) power to the AC grd [18]. Moreover, the VSC can control P S and Q S by adjustng the modulaton factor m a and the converter output voltage U DC. The power flow models of the VSC statons can be wrtten nto the followng constrants: P S ¼ e s þ f s ð 1c þ c Þ ð8þ ½ 1c ðe s e c þ f s f c Þ B 1c ðe s f c e c f s ÞŠ Q S ¼ e s þ f s ð B1c þ B c Þ ð9þ þ ½B 1c ðe s e c þ f s f c Þþ 1c ðe s f c e c f s ÞŠ P C ¼ e c þ f s ð 1c þ 3c Þ ð10þ þ ½ 1c ðe s e c þ f s f c ÞþB 1c ðe s f c e c f s ÞŠ where Y 1c, Y c, and Y 3c are the sets accordng to the Y D transformaton of the components n the VSC staton. Ther calculatons can be found n Appendx B. In addton, the operatng constrants are enforced as: e c þ f s m au DC ð11þ m a;mn m a m a;max ð1þ P S ;mn PS PS ;max ð13þ Q S ;mn QS QS ;max ð14þ P S þ Q S S ;conv ð15þ where m a s the modulaton factor; U DC s the DC voltage; S ;conv s the maxmum capacty of VSC staton. In general, converters have actve power losses. Snce the VSC statons are nstalled wth reactve power compensators, a quadratc functon of the AC current magntude to represent the actve power losses of converters s proposed n []. Heren, a smplfed power loss model shown n [] s adopted, yeldng P C bp C ¼ P DC ð16þ where b s a constant represents the coeffcent of actve power loss n VSC staton. To elucdate the couplng between the VSC staton and the DC grd, we consder the lnear equvalent crcut of a DC grd shown n Fg. 3. Usng the notatons adopted n the fgure, the calculaton of real power P DC P DC U DC X N ¼ U DC DC j¼1 ¼ U DC j P loss j ¼ PDC j Uj DC and DC voltage U DC U DC j R DC j PDC j Uj DC! R DC j R DC j can be expressed as: ð17þ ð18þ ð19þ It s noteworthy that the real power P DC of all DC buses are njectons; thus, the power balance of DC grds can be expressed as: X N DC P DC ¼1 X P loss ð;jþl DC j ¼ 0 ð0þ where N DC s the number of DC buses; L DC s the set of DC lnes.. Natural gas system formulatons consderng gas-supply uncertanty..1 SOCP formulaton of natural gas systems A general natural gas system conssts of gas producers, storage facltes, compressor statons, ppelnes, and customers [7, 3]. In the steady state, ppelnes and compressor statons can be modeled by branches, and the nterconnecton ponts can be represented by nodes n a network graph [3]. Furthermore, the steady-state gas flow through a ppelne s assumed to be the functons of the upstream and downstream pressures. as typcally flows from the upstream node to the downstream node. For example, f the gas flows from node to node j, we obtan: fj k ¼ C j p k p k j otherwse, fj k ¼ C j p k j p k f k j [ 0 f k j \0 ð1þ ðþ P DC U DC P loss j R DC j Fg. 3 Equvalent crcut of DC grd 13 U DC j P DC j where k s the ndex of scenaros; fj k s the gas flow n ppelne ð; jþ; p k s the gas pressure at node ; C j s a constant determned by the length, dameter, absolute rugosty, and the gas composton of the ppelne [3]. Equatons (1) and () are exactly n the form of a rotated quadratc cone by convex relaxatons. For example, usng convex relaxaton, (1) can be rewrtten as:
Unfed optmal power flow model for AC/DC grds ntegrated wth natural gas systems... fj k þ Cj p k j Cj p k ð3þ Subsequently, we can obtan the SOCP formulaton of gas flow models as follows. If the gas flows from node to node j, we have: fj k C j p k C j p k ð4þ j otherwse, fj k C C j p k j p k j ð5þ where kk denotes the two norms. The gas-system nodal balance can be expressed as: X N s Ag XN l BL k X fj k ¼ 0 ð6þ ¼1 ¼1 jðþ where N s s the number of gas supplers; N l s the number of gas loads; A s the node-gas suppler ncdence matrx; B s the node-gas load ncdence matrx; g s the gas generaton of gas suppler ; L k s the gas load at node. Partcularly, the gas loads at the nodes connected wth gas-fred generators are stated as: L k ¼ q g P R;k ð7þ where q g s a constant represents the fuel functon of gasfred generator; P R;k s the actve power generated by the th gas-fred generator n the kth scenaro. Natural gas systems are also constraned by the operatng lmts: f j;mn fj k f j;max p ;mn p k p ;max g ;mn g g ;max.. Modelng of gas-supply uncertanty of FUs ð8þ ð9þ ð30þ Accordng to a report [1], the gas supply of FUs could not be satsfed especally n areas where the gas demand occupes the major proporton of the total gas demand. In [11], the varablty of gas ppelne capactes was adopted to represent the gas-supply uncertanty of FUs. However, n practce, the gas ppelne capactes are typcally fxed. Heren, the gas-supply uncertantes are represented by the day-ahead gas-load forecast errors. The latter s modeled by a normal dstrbuton functon, e L Nð0; r L Þ, where the standard devaton r L s 5% for the gas-load forecast value. Subsequently, a set of possble scenaros representng gasload varabltes can be generated by Monte Carlo smulatons. Followng that, the gas demands of the resdental, commercal, and ndustral customers (RCI) can be expressed as: L k ¼ L 0 þ e k L; ð31þ where L 0 s the day-ahead gas demand forecast; e k L; represents the forecast errors. 3 Soluton methodology The MS-OPF model proposed heren ams at fndng an optmzed schedule for thermal power plants and gas-fred generators under gas-load uncertantes. It s noteworthy that dfferent scenaros of the natural gas system may affect the fnal soluton of the MS-OPF. In ths regard, the BD [4, 5] s adopted owng to ts capablty n handlng uncertantes. Usng the BD method, the MS-OPF s decomposed nto a master problem,.e., a nonlnear programmng AC/DC OPF, and S subproblems, where each problem corresponds to a natural gas network load scenaro. Frst, the natural gas system operators would determne the schedule of gas sources and storages accordng to the gas-load forecast. However, because of the low cost and the envronmental frendly characterstcs of gas-fred generators, the full use of gas turbnes s preferred. In other words, gas-fred generators are run at the maxmum power, yeldng the gas consumpton of gas-fred generators shown n (7). Nevertheless, the detaled formulatons of the master problem and subproblems are shown as follows. 1) Master problem (NLP) mn XN t ¼1 f P s:t: ðþ ð0þ ) Benders cut x k þ XN g k k ¼1 þ XNg ¼1 P R P R;k f R P R 0 3) The k th subproblem (SOCP) mn x k ¼ XN g ¼1 s:t: ð4þ ð31þ u R ;þ þ ur ; ð3þ ð33þ ð34þ ð35þ ð36þ P R;k ¼ P R ; þ ur ;þ ur ; : kk ð37þ where P R ; s the soluton calculated from the master problem; u R ;þ and ur ; are the auxlary varables; k s the Lagrangan coeffcent of (37). 13
Jale FAN et al. The detaled procedures of adoptng the BD method to solve the proposed formulaton can be summarzed n the followng steps. Step 1: Solve the master problem,.e., the real-tme OPF problem for AC/DC grds. For a large-scale nonlnear programmng (NLP) problem, ths paper adopted the IPOPT [6] to obtan the soluton of P R ;. Step : Check the feasblty of subproblem 1, whch s based on gas-load scenaro 1, and substtute P R ; nto the subproblem (as referred to n (37)). Snce the gas flow constrants have been transformed nto SOCP formulatons, we use the urob [7] to calculate ths convex optmzaton problem and obtan the optmzed soluton of x 1. Step 3: If x 1 ¼ 0, go back to Step and check the feasblty of the nd subproblem untl the termnaton crtera s valdated for all subproblems. Otherwse, add the Benders cut (34) constrant to the master problem and go to Step 1; subsequently, recalculate the soluton of P R ; and check the feasblty of the next subproblem. Step 4: Once each of the total S subproblems s feasble, output the optmzed schedule of the thermal generators and FUs. It s noteworthy that the coordnated system s dspatched accordng to dfferent gas demand scenaros. The maxmum order of constrants n the master problem s two, and they are calculated n parallel. If the bus number of the AC grds s n, the tme complexty of the NLP model s Oðn Þ. Nevertheless, all the constrants n the subproblem are lnear and can be calculated n parallel. If the gas node number s m, the tme complexty of the SOCP model s OðmÞ. In summary, the tme complexty of the soluton algorthm s Oðn Þ. Although the master problem can be smplfed to be lnear as well, the guarantee of computatonal accuracy for practcal applcatons may be dffcult. 4 Case studes 4.1 Test systems The proposed MS-OPF was tested on a benchmark AC/ DC grd coupled wth a natural gas system. The overall system s shown n Fg. 4, whch conssts of a modfed verson of the IEEE 118-bus test system and the Belgum 0-node gas network. As shown n Fg. 4, some AC lnes n the IEEE 118-bus system are substtuted for the multtermnal HVDC systems. There are 54 generators n total. To nterconnect the two systems, thermal generators 10, 4, 5, 7, and 87 of the IEEE 118-bus system are replaced by gas-fred generators, and these generators are connected to nodes 16, 9, 6, 4, and 1 of the natural gas system, respectvely. The total actve power loads decreased to 000 MW/h; the capacty of each gas-fred generator s 100 MW, whle the remanng 49 thermal power generators maxmum actve power outputs are modfed to 50 MW. In ths condton, the FUs account for approxmately 1/6 of the total power generaton capacty. The remanng detaled nformaton of the IEEE 118-bus test system, ncludng the power cost functons and the varables boundares can be obtaned from the MAT- POWER lbrary [8]. As for the DC part of the modfed system, sx DC buses are avalable, namely DC1,..., DC6, whch are connected to AC buses 33, 39, 40, 6, 65, and 81 through the VSC statons, respectvely. The nomnal value of the DC voltage s assumed to be 30 kv and the resstances of all HVDC lnes are 0.06 pu. The parameter settngs of the VSC staton are shown n Table 1. The slack bus of the DC network s DC3 and the reference DC voltage U3 DC ¼ 0:98; the reference bus of the AC grds s bus 69. Although the converters n the AC/DC grd can have varous control modes, ths paper only consders the boundary condton on the converter-controllng varables. Nevertheless, for smplcty, the parallel ppelnes of the Belgan 0-node natural gas system are modeled as a sngle equvalent ppelne. The total forecasted RCI gas loads are changed to 50 mm 3 =h. The gas fuel capactes of gas supplers at nodes 1,, 5, 8, 13, and 14 are 18, 16, 9, 7, 7.5, and 7.5 mm 3 =h, respectvely. The gas fuel cost of gas storages at nodes, 5, 13, and 14 s 10 $/h and that of gas sources at nodes 1 and 8 s 50 $/h. q g shown n (7) s 0.05 n the smulaton. The nodal pressure bounds and the ppelne maxmum capactes can be found n [3]. All tests are carred out on a PC (Intel 5-410 Quad Core CPU,.90 Hz, 4-B RAM). The problems are conducted on the Yalmp [9] platform bult n MATLAB. 4. Smulaton results and analyss Based on the test system nformaton shown n Secton 4.1, Case 1 s desgned to demonstrate the computatonal performance of the SOCP formulaton for gas systems. The effect of gas-load uncertantes on the coordnated systems operaton s studed by comparng Case wth Case 3. Case 1: Optmzng gas flow (OF) of the Belgan natural gas system to determne the most economcal schedule of gas supplers. The SOCP formulaton of the OF proposed heren s compared wth the NLP formulaton of the OF model adopted n most publshed papers [3, 8]. The NLP formulaton s solved by the IPOPT and the SOCP formulaton s solved by the urob. The smulaton results are shown n Table and Table 3. 13
Unfed optmal power flow model for AC/DC grds ntegrated wth natural gas systems... D3 40 41 4 53 54 56 55 59 1 D 5 3 117 D1 58 4 11 33 39 63 44 60 14 37 5 43 34 48 57 64 6 7 13 15 45 51 16 67 46 50 61 19 35 36 47 49 6 113 17 66 31 38 69 116 D4 30 0 8 73 3 68 1 71 65 7 D5 9 4 70 D6 9 79 114 115 3 6 78 81 10 8 74 99 7 5 75 118 76 80 98 77 97 106 8 W S 96 1 3 4 7 6 5 S 83 95 94 84 93 100 14 104 105 107 85 88 89 9 16 15 W 108 10 9 8 13 S 86 103 90 91 10 101 1 109 S 11 17 18 19 0 87 110 11 111 enerator; as-fred generator; VSC-staton; DC lne; Power load; W as source; S as storage; as load Fg. 4 AC/DC grd coordnated wth natural gas systems Table 1 Parameter settngs of VSC staton VSC parameters Value Table Smulaton results of Case 1 Formulaton Tme (s) Cost ($=h) R Tf 0.0005 X Tf 0.015 B f 0. R pc 0.0005 X pc 0.04 m a 1.15 U DC [ 0.9,1.1 ] P S [ - 0.5,0.5 ] Q S [ - 0.1,0.1 ] These results ndcate that the computaton tme of the SOCP formulaton s much shorter than that of the NLP models. In addton, we found that to obtan the lowest gas fuel cost, the gas storages of nodes, 5, 13, and 14 have to account for more gas supply. The msmatch n the gas supplers generaton schedule n the two formulatons s prmarly because the SOCP formulaton transforms the NLP 0.018 1:09 10 4 SOCP (proposed formulaton) 0.0094 1:09 10 4 ntal NLP problem nto a convex optmzaton problem, and the global optmal soluton can be found. By contrast, the non-convex NLP problem only obtans the local optmal soluton. Case : The test for a determnstc OPF of the AC/DC grd ntegrated wth the natural gas system,.e., optmzng the schedule of generators and gas sources smultaneously but the gas-load uncertantes are not consdered. Ths case was solved by the IPOPT solver. Case 3: The test for the two-stage MS-OPF consderng the gas-system uncertanty. The optmzed schedule of the gas supplers and FUs are shown n Table 4 and Table 5. 13
Jale FAN et al. Table 3 as supplers output n Case 1 Formulaton Output (mm 3 =h) Node 1 Node Node 5 Node 8 Node 13 Node 14 NLP 0.000 16.0000 9.0000 9.9998 7.5000 7.5000 SOPC (proposed formulaton).6818 16.0000 9.0000 7.318 7.5000 7.5000 Table 4 Optmzed generaton of gas supplers Node eneraton (mm 3 =h) Case Case 3 (proposed method) 1 10.343 15.0674 16.0000 16.0000 5 9.0000 9.0000 8 4.7657 19.936 13 7.5000 7.5000 14 7.5000 7.5000 Actve power generaton of thermal generators (MW) 60 50 40 30 0 10 Case Case 3 Table 5 Optmzed generaton of FUs (MW=h) Bus eneraton (MW/h) Case 10 100 78.6346 4 100 88.868 5 100 86.916 7 100 87.3553 87 100 86.4071 Case 3 (proposed method) The total system operaton cost s 6:4045 10 4 $=h and 6:6888 10 4 $=h for Case and Case 3, respectvely. Ths ndcates that when the gas-load uncertantes are consdered, the proposed MS-OPF offers a hgher system operaton cost than by optmzng the coordnated system smultaneously. Ths s because the gas-load varablty may cause the supply shortage of gas-fred generators, and thermal generators are forced to provde more power to meet the electrcty demands. As shown n Table 5 and Case, f the gas supply for FUs s suffcent; as a result, gas-fred generators would run at ther maxmum capactes as ther operaton costs are much lower than those of thermal generators. In Case 3, the gas supply for the FUs s constraned owng to the uncertanty of gas demands, resultng n the reduced generaton of FUs. It s noteworthy that once the gas supplers schedule s determned, t s fxed for hours snce the gas transmsson s very slow. Hence, when gas loads are ncreased n a short tme, such as durng wnter, the FUs wll experence 0 1 supply shortages, yeldng power system securty problems. Furthermore, as observed from Fg. 5, the thermal generators n Case 3 ncrease ther generaton to provde addtonal power owng to the decrease n FU generaton compared to Case. These results ndcate that f the gasload uncertanty s not consdered when the generator schedules are developed, once the FU gas supply s not suffcent, the thermal generators would not satsfy the current power demand and the power system securty s threatened. Thus, the proposed MS-OPF ensures the balance between the supply and demand of the ntegrated system and reduces the varablty mpact n natural gas systems on the safety of power systems. In addton, Fg. 6 shows that the gas flow n ppelnes s redstrbuted when the gas-load uncertanty s consdered. It s noteworthy that the addtonal flexblty provded by VSC statons and DC connectons does not vary consstently n the optmal soluton of the two cases. Ths s prmarly due to the acton of the storage devces ncluded n the gas network, whch effectvely decouples the gas operaton from the power grd. Clearly, the hgher the control capablty of the electrcal system, the lower the topologcal changes. 4.3 Senstvty analyss 4 6 8 AC bus number Fg. 5 Actve power generaton of thermal generators 1 15 The senstvty of the thermal generators actve power generaton wth respect to the proporton of the FUs 13
Unfed optmal power flow model for AC/DC grds ntegrated wth natural gas systems... sd flow (mm 3 /h) 35 Case 30 Case 3 5 0 15 10 5 0-5 -10 0 Fg. 6 as flow n ppelnes 0.16 0.14 0.1 0.10 4 5% 10% 15% 6 8 10 1 14 16 18 0 as ppelne number of the gas system uncertanty. The amount of the thermal generators generaton adjustment s found to be proportonal to the gas system uncertanty level. Thus, the gasload uncertantes are to be consdered n the presence FUs. 4.4 Evaluatng robustness of proposed method In ths secton, to evaluate the robustness of the soluton obtaned by the BD technque n Case 3, we utlze the scenaro-based approach [15, 30] to solve the OPF problem of the test system consderng gas-load uncertanty. The probablty dstrbuton functon of gas load s assumed to be a normal dstrbuton and the mean value of the RCI gas loads s 50 mm 3 =h, whle ts standard devaton s.5 mm 3 =h. We generate nne scenaros for comparson, and the results are shown n Table 6. Table 6 shows that the expected value of the total system operaton cost calculated by the SB approach s 6470 $/h, whch s lower than 66888 $/h obtaned by the BD method. Comparng wth the SB method gves varable optmzed solutons; the BD approach adopted n ths paper gves an approprate schedule plan for power system operators. ΔP 0.08 5 Concluson 0.06 0.04 0.0 10 Fg. 7 Senstvty analyss 0 30 40 50 Proporton (%) generaton n total actve power generaton, and the gasload uncertanty s dscussed n ths secton. The senstvty ndex s defned as: DP ¼ P P P ð38þ where DP s the senstvty ndex for the thermal generators; P s the total power generaton of thermal generators consderng the gas-load uncertanty; P s the total power generated by the thermal generators wthout consderng gas-load uncertanty. The smulaton results are shown n Fg. 7, where the X- axs represents the proporton of the FUs generaton n the total power generaton. The dfferent-colored lnes represent the dfferent standard devatons of the gas-load forecast errors (.e., the percent of gas-load forecast value). The greater the standard devaton, the hgher s the degree Ths paper presents a unfed OPF model of AC/DC grds ntegrated wth natural gas systems for real-tme schedulng of power systems. The formulaton s desgned as a two-stage stochastc optmzaton problem to study the effect of gas-supply uncertantes on the fnal optmzaton solutons. The salent features of the proposed approach are summarzed as follows: 1) The SOCP formulaton of the OF problem proposed heren can effectvely mprove the computatonal effcency for solvng the subproblem. Table 6 as load scenaros Scenaro as load Probablty System cost (mm 3 =h) ($/h) 1 49.61 0.1577 63948 50.887 0.1498 66314 3 5.717 0.0884 67759 4 47.77 0.1055 63477 5 45.731 0.0371 6978 6 50.555 0.1557 64478 7 49.778 0.1590 63990 8 48.170 0.11 63588 9 54.84 0.048 67846 13
) Although the total fuel cost poses a slght ncrease when the gas-load uncertantes are consdered, thermal generators are able to ncrease ther generaton to mtgate the varablty of gas demands. Thus, the proposed MS-OPF provdes a tradeoff between economy and safety. 3) The thermal generators adjustment of power generaton shows a postve correlaton wth the proporton of FUs n the total power generaton and the uncertanty degree of the gas supply. 8 >< >: 8 >< >: Y c ¼ 1 Z c Z c ¼ Z Tf j B f jz Tf Z pc B f Y 3c ¼ 1 Z 3c Z 3c ¼ Z pc j B f jz pc Z Tf B f Jale FAN et al. ðbþ ðb3þ The technologcal progress n power-to-gas (Pt) has made the power and natural gas systems more closely lnked, whle the ntegraton of renewable energy resources causes securty problems to the power system. For future work, the authors plan to study the statc stablty of powergas bdrectonal coupled systems wth renewable resource penetraton. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense (http:// creatvecommons.org/lcenses/by/4.0/), whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made. Appendx A Taxonomy of OPF model for coordnated systems s shown n Table A1. Table A1 Taxonomy of OPF model for coordnated systems Reference OPF DC connecton Uncertanty modelng [3] Y N N [9] Y N Y [4] Y N Y [11] N N Y Ths paper Y Y Y Appendx B 8 < Y 1c ¼ 1 Z : 1c Z 1c ¼ Z Tf þ Z pc þ jb f Z Tf Z pc : ðb1þ References [1] EIPC (015) Evaluate the capablty of the natural gas systems to satsfy the needs of the electrc systems. as-electrc system nterface Target Report, Department of Energy, Natonal Energy Technology laboratory [] Mohamadreza B, Mehrdad (013) A mult-opton unfed power flow approach for hybrd AC/DC grds ncorporatng mult-termnal VSC-HVDC. IEEE Trans Power Syst 8(3):376 383 [3] Clodomro U, Lma JWM, Souza ACZD (007) Modelng the ntegrated natural gas and electrcty optmal power flow. In: Proceedngs of 007 IEEE power and energy socety general meetng, Tampa, USA, 4 8 June 007, 7 pp [4] Sun, Chen S, We Z et al (017) Mult-perod ntegrated natural gas and electrc power system probablstc optmal power flow ncorporatng power-to-gas unts. J Mod Power Syst Clean Energy 5(3):41 43 [5] Alberto M, Claudo RF (01) A unfed gas and power flow analyss n natural gas and electrcty coupled networks. IEEE Trans Power Syst 7(4):156 166 [6] Fang J, Zeng Q, A X et al (017) Dynamc optmal energy flow n the ntegrated natural gas and electrcal power systems. IEEE Trans Sustan Energy 9(1):188 198 [7] Aghtae M, Abbaspour A, Fruzabad F et al (014) A decomposed soluton to multple-energy carrers optmal power flow. IEEE Trans Power Syst 9():707 715 [8] Cong L, Mohammad S, Yong F et al (009) Securty-constraned unt commtment wth natural gas transmsson constrants. IEEE Trans Power Syst 4(3):153 1536 [9] Ahmed A, Abdullah A, Zhang X et al (015) Coordnaton of nterdependent natural gas and electrcty nfrastructures for frmng the varablty of wnd energy n stochastc day-ahead schedulng. IEEE Trans Sustan Energy 6():606 615 [10] Zhang X, Mohammad S, Ahmed A et al (016) Hourly electrcty demand response n the stochastc day-ahead schedulng of coordnated electrcty and natural gas networks. IEEE Trans Power Syst 31(1):59 601 [11] Bnng Z, Antono JC, Ramteen S (017) Unt commtment under gas-supply uncertanty and gas-prce varablty. IEEE Trans Power Syst 3(3):394 405 [1] Jef B, Stjn C, Ronne B (01) eneralzed steady-state VSC MTDC for sequental AC/DC power flow algorthms. IEEE Trans Power Syst 7():81 89 [13] Cha R, Zhang B, Dou J et al (016) Unfed powe flow algorthm Based on the NR method for hybrd AC/DC grds ncorporatng VSCs. IEEE Trans Power Syst 31(6):4310 4318 [14] Meng K, Zhang W, L Y et al (017) Herarchcal SCOPF consderng wnd energy ntegraton through mult-termnal VSC-HVDC grds. IEEE Trans Power Syst 3(6):411 41 13
Unfed optmal power flow model for AC/DC grds ntegrated wth natural gas systems... [15] Abbas R, Alreza S (014) Stochastc multperod OPF model of power systems wth HVDC-connected ntermttent wnd power generaton. IEEE Trans Power Delv 9(1):336 344 [16] Javad L, Steven HL (01) Zero dualty gap n optmal power flow problem. IEEE Trans Power Syst 7(1):9 107 [17] Ramtn M, Somayeh S, Havad L (015) Convex relaxaton for optmal power flow problem: mesh networks. IEEE Trans Power Syst 30(1):199 11 [18] Mohamadreza B, Mohammad RH, Mehrdad (013) Secondorder cone programmng for optmal power flow n VSC-type AC DC grds. IEEE Trans Power Syst 8(4):48 490 [19] Shahab B, Francs T, Vncent WS et al (017) Semdefnte relaxaton of optmal power flow for AC DC grds. IEEE Trans Power Syst 3(1):89 304 [0] Burak K, Santanu SD, Xu AS (016) Inexactness of SDP relaxaton and vald nequaltes for optmal power flow. IEEE Trans Power Syst 31(1):64 651 [1] Zhang X (004) Multtermnal voltage-sourced converter-based HVDC models for power flow analyss. IEEE Trans Power Syst 19(4):1877 1884 [] Daelemans (008) VSC HVDC n meshed networks. Dssertaton, Katholeke Unverstet Leuven [3] Danel DW, Smeers Y (000) The gas transmsson problem solved by an extenson of the smplex algorthm. Manag Sc 46(46):1454 1465 [4] Olvera F, rossmann IE, Hamacher S (014) Acceleratng benders stochastc decomposton for the optmzaton under uncertanty of the petroleum product supply chan. Comput Oper Res 49(014):47 58 [5] Ragheb R, Teodor C, Mchel et al (017) The benders decomposton algorthm: a lterature revew. Eur J Oper Res 59(017):801 817 [6] Lorenz TB (009) Large-scale nonlnear programmng usng IPOPT: an ntegratn framework for enterprse-wde dynamc optmzaton. Comput Chem Eng 33(3):575 58 [7] urob optmzer 7.0. http://www.gurob.com. Accessed 5 Dec 017 [8] Ray DZ, Carlos E, Robert JT (011) MATPOWER: steady-state operatons, plannng, and analyss tools for power system research and educaton. IEEE Trans Power Syst 31(1):64 651 [9] Lofberg J (004) Yalmp: a toolbox for modelng and optmzaton n MATLAB. In: Proceedngs of IEEE nternatonal symposum on computer aded control systems desgn, New Orleans, USA, 4 September 004, 6 pp [30] Abbas R, Alreza S, Behnam M et al (014) Correctve voltage control scheme consderng demand response and stochastc wnd power. IEEE Trans Power Syst 9(6):965 973 Jale FAN receved her bachelor degree from Zhengzhou Unversty, Zhengzhou, Chna, n 015. She s currently pursung the Ph.D. degree at the School of Electrcal Engneerng, Southwest Jaotong Unversty, Chna. Her research nterests nclude optmal power flow, power system modelng and plannng. Xaoyang TON receved hs B.S, M.S and Ph.D. degrees from Southwest Jaotong Unversty (SWJTU), Chengdu, Schuan, Chna, n 1993, 1996 and 007 respectvely. He s an assocated professor of school of electrcal engneerng, at SWJTU. Hs research nterests nclude wde area protecton, power system fault dagnoss, optmzaton and plannng, smart substaton. Junbo ZHAO has been workng toward the Ph.D. degree n the Bradley Department of Electrcal and Computer Engneerng, Vrgna Polytechnc Insttute and State Unversty, Blacksburg, VA, USA, snce 015. Hs research nterests are n the theoretcal and algorthmc studes n power system state estmaton, power system operaton and cyber securty, robust statstcs, and sgnal processng. 13