Second-order cone AC optimal power flow: convex relaxations and feasible solutions

Transcription

1 J. Mod. Power Syst. Cean Energy (19) 7(): Second-order cone AC optima power fow: convex reaxations and feasibe soutions Zhao YUAN 1, Mohammad Reza HESAMZADEH 1 Abstract Optima power fow (OPF) is the fundamenta mathematica mode to optimize power system operations. Based on conic reaxation, Tayor series expansion and McCormick enveope, we propose three convex OPF modes to improve the performance of the second-order cone aternating current OPF (SOC-ACOPF) mode. The underying idea of the proposed SOC-ACOPF modes is to drop assumptions of the origina SOC-ACOPF mode by convex reaxation and approximation methods. A heuristic agorithm to recover feasibe ACOPF soution from the reaxed soution of the proposed SOC-ACOPF modes is deveoped. The proposed SOC-ACOPF modes are examined through IEEE case studies under various oad scenarios and power network congestions. The quaity of soutions from the proposed SOC-ACOPF modes is evauated using (oca optimaity) and LINDO- GLOBAL (goba optimaity). We aso compare numericay the proposed SOC-ACOPF modes with other two convex ACOPF modes in the iterature. The numerica resuts show robust performance of the proposed SOC- ACOPF modes and the feasibe soution recovery agorithm. CrossCheck date: 19 Juy 18 Received: 19 March 18 / Accepted: 19 Juy 18 / Pubished onine: 1 October 18 Ó The Author(s) 18 & Zhao YUAN yuanzhao@kth.se Mohammad Reza HESAMZADEH mrhesamzadeh@kth.se 1 Department of Eectric Power and Energy Systems, KTH Roya Institute of Technoogy, Teknikringen 33, 1148 Stockhom, Sweden Keywords Optima power fow, Conic reaxation, McCormick enveope, Tayor series expansion, Feasibe soution 1 Introduction Optima power fow (OPF) is an indispensabe too in fairy wide areas of power system operations and the appications are sti expanding [1 4]. The chaenges of integrating arge amount of renewabe energy, increasing muti-termina high votage direct current (HVDC) connections and growing number of prosumers in distribution grid are now pushing the eectricity industry to seek more accurate, reiabe and efficient OPF toos. Since the aternating current (AC) power fow constraints are compex, noninear and nonconvex in nature, enormous research efforts have been put into deveoping efficient agorithms to sove OPF during the past decades. References [1] and [] summarize methods to sove OPF in the eary stages ranging from inear, noninear and quadratic programming to Newton-based agorithm and interior point method (IPM). Heuristic optimization agorithms based on evoutionary and inteigence approaches to sove OPF can be found in [] and [6]. Traditionay, the direct current OPF (DCOPF) as an estimation of fu aternating current OPF (ACOPF) is pervasivey empoyed for arge-scae power system cacuations [7]. With fast deveopment of smart grids [8], distribution network is now in the unprecedented interest of advanced monitoring and contro [9]. Distribution networks have arger resistance to reactance (R/X) ratio as compared to the transmission networks. Accordingy, the DCOPF resuts of distribution networks need to be carefuy examined. Besides, the operation points and noda prices obtained from soving ACOPF in transmission

2 Second-order cone AC optima power fow: convex reaxations and feasibe soutions 69 and distribution networks are more accurate [1, 11]. Accordingy, accurate and fast methods to sove ACOPF are in demand. Mathematica modeing techniques to convexify the nonconvex ACOPF mode by convex reaxations can give usefu bound of ACOPF objective function. Convexity aso guarantees finding goba soution by using mature soution agorithms (e.g. IPM). Considering abundant commerciay avaiabe sovers (such as MOSEK [1]) for soving convex optimization probems, the remaining task is to tighten the reaxations used in the convexification. Though assuming no ower bound for active power generation to exacty convexify mesh networks is not reaistic, the proposed branch fow mode in [13] as a convex reaxation of ACOPF is very important and usefu. Reference [14] gives exact convex reaxations of OPF under some assumptions on network parameters. Based on the branch fow approach, authors in [1] present an ACOPF mode using second-order cone programming (SOCP) and it shows accurate soutions for severa IEEE test cases when the objective is transmission oss minimization. The SOCP based convex ACOPF mode in [1] serves as the starting point of deriving our second-order cone ACOPF (SOC- ACOPF) modes in this paper. A cone-programming-based OPF for radia distribution networks is proposed in [16]. Reference [9] continues to improve its cone reaxation by generating tight cutting panes. For radia networks, sufficient conditions regarding network property and votage upper bound under which the proposed reaxed ACOPF can give goba ACOPF soution are derived in [14]. Recent appications of SOCP based convex ACOPF mode in distributiona ocationa margina pricing (DLMP), transmission-distribution coordination and decentraized power system operation can be found in [17 19]. Semi-definite programming (SDP) is another promising convexification approach for ACOPF [, 1]. The computationa imits of SDP are shown in []. Efficient agorithms for soving SDP-based ACOPF mode remain to be found []. Regarding soving ACOPF in arge-scae power networks, SDP-based ACOPF takes much more CPU time than SOCP-based ACOPF. Using matrix combination and decomposition techniques, authors in [3] acceerate SDP-based ACOPF and show that more than 1 seconds sover time are required to compute cases with around 3 buses. Important anaysis and resuts from [4 6] show that SDP reaxations are exact ony for imited types of probems. Even for a -bus 1-generator power system, SDP-based ACOPF can be infeasibe and inexact [7]. In cases where the exactness is not guaranteed, soutions of SDP-based ACOPF rarey have physica meanings. Resuts from [8] show that quadratic convex (QC) reaxation of ACOPF may produce some improvements in accuracy over SOCP-based ACOPF but with reduced computationa efficiency. The feasibe region reationship of SOCP, SDP and QC approaches are anayzed in [8]. The SDP and QC approaches give tighter reaxations than the SOCP approach but they are not equivaent to each other. In terms of computationa performance, the QC and SOCP approaches are much faster and reiabe than the SDP approach [8]. Based on first-order Tayor series expansion, a current votage (IV) formuation for ACOPF is proposed in [9, 3]. The advantages of the currentvotage formuation are that the approximated ACOPF probem is inear and much faster to be soved than noninear formuations. However, an iterative agorithm is required to check the vioations of nonconvex ACOPF constraints and then to construct the inner or outer bounds of the approximations. Authors in [31] propose three different reaxation methods to improve SOCP-based ACOPF. The arctangent constraints in the rectanguar formuation of ACOPF are convexified by McCormick reaxations, poyhedra enveopes and dynamicay generated inear inequaities [31]. The resuts show prominent computationa efficiency of the SOCP approach over the SDP approach [31]. Compared with the formuations in [31], the proposed SOC-ACOPF modes in this paper dea with the nonconvex ACOPF constraints directy and the set of decision variabes used in our modes are different. Regarding the feasibiity of the reaxed soutions, three types of sufficient conditions about power injections, votage magnitudes and phase anges to guarantee obtaining exact soutions are proposed in [6]. Authors in [31] strengthen the reaxations of SOCP-based ACOPF mode by dynamicay generating inear vaid inequaities to separate soutions of SOCP-based ACOPF from other reaxed constraints. However, it is not guaranteed that feasibe soution can be aways recovered by this approach. The compementarity conditions in the Karush Kuhn Tucker (KKT) system of DCOPF are used in [3] to recover feasibe soution of ACOPF. A sequentia agorithm to improve the tightness of some reaxed constraints of a SOCP-based ACOPF mode is proposed in [17]. The performance of this agorithm for arge power networks remains to be improved. The main contributions of the current paper are threefod: 1) Three SOC-ACOPF modes based on second-order cone reaxations, Tayor series expansion and McCormick enveops are proposed. ) A heuristic agorithm is proposed to recover feasibe soutions of ACOPF from the reaxed soutions obtained from the proposed SOC-ACOPF modes. 3) A computationa comparison with other SOCP-based ACOPF formuations in the iterature is conducted.

3 7 Zhao YUAN, Mohammad Reza HESAMZADEH Note we are caiming that the derived SOC-ACOPF modes are the origina contributions of this paper (not the SOCP method). The rest of this paper is organized as foows. Section presents mathematica formuations of the proposed SOC- ACOPF modes. Section 3 proposes the heuristic agorithm to recover feasibe soutions. Section 4 discusses numerica resuts for various IEEE test cases under different oad scenarios and power network congestions. A numerica comparison with two other SOCP based convex ACOPF modes in the iterature is conducted. Section concudes the improvements of the proposed SOC-ACOPF modes. SOC-ACOPF It is assumed that the eectric networks are three phase and baanced (this assumption is generay used by many OPF modes such as the DCOPF mode). Our proposed SOC-ACOPF modes use ess approximations or assumptions than the DCOPF mode. Considering the wide appication of DCOPF, the proposed SOC-ACOPF modes are more appicabe. The proposed SOC-ACOPF modes are derived using ine sending-end power injections and votage phase ange difference variabes. In this way, we can directy obtain votage phase ange soutions from the modes. Note that in some of the derived modes, instead of using votage magnitude variabes v n ; v s ; v r, votage magnitude square variabes V n,v n ; V s,v s ; V r,v r are incuded (votage magnitude can be recovered from the mode by taking the root of the votage magnitude square soutions). n N is the index of node set N, is the index of transmission ine set L. v n is the votage magnitude at node n. v s ; v r are the votage magnitudes at the sending end and receiving end of transmission ine. The convexity of the proposed SOC-ACOPF modes are further vaidated numericay by MOSEK sover which can ony sove convex programming modes in genera agebraic modeing system (GAMS). We want to emphasize that the numerica vaidation by MOSEK is a doube-check of the convexity of the proposed SOC-ACOPF modes. We have detaied anaytic expanations in foowing sections about why our derived mode are convex. In short, the fundamenta reason is because a the derived constraints are either inear or in the form of second-order cone which are definitey convex constraints..1 SOC-ACOPF: Mode P Mode P of the SOC-ACOPF is set out in (1) (1) [1]. Note p s ; q s represent receiving end power fows in [1] which are different in our formuation (sending end active and reactive power fows). So some constraints are accordingy different. The term s in p s ; q s ; v s ; V s is not an index but ony to impy the meaning of sending end of ine. The term r in v r ; V r is not an index but ony to impy the meaning of receiving end of ine. The term d in p dn ; q dn is not an index but ony to impy the meaning of power demand. p dn ; q dn are active and reactive power demand (parameters). Simiar reasoning hods for the term o in p o ; q o which is to denote the meaning of power oss. p o ; q o are active and reactive power oss. min X s:t: p n P dn q n Q dn f ðp n ; q n ; p o ; q o Þ ð1þ ¼ X ða þ n p s A n p o ÞþG n V n 8n N ðþ L ¼ X ða þ n q s A n q o Þ B n V n 8n N ð3þ L K p o p s þ q s R 8 L ð4þ V s p o X ¼ q o R 8 L ðþ V s V r ¼ R p s þ X q s R p o X q o 8 L ð6þ h ¼ X p s R q s 8 L ð7þ V min;n V n V max;n 8n N ð8þ p min;n p n p max;n 8n N ð9þ q min;n q n q max;n 8n N ð1þ where X ¼fp n ; q n ; p s ; q s ; p o ; q o ; V n ; h gr is the set of decision variabes. Equations () and (3) represent the active and reactive power baance. A þ n and A n are eements of incidence matrix of the network with A þ n ¼ 1; A n ¼ ifnis the sending end of ine and A þ n ¼ 1; A n ¼ 1ifn is the receiving end of ine. Since we aow the variabes p s ; q s to take both positive and negative vaues, the defaut power fow direction of each ine does not affect the fina resuts (negative vaues of these variabes mean the actua power fow direction is in the reverse direction of the defaut power fow direction). R is the resistant of ine. X is the reactance of ine. V min;n ; V max;n are the ower and upper bounds of V n. p min;n ; p max;n are the ower and upper bounds of p n. q min;n ; q max;n are ower and upper bounds of q n. K is the upper bound of p o. Mode P is convex because: À constraints (), (3) and () (1) are inear; ` constraint (4) is in the form of second-order cone. Constraints (4) and () represent active power and reactive power oss. The eft side of (4) bounds p o (which equivaenty bounds capacity of ine ). h,h s h r is defined as the votage phase ange difference of ine. h s ; h r

4 Second-order cone AC optima power fow: convex reaxations and feasibe soutions 71 are the votage phase anges at the sending end and receiving end of transmission ine. Equation (7) is approximated from the nonconvex constraint (11) as beow: v s v r sin h ¼ X p s R q s 8 L ð11þ This approximation is based on foowing assumptions: À assumption 1, votage magnitude product v s v r is approximatey equa to 1 per-unit in (11); ` assumption, votage phase ange difference across each ine is sma enough such that sin h h. It is worth to mention that assumption 1 is ony used to inearize constraint (11). Votage soutions from Mode P are aowed to take other vaues which do not satisfy assumption 1. In this paper, we refer Mode P summarized in (1) (1) as the origina SOC-ACOPF mode. We wi graduay improve Mode P by dropping assumptions 1 and in foowing sections. Constraint (7) in Mode P is repaced by different formuations in the new SOC-ACOPF modes. Assumptions 1 and are vaid for both transmission and distribution network in norma operations. However, there can be abnorma situations when these assumptions do not hod. This is one of the reasons that we want to drop assumptions 1 and, and propose new SOC- ACOPF modes in order to improve the soution accuracy.. SOC-ACOPF: Mode R To drop assumption 1 in Mode P, we use the foowing biinear transformation: v s v r ¼ 1 4 ½ðv s þ v r Þ ðv s v r Þ Š 8 L ð1þ If we introduce auxiiary variabe v m as (note the term m in the subscript of v m is not an index): v m ¼ v s v r 8 L ð13þ And repeat transformation (1) then the eft side of (11) can be repaced by: v m h ¼ 1 4 ½ðv m þ h Þ ðv m h Þ Š 8 L ð14þ Introducing new variabes u x ; w x ; u v and w v as (note the terms x and v in the subscripts are not indexes but are ony to distinguish different variabes): u x ¼ v s þ v r 8 L ð1þ w x ¼ v s v r 8 L ð16þ u v ¼ v m þ h 8 L ð17þ w v ¼ v m h 8 L ð18þ Equations (1) and (14) can be expressed by the new variabes u x ; w x ; u v and w v as: v s v r ¼ 1 4 ðu x w x Þ 8 L ð19þ v m h ¼ 1 4 ðu v w v Þ 8 L ðþ The quadratic functions u x ; w x ; u v and w v are reaxed as foowing second-order cones: u xa u x 8 L ð1þ w xa w x 8 L ðþ u va u v 8 L ð3þ w va w v 8 L ð4þ where u xa ; w xa ; u va ; w va are auxiiary approximation variabes. Note the terms xa and va in the subscripts of the corresponding variabes are not indexes but are ony to distinguish different approximation variabes. The upper bounds are expressed ineary: u xa ðu x þ u x Þu x u x u x 8 L ðþ w xa ðw x þ w x Þw x w x w x 8 L ð6þ u va ðu v þ u v Þu v u v u v 8 L ð7þ w va ðw v þ w v Þw v w v w v 8 L ð8þ where constraints (1) (4) are second-order cones and constraints () (8) are McCormick enveopes. u x ; w x, u v and w v are upper bounds of the corresponding variabes. u x ; w x ; u v and w v are ower bounds of the corresponding variabes. The variabes v s and v r are inked to their squares V s and V r by foowing convex constraints: V n v n 8n N ð9þ V n ðv n þ v n Þv n v n v n 8n N ð3þ where v n and v n are upper and ower bounds of votage magnitude. Constraint (3) tightens the cone reaxations in (9). In Mode R of SOC-ACOPF, we repace constraint (7) of optimization probem (1) (1) by constraints (1) (18), (1) (3) and constraints (19), () where the quadratic functions u x ; w x ; u v and w v are repaced by u xa ; w xa ; u va and w va, the term v s v r is repaced by v m, the term v m h is repaced by X p s R q s. Mode R is summarized as foowing: min f ðp n ; q n ; p o ; q o Þ X s.t. () (6), (8) (1), (1) (18), (1) (3) ð31þ v m ¼ 1 4 ðu xa w xa Þ 8 L ð3þ

5 7 Zhao YUAN, Mohammad Reza HESAMZADEH X p s R q s ¼ 1 4 ðu av w av Þ 8 L ð33þ Mode R is convex because: À constraints (1) (18), (1) (8), (3) (33) are inear; ` constraint (9) is in the form of second-order cone. For the convexity of other constraints, pease refer to the expanations of Mode P in Section.1..3 SOC-ACOPF: Mode T To drop assumption of Mode R, we propose to appy Tayor series expansion to approximate sine function: sin h ¼ h h3 6 þ h 1 þ Oðh7 Þ 8 L ð34þ As a trade-off between mode compexity and accuracy, fifth-order Tayor series expansion is seected. The approximation error is ess than.4% for jh j\p=. Repeating the biinear transformation procedure simiar in Section. of this paper, we have: 3 ¼ h 8 L ð3þ ¼ h 3 8 L ð36þ ¼ h 8 L ð37þ h 3 ¼ 1 4 ½ðh þ Þ ðh Þ Š 8 L ð38þ h ¼ 1 4 ½ð þ u h 3 ðu h u h 3 Š 8 L ð39þ We introduce auxiiary variabes to formuate reaxations of quadratic equations: h x ¼ h þ 8 L ð4þ y x ¼ h 8 L ð41þ h v ¼ þ u h 3 8 L ð4þ y v ¼ u h 3 8 L ð43þ u v ¼ v m þ h 3 6 þ u h 8 L ð44þ 1 w v ¼ v m h 3 6 þ u h 1 8 L ð4þ Higher order terms of h can be expressed simiary by introducing new auxiiary variabes. Again, as in Mode R, using variabes u v and w v, we have: 3 ¼ 1 4 ðh x y x Þ 8 L ð46þ ¼ 1 4 ðh v y v Þ 8 L ð47þ v s v r sin h 1 4 ðu v w v Þ 8 L ð48þ Simiary, auxiiary variabes h xa ; y xa ; h va and y va are proposed: h xa h x 8 L ð49þ y xa y x 8 L ðþ h va h v 8 L ð1þ y va y v 8 L ðþ The quadratic functions are upper bounded by: h xa ðh x þ h x Þh x h x h x 8 L ð3þ y xa ðy x þ y x Þy x y x y x 8 L ð4þ h va ðh v þ h v Þh v h v h v 8 L ðþ y va ðy v þ y v Þy v y v y v 8 L ð6þ where h x ; y x ; h v and y v are upper bounds of the corresponding variabes; parameters h x ; y x ; h v and y v are ower bounds of the corresponding variabes. is bounded as foows: h 8 L ð7þ h 8 L ð8þ where h is the upper bound of h. In Mode T of SOC- ACOPF, we repace constraint (7) of optimization probem (1) (1) by constraints (1), (16), (1) (8), (4) (4), (49) (8) and constraints (19), (46) (48) where quadratic functions u x ; w x ; h x ; y x ; h v ; y v ; u v and w v are repaced by u xa ; w xa ; h xa ; y xa ; h va ; y va ; u va and w va, the term v s v r is repaced by v m, the term v s v r sin h is repaced by X p s R q s. Mode T is summarized as foowing: min X f ðp n ; q n ; p o ; q o Þ ð9þ s.t. () (6), (8) (1), (1) (16), (1) (8), (3), (33), (4) (4), (49) (8) 3 ¼ 1 4 ðh xa y xa Þ 8 L ð6þ ¼ 1 4 ðh va y va Þ 8 L ð61þ Mode T is convex because: À constraints (4) (4), (3) (6), (8) (61) are inear; ` constraints (49) () and (7) are in the form of second-order cone. For the convexity

6 Second-order cone AC optima power fow: convex reaxations and feasibe soutions 73 of other constraints, pease refer to the expanations of Mode P in Section.1 and Mode R in Section...4 SOC-ACOPF: Mode E The Nonconvex term in the eft side of (11) can be directy repaced by McCormick enveopes described in [33]. Empoying (13) and introducing new variabes z h ¼ sin h and z h ¼ v s v r sin h (note the term h in the subscript of z h and the term h in the subscript of z h are not indexes but are ony to distinguish different variabes), we have: v s v r sin h ¼ v m z h 8 L ð6þ z h ¼ X p s R q s 8 L ð63þ McCormick enveopes for z h ¼ v m z h are: z h > v m z h þ z h v m v m z h 8 L ð64þ z h > v m z h þ z h v m v m z h 8 L ð6þ z h 6 v m z h þ z h v m v m z h 8 L ð66þ z h 6 v m z h þ z h v m v m z h 8 L ð67þ where v m ; z h are ower bounds and v m ; z h are upper bounds for their corresponding variabes. McCormick enveopes for v m ¼ v s v r are: v m > v s v r þ v r v s v s v r 8 L ð68þ v m > v s v r þ v r v s v s v r 8 L ð69þ v m 6 v s v r þ v r v s v s v r 8 L ð7þ v m 6 v s v r þ v r v s v s v r 8 L ð71þ where v s and v r are ower bounds and v s and v r are upper bounds for their corresponding variabes. McCormick enveopes for z h are: z h > cos h h þ h sin h 8 L ð7þ z h 6 cos h h h þ sin h 8 L ð73þ where h is the upper bound of h. Constraints (64) (73) are inear. Constraints (6) and (73) are vaid for \h \p=. Bounds of the variabes can be determined a priori. In Mode E of SOC-ACOPF, we repace constraint (7) of optimization probem (1) (1) by constraints (63) (73). Accordingy, assumptions 1 and are not required in Mode E. Mode E is summarized as foowing: ( min f ðp n ; q n ; p o ; q o Þ X s:t: ðþ ð6þ; ð8þ ð1þ; ð63þ ð73þ ð74þ Mode E is convex because constraints (63) (73) are inear. For the convexity of other constraints, pease refer to the expanations of Mode P in Section.1. 3 Feasibiity The proposed SOC-ACOPF modes give reaxed soutions of the ACOPF probem. In case the AC feasibiity is vioated by the soutions of the derived SOC-ACOPF modes, we propose here a heuristic agorithm to recover feasibe soutions from the reaxed soutions. The heuristic technique is summarized in Agorithm 1. We use the reaxed soutions of the active power generation p n from the SOC-ACOPF modes. If p n are feasibe for a the ACOPF constraints, we can confirm that we have found the goba optima soution of ACOPF. Otherwise, we propose to fix p n ¼ p n for the cheap cost generators and take the p n of the margina generator (the most expensive generator in N ) as a variabe. Where N N is a dynamic set initiated as 8n N ; p n [. N is updated by removing the eement n which is the index of the dispatched generator with the highest margina cost in each iteration. This process is repeated unti the ACOPF is feasibe in Agorithm 1. c n is the margina cost of generator at node n. i max is the maximum number of aowed iterations. We show in Section 4 of this paper that the feasibe soution can be recovered by this agorithm normay in few iterations. Agorithm 1: Feasibe soution recovery agorithm Input: soution of SOC-ACOPF p n ; Output: feasibe soution of ACOPF p n ; Initiaization; i =1; Define N N such that n N,p n >; p n = p n ; do if c n =max{c n }, n, n N then Repace p n = p by n p min,n <p n <p max,n ; N = N \ n ; Sove nonconvex ACOPF; i = i +1; whie ACOPF is not feasibe and i<i max ;

7 74 Zhao YUAN, Mohammad Reza HESAMZADEH 4 Numerica resuts A the proposed SOC-ACOPF modes (Mode P, Mode R, Mode T and Mode E) are impemented in GAMS and soved by MOSEK. A computer with.4 GHz CPU and 8 GB RAM is depoyed for the computations (except the computations in Section 4.). It is worth to mention that votage phase ange soutions can be obtained from soving the proposed SOC-ACOPF modes directy since votage phase ange is one of the decision variabes in a the SOC- ACOPF modes (see constraint (7) in Mode P, constraints (17), (18) in Mode R, constraints (4) (4) in Mode T and constraints (7), (73) in Mode E). The convergence of MOSEK sover is guaranteed by the convexity of a the proposed SOC-ACOPF modes. Soutions of ACOPF from [34] and LINDOGLOBAL are set as the benchmarks. The LINDOGLOBAL sover empoys branch-and-cut methods to find the goba optima soution. uses MATLAB buit-in interior point sover (MIPS) to sove nonconvex ACOPF. If a soution is not found, we denote the corresponding resut as NA (the LINDOGLOBAL sover in GAMS currenty cannot sove optimization modes with over 3 variabes and constraints). To compare with other SOC-ACOPF modes in the iterature, we aso impement the mode in reference [13] (denoted as Mode 1 in the tabes) and the mode in reference [8] (denoted as Mode in the tabes). The data for test case of 134pegase and 869pegase are from reference [3]. Since a the proposed SOC-ACOPF modes are convex, soutions from MOSEK sover in GAMS for the modes are goba optima. Pease note the soutions of the proposed SOC-ACOPF modes are goba optima for the corresponding SOC-ACOPF modes. We do not mean that the optima soutions from SOC-ACOPF modes are goba optima for the nonconvex ACOPF mode. Ony the soutions from LINDOGLOBAL here can be regarded as the goba optima soution for the nonconvex ACOPF mode. The soutions from are oca optima and are not guaranteed to be goba optima for the nonconvex ACOPF mode. The sight differences of the soutions from the proposed SOC-ACOPF modes are because the formuations are different and they have different feasibe regions. In other words, the proposed SOC-ACOPF modes are not equivaent with each other. This is why we use LINDOGLOBAL as a benchmark for the comparisons. The reative gap termination toerance of the soution agorithm using by MOSEK to sove SOC-ACOPF modes is 1 7.If the GAMS soution report of the MOSEK sover caims norma_competion, which means optima soution is found, we wi report the numerica resuts. The convexity of the proposed SOC-ACOPF modes are further vaidated numericay here since the MOSEK sover in GAMS can ony sove convex optimiza probems. 4.1 Performance of SOC-ACOPF modes Base case The SOC-ACOPF objective vaues are isted in Tabe 1. The best resuts compared with LIDOGLOBAL are in bod. When LINDOGLOBAL cannot converge, we use the recovered best feasibe soutions from Section 4. as the benchmark. The LINDOGLOBAL sover is abe to find goba soutions for IEEE 14-bus, IEEE 7-bus and IEEE 118-bus cases. For arger power networks, the number of variabes and constraints exceed the imits of LINDO- GLOBAL. The resuts of and LINDO- GLOBAL are very cose. A proposed SOC-ACOPF modes give very cose resuts as compared to MAT- POWER and LINDOGLOBAL soutions. Normay, since our SOC-ACOPF modes are reaxed modes, the objective soutions are sighty ower than the soutions from LIN- DOGLOBAL which soves the nonconvex ACOPF mode. Compared to LINDOGLOBAL resuts, the objective vaues of IEEE 14-bus case from Mode T and Mode E are bit higher. The reason is that the votage phase ange constraint \h \p= is incuded in Mode E whie this is not Tabe 1 Objective vaue Case Objective vaue ($) Mode P Mode R Mode T Mode E Mode 1 Mode LINDOGLOBAL IEEE IEEE NA IEEE IEEE NA 134pegase NA 869pegase NA

8 Second-order cone AC optima power fow: convex reaxations and feasibe soutions 7 Tabe Computation CPU time Case CPU time (s) Mode P Mode R Mode T Mode E Mode 1 Mode LINDOGLOBAL IEEE IEEE NA.1.31 IEEE IEEE NA 134pegase NA 869pegase NA necessary for the nonconvex ACOPF mode in GAMS soved by LINDOGLOBAL (this can be aso due to the accuracy toerance differences of different optimization sovers). The computation time resuts are isted in Tabe. Mode P requires east computation time whie Mode T requires the most. This is because mode compexity increases as we increase mode accuracy. The accuracy improvement is vaidated by the soution resuts in Tabe 1. A proposed SOC-ACOPF modes are computationay competitive with. The proposed SOC- ACOPF modes require much ess computation time as compared to for arge-scae network cases (see the resuts of 134pegase and 869pegase cases in Tabe ). Our proposed SOC-ACOPF modes are vaid for both radia and mesh power networks. To compare the computationa performance of different SOCP-based ACOPF formuations, we have aso impemented the ACOPF modes in [13] and [8]. The resuts are isted in Tabes 1 and. It is worth to mention that the mode in reference [8] takes much onger time than our proposed SOC-ACOPF modes for GAMS to generate the executabe mode format to the sover though the sover CPU time is short. In genera, the mode in [13] has the east number of constraints and requires ess computationa time. However, this mode is not vaid for mesh networks because there is no votage phase ange constraint in this mode (the resuts from the mode in [13] are reaxed soutions for mesh networks). For the mode in [8], MOSEK in GAMS cannot converge for the IEEE 7-bus test case. These resuts show stronger robust performance of our SOC-ACOPF modes compared with the other two convex ACOPF modes in the iterature Power oad scenarios We evauate the performance of the proposed SOC- ACOPF modes under different power oad scenarios and compare the resuts with resuts. The incrementa power oad scenarios are generated from 1% to 1% of the power oads (both active power and reactive power) in the base case. The SOC-ACOPF and ACOPF modes are soved repeatedy by using different power oad parameters. To demonstrate the reaxation performance of the proposed SOC-ACOPF modes, we aso cacuate and pot the maximum vaue of reaxation gaps G o. The maximum reaxation gap G o is cacuated by: G o ¼ max p o p s þ q s V s R ð7þ Sma vaues of G o mean better AC feasibiity. The resuts are shown in Figs. 1,, 3, 4,, 6, 7, 8, 9, 1, 11 and 1. For IEEE 14-bus, IEEE 7-bus, IEEE 118-bus and IEEE 3-bus cases, G o \1 6 is vaid for a the power oad scenarios. For power oad scenarios when cannot converge, the MOSEK sover is convergent for our proposed modes however with arge reaxation gaps (see in Fig. 1 the resuts of reaxation gap for power oad ratio 1%). These resuts show strong robustness of the proposed SOC-ACOPF modes over the power oad scenarios Congested power networks The network congestions are caused by reducing the capacity of transmission ines to be 8% of the power fow soutions in the base case. Each ine is congested sequentiay. We summarize the statistics of the resuts in Tabe 3. Objective (k$) Mode P.6. Mode R.4 Mode T.3 Mode E..1 Mode Fig. 1 Objectives of IEEE 14-bus for different power oad scenarios

9 76 Zhao YUAN, Mohammad Reza HESAMZADEH Gap (1-7 p.u.) 3 1 Mode P Mode R Mode T Mode Fig. Reaxation gaps of IEEE 14-bus for different power oad scenarios.8.9 Objective (k$) 1 1 Mode P Mode R Mode T Mode E Mode Fig. Objectives of IEEE 118-bus for different power oad scenarios Objective (k$) 6 4 Mode P Mode R Mode T Mode E Mode Fig. 3 Objectives of IEEE 7-bus for different power oad scenarios Gap (1-7 p.u.) Mode P Mode R Mode T Mode Fig. 6 Reaxation gaps of IEEE 118-bus for different power oad scenarios Gap (1-7 p.u.) 4 Mode P Mode R Mode T Mode.1..3 When some critica ines are congested, both and GAMS cannot converge. A the proposed SOC- ACOPF modes are more robust than the nonconvex ACOPF mode in. The convergence performance of Mode P is very cose to the nonconvex ACOPF mode in. These resuts show that Fig. 4 Reaxation gaps of IEEE 7-bus for different power oad scenarios.8.9 Objective (k$) Mode P Mode R Mode T Mode E Mode the improvements (robustness) of the proposed SOC- ACOPF modes compared with Mode P are prominent. 4. Feasibe soution Fig. 7 Objectives of IEEE 3-bus for different power oad scenarios The heuristic agorithm described in Section 3 for recovering a feasibe soution is vaidated numericay in

10 Second-order cone AC optima power fow: convex reaxations and feasibe soutions 77 Gap (1-7 p.u.) 4 Mode P Mode R Mode T Mode Fig. 8 Reaxation gaps of IEEE 3-bus for different power oad scenarios.8.9 Objective (k$) 1 1 Mode P Mode R Mode T Mode E Mode Fig. 11 Objectives of 869pegase for different power oad scenarios Objective (k$) Mode P Mode R Mode T Mode E Mode Fig. 9 Objectives of 134pegase for different power oad scenarios Gap (p.u.) 1 Mode P Mode R Mode T Mode.1..3 this section. We use the IPOPT sover in GAMS to sove the nonconvex ACOPF mode. A desktop with 3.4 GHz CPU and 3 GB RAM is used to impement Agorithm 1. For a the reaxed soutions of SOC-ACOPF modes isted in Tabe 1, the feasibe soutions are recovered within two iterations. The objective function vaues of the recovered Fig. 1 Reaxation gaps of 134pegase for different power oad scenarios.8.9 Gap (p.u.) 1 1 Mode P Mode R Mode T Mode.1..3 feasibe soutions are isted in Tabe 4. The CPU time required for the computation of the feasibe soutions are reported in Tabe. The feasibe soutions from the modes in [13] and [8] are aso recovered and reported. Feasibe soutions recovered from our SOC-ACOPF modes are better than [13] and [8]. For 134pegase and 869pegase test cases, feasibe soutions with ower objective function vaues than the soutions from are recovered. The disadvantage of the proposed heuristic feasibe soution recovery agorithm is that the computation time is arger than soving the nonconvex ACOPF mode by. However, this agorithm is sti usefu when computation time is not stricty constrained considering some recovered feasibe soutions are better than the soutions obtained from. It is aso worth to mention that, since IPOPT is an open source sover, it may not be suitabe to compare the CPU time of IPOPT with the MIPS sover in MATLAB which is commercia software. The major reason of arger CPU time is due to the iterative nature of the heuristic feasibe soution recovery agorithm Fig. 1 Reaxation gaps of 869pegase for different power oad scenarios.8.9

11 78 Zhao YUAN, Mohammad Reza HESAMZADEH Tabe 3 Convergence performance in congested power networks Case Number of convergent soutions Tota number of transmission ines Mode P Mode R Mode T Mode E IEEE IEEE IEEE IEEE pegase pegase Tabe 4 Objective vaues of recovered feasibe soution Case Objective vaue ($) Mode P Mode R Mode T Mode E Mode 1 Mode LINDOGLOBAL IEEE IEEE NA IEEE IEEE NA 134pegase NA 869pegase NA Tabe Computation time of feasibe soution recovery agorithm Case CPU time (s) Mode P Mode R Mode T Mode E Mode 1 Mode LINDOGLOBAL IEEE IEEE NA.1.31 IEEE IEEE NA 134pegase NA 869pegase NA Concusion Three second-order cone modes (Mode R, Mode T and Mode E) for ACOPF are proposed using convex reaxation and approximation techniques. Compared with other SOCP-based ACOPF formuations in the iterature, our formuations are vaid for both mesh and radia power networks. The numerica resuts show that the proposed SOC-ACOPF modes can give accurate resuts. Though the biinear transformation based derivations ead to more constraints in the proposed SOC-ACOPF modes compared with Mode P, the accuracy improvement is achieved with simiar computation time as compared to. The quaity of resuts with respect to the goba optima soution is aso checked using LINDOGLOBAL sover in GAMS. The computation resuts for various power oad scenarios show robust performance of the proposed SOC- ACOPF modes. An interesting observation from the resuts of test cases with ow power oad eves when cannot converge is that high reaxation gap vaues from the soutions of the proposed SOC-ACOPF modes can actuay be regarded as indicators of non-convergence of the origina nonconvex ACOPF mode. In these cases, the objective vaue soutions from the proposed SOC-ACOPF modes can serve as ower bounds of the origina nonconvex ACOPF mode. A computationa comparison of different SOCP-based ACOPF formuations shows the strong convergence

12 Second-order cone AC optima power fow: convex reaxations and feasibe soutions 79 performance of the proposed SOC-ACOPF modes. To recover feasibe soutions from the reaxed soutions of the proposed SOC-ACOPF modes, we deveop a heuristic feasibe soution recovery agorithm. This agorithm is capabe of recovering the feasibe soutions from a the reaxed soutions of the proposed SOC-ACOPF modes in the test cases. Another key observation from our research is that, instead of seeking one singe mode to satisfy a test cases under a scenarios, it is perhaps more reaistic to provide mutipe modes or methods, which can be vauabe or appicabe for the power network operators to dea with many different test cases or operationa situations. This is reasonabe considering the arge-scae changing nature of power network parameters (such as power oads and network congestions) in the rea-word operations. Considering uncertainties for the proposed SOC-ACOPF modes can be good future work. 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13 8 Zhao YUAN, Mohammad Reza HESAMZADEH [9] O Nei RP, Castio A, Cain MB (1) The IV formuation and inear approximations of the AC optima power fow probem. FERC Staff Technica Paper [3] O Nei RP, Castio A, Cain MB (1) The computationa testing of AC optima power fow using the current votage formuations. FERC Staff Technica Paper [31] Kocuk B, Dey SS, Sun XA (16) Strong SOCP reaxations for the optima power fow probem. Oper Res 64(6): [3] Ghamkhari M, Sadeghi-Mobarakeh A, Mohsenian-Rad H (16) Strategic bidding for producers in noda eectricity markets: a convex reaxation approach. IEEE Trans Power Syst 3(3): [33] McCormick GP (1976) Computabiity of goba soutions to factorabe nonconvex programs: Part I convex underestimating probems. Math Program 1(1): [34] Zimmerman RD, Murio-Sánchez CE, Thomas RJ (11) : steady-state operations, panning, and anaysis toos for power systems research and education. IEEE Trans Power Syst 6(1):1 19 [3] Fiscounakis S, Panciatici P, Capitanescu F (13) Contingency ranking with respect to overoads in very arge power systems taking into account uncertainty, preventive, and corrective actions. IEEE Trans Power Syst 8(4): eectrica engineering from Huazhong University of Science and Technoogy and in renewabe energy from ParisTech in 14. He was awarded the nationa second prize in 1 China Undergraduate Contest of Mathematica Modeing by China Society for Industria and Appied Mathematics. He is currenty pursuing the Ph.D. degree in the Department of Eectric Power and Energy Systems of KTH Roya Institute of Technoogy, Sweden. His research work initiates and contributes to two technica patents in Sweden. His research interests incude optima power fow, power system operation and eectricity market. Mohammad Reza HESAMZADEH received his Docent degree from KTH Roya Institute of Technoogy, Sweden, and his Ph.D. degree from Swinburne University of Technoogy, Austraia, in 13 and 1 respectivey. He was a post-doctora feow at KTH in 1 11 where he is currenty a facuty member. He is a member of Internationa Association for Energy Economics (IAEE) and a Member of CIGRE, Sweden Section. His specia fieds of interests incude eectricity market modeing, anaysis, and design and mathematica modeing and computing. Zhao YUAN received B.E. degree in eectrica engineering from Hebei University of Technoogy in 11, and doube M.E. degrees in