Robust optimization for improving resilience of integrated energy systems with electricity and natural gas infrastructures

Transcription

1 J. Mod. Power Syst. Clean Energy (2018) 6(5): Robust optmzaton for mprovng reslence of ntegrated energy systems wth electrcty and natural gas nfrastructures Hao CON 1, Yang HE 2, u WAN 3, Chuanwen JIAN 1 Abstract The ntegraton of natural gas n electrcty network requres a more relable operatng plan for ncreasng uncertantes n the whole system. In ths paper, a threestage robust optmzaton model s proposed for reslent operaton of energy system whch ntegrates electrcty and natural gas transmsson networks wth the objectve of mnmzng load curtalments caused by attacks. Nonconvex constrans are lnearzed n order to formulate the dual problem of optmal energy flow. Then, the proposed three-stage problem can be reformulated nto a two-stage mxed nteger lnear program (MILP) and solved by Benders decomposton algorthm. Numercal studes on IEEE 30-bus power system wth 7-node natural gas network and IEEE 118-bus power system wth 14-node natural gas network valdate the feasblty of the proposed model for mprovng reslence of ntegrated energy system. Energy CrossCheck date: 23 November 2017 Receved: 9 December 2016 / Accepted: 23 November 2017 / Publshed onlne: 12 February 2018 Ó The Author(s) Ths artcle s an open access publcaton & Hao CON babyconghao@gmal.com Yang HE hugesea78@sna.com u WAN xwang233@t.edu Chuanwen JIAN jangcw@sjtu.edu.cn Shangha Jao Tong Unversty, Shangha , Chna State rd Henan Electrc Power Company, Zhengzhou , Henan, Chna Illnos Insttute of Technology, 10 W 35th Street, #1600, Chcago, IL, USA storage facltes are also consdered for the reslency analyss. Keywords Reslence, Robust optmzaton, Integrated energy systems, Natural gas networks, Energy storage systems 1 Introducton In recent years, renewable energy generaton gans rsng attenton due to the lack of tradtonal resources. However, the electrcty qualty and the relablty of power grds are sgnfcantly affected by the ntermttency and nstablty of renewable energy resources such as wnd and solar. By contrast, natural gas s a more stable and relable sort of resource whch can provde contnuous energy for both gas and electrcty loads by gas-fred generators [1]. Due to the clean, effcent and hgh-qualty characterstcs of natural gas, t has been wdely used as the man energy resource n some areas and the coordnated operaton of natural gas and electrcty system has been researched n many prevous studes [2]. A basc model for ntegratng natural gas and electrcty networks s presented n [3], whch shows the fundamentals of natural gas network and descrbes the constrants for the energy transmsson between electrcty and gas systems. In [4], a steady state power flow model s presented for solvng the combned optmzaton operatng problem of dfferent energy facltes based on the new concept of energy hubs. A decomposton method s appled to solve the securtybased model proposed n [5] for the soluton of SCUC problem consderng natural gas transmsson system. In [6], an mxed nteger lnear program (MILP) method s presented to formulate the optmal power flow n mult-

2 Robust optmzaton for mprovng reslence of ntegrated energy systems wth electrcty 1067 carrer energy systems. The non-convex constrants of natural gas transmsson system are lnearzed, so that the problem s reformulated as an MILP problem whch can be solved by tradtonal optmzaton methods. However, the operatng stablty and relablty are stll affected by ncreasng uncertantes n energy systems. Dsruptons n an energy system are sometmes nevtable, uncontrollable and unpredctable [7]. As servce ndustry, the energy system must guarantee the contnuty of energy supply for customers. Therefore, mprovng reslence of energy systems s of vtal mportance. Reslence s defned as the ablty to provde and mantan an acceptable level of servce n the face of faults and challenges to normal operaton [8]. The contngences and challenges for servces range from natural dsaster to terrorst attacks. As a role of defender, the system tself wll take preventve measures for attacks before dsruptons occur and respond to the damage after attacks. The reslency analyss n electrcty system has been studed by several researchers. In [9], a reslency-orented mcrogrd optmal schedulng model s proposed for mnmzng the load curtalments when the servce of man grd s nterrupted. A non-cooperatve game-theoretc framework s presented n [10] to study the strategc behavor of mcrogrds. The framework ncorporates economc factors and stablty and effcency of mcrogrds, whch s solved by fully dstrbuted phasor measurement unt (PMU)-enabled algorthm to ensure the reslency of the proposed method. In [11], a reslent dstrbuton network plannng problem s presented and formulated as a two-stage robust optmzaton model to mnmzng the system damage by coordnatng the hardenng and dstrbuted resource allocaton. In [12], a dstrbuton system operatng method by mcrogrd formaton after natural dsaster s proposed to restore mportant loads from power outage. A tr-level optmzaton model for electrc power system defense s presented n [13] whch can dentfy crtcal elements n power grd to defend aganst unpredctable attacks. A column-and-constrant generaton algorthm s appled to solve a two-stage robust optmzaton problem n [14]. In [7], a rsk assessment model s proposed to determne potental vulnerabltes of power system and provde feasble plans for enhanced protectons accordng to the budgets for power grd constructon. There are few work whch has been done to analyze reslence of ntegrated energy system whch ncludes electrcty, gas and other forms of energy. As the operatonal feature of natural gas system s dfferent from that n power grds and the operatng condtons have mpacts on electrcty networks, the reslency analyss s an essental topc to be researched. In [15], a methodology s proposed to locate the most vulnerable components to make sure the reslent operaton of multple energy carrer mcrogrds when terrorsts attack the network. The model s formulated as a b-level optmzaton problem to solve the optmal operaton for mult-energy mcrogrds n consderaton of securty and reslency. In [16], a methodology s proposed to dentfy and protect vulnerable components of ntegrated electrc and gas nfrastructures. The reslence s guaranteed by solvng a tr-level optmzaton problem. A mxed nteger lnear programmng and nested column-andconstrant generaton algorthm s appled to solve the proposed model. A novel mxed nteger lnear programmng for securty-constraned power and gas flow s presented n [17]. The proposed model allows the ntegrated system operates n both normal and contngency condtons wth the least volatons. In [18], three models are proposed for dentfyng optmal energy flow solvablty to ensure secure operatng condtons wth correctve controls. In ths paper, a coordnated operaton model of energy system whch ntegrates electrcty and natural gas nfrastructures s formulated. The model ncludes constrants for both electrcty and natural gas transmsson networks. Then, a three-stage robust optmzaton algorthm s presented to solve the defender-attacker-defender problem of the ntegrated energy system when contngences occur. The constrants and operatonal features of natural gas system and the couplng of electrcty and natural gas networks make t more complcated to settle the plans for both attackers and defenders. In the frst stage, as defender, the ntegrated energy system must make plans for network enhancement to mnmze the damage caused by unpredctable attacks. In the second stage, attackers wll attack vulnerable components of electrcty and natural gas networks to cause maxmum damage to the entre energy system. In the last stage, defender responds to the results of dsruptons, whch s formulated as optmal energy flow of ntegrated electrcty and natural gas system. As some of the constrants of electrcty and natural gas transmsson networks are nonlnear, Taylor seres expanson algorthm s appled to realze the lnearzaton. Then, the proposed three-stage robust optmzaton problem s reformulated nto a two-stage optmzaton problem by the applcaton of dualty theory. The two-stage problem can be solved as an MILP by decomposton algorthms. The man contrbutons of our paper are: 1) A nested Benders decomposton algorthm s appled to solve the proposed defender-attacker-defender problem. It s more effectve to be appled for largescale problems. 2) Reslency of ntegrated energy systems s analyzed n the presence of energy storage system. By addng proper electrcty and gas storage facltes, relablty of ntegrated energy system s re-analyzed.

3 1068 Hao CON et al. Compared wth [15], we ncorporate the hardenng plans before attacks, whch the reslence of the whole system can be better mproved. That s to say, a defender-attacker defender model s presented n comparson wth the attacker-defender model n [15]. Compared wth [16], we apply a nested Benders decomposton algorthm to solve the nested two-stage proposed model. Moreover, Taylor seres expanson s appled to lnearze the quadratc polynomals of cost functons and the gas flow square for gas ppelnes and compressors. In addton, energy storage systems are also consdered n the end of ths paper to analyze the reslence of ntegrated energy systems. The remander of ths paper s organzed as follows. Secton 2 descrbes the mathematcal formulatons for the optmal energy flow and the three-stage robust optmzaton model of the ntegrated electrcty and natural gas system. Secton 3 provdes the soluton methodologes to lnearze and decompose the presented three-stage problem. Secton 4 presents and dscusses the numercal results of the proposed method and analyze the mpact of the presence of energy storage devces. Fnally, the concluson s gven n Secton 5. 2 Mathematcal formulaton In ths secton, a power flow model for coordnated electrcty and gas network s presented. Based on ths ntegrated system, a robust optmzaton model s appled for reslency analyss. Any contngency occurred n ether electrcty system or gas network may cause coordnaton problems or safety and stablty problems of both systems. Models are presented to fgure out the optmal defense and operaton plans aganst contngences. 2.1 Integrated electrcty and gas network optmal flow model Fgure 1 shows a general network of natural gas transmsson system. as supplers, compressors, storage systems, ppelnes and gas loads are essental components n gas networks. Natural gas s transferred from producers to as suppler s m m f mn Ppelne Compressor n p as storage Fg. 1 eneral network of natural gas transmsson system f sm q b a as load e b da end customers through ppelnes and compressors. Unlke electrcty, natural gas can be njected nto certan gas storage facltes durng off-peak perods and used durng hgh-demand perods [19]. Therefore, steady flow over ppelnes can be easly mantaned. as n ppelnes has much lower speed than electrcty and can be stored n ppelnes n a short perod of tme due to ts compressblty [20]. So the dynamc process s a key feature of gas network, and usage of steady-state gas flow models could result n sub-optmal results or even nsecure operaton decson. For smplcty, n ths paper, we only analyze steady state behavor for ntegrated electrcty and gas system. Therefore, for modelng natural gas system, we assume that the natural gas system operates n steady states and the lne pack s gnored [16]. For ppelnes, there are two major varables whch are gas pressure p m at each node m and gas flow f mn between two end nodes m and n. Smlar to the voltage n power system, gas pressure nsures that gas can transfer from one gas node to another. In general, gas can only be delvered from hgher pressure nodes to lower ones. For each node, there are gas flow njecton and gas loads. Mathematcally, the gas flow balance equaton s expressed as: s m þ f mn þ f sm ¼ d m þ e m ðp m Þ ð1þ ðm;nþ2u where s m s the gas supply from producers at node m; f sm s the gas storage at node m; d m s the non-electrcal gas demand at node m; e m s the gas demand for gas-fred generator at node m; P m s the power output of the generator node m. For each ppelne, the amount of gas flow s assocated wth end nodes pressures and propertes of each ppelne [5], whch s presented n (2). qffffffffffffffffffffffffffffffffffff f mn ¼ sgnðp m p n ÞC mn p 2 m p2 n ð2þ where sgnðþ s the sgn functon; C mn s a constant whch depends on propertes of ppelnes such as length, dameter, temperature, frcton and gas composton. For ppelnes whch have compressors, natural gas flows from the lower pressure node to hgher ones because of the exstence of compressors. Inevtable frcton between gas and ppelnes wll result n pressure loss whch can be compensated by compressors [5]. The gas flows through compressors can be calculated n (3). f cp mn ¼ sgnðp m p n Þ H mn h a ð3þ maxðp k mn2 k m ;p n Þ mn1 mnðp m ;p n Þ where H mn s the power nput of compressor between nodes m and n; k mn1 ; k mn2 ; a are emprcal parameters that depend on propertes of compressors. as consumed by

4 Robust optmzaton for mprovng reslence of ntegrated energy systems wth electrcty 1069 compressors s treated as transmsson loss of natural gas network, whch s a quadratc functon of H mn : f Hmn ðh mn Þ¼a mn H 2 mn þ b mnh mn þ c mn ð4þ It s a smplfed form of expresson whch can be found n [5]. The optmal flow for ntegrated electrcty and gas system takes the mnmum total cost of operaton as objectve functon: mn 2N ða P 2 þ b P þ c Þþ m2n S K m s m ð5þ where N s the set of non-gas-fred generator nodes; N S s the set of natural gas nodes; K m s the prce of gas at node m. The constrants of electrcty system are shown n (6) (12). P Re S j ðv; hþ ¼ P L ð6þ Q Im S j ðv; hþ ¼ Q L ð7þ P ;mn P P ;max Q ;mn Q Q ;max V ;mn V V ;max h ;max h h ;max P j ðv; hþ Pj;max ð8þ ð9þ ð10þ ð11þ ð12þ Equatons (6) and (7) are the power balance formulaton; (8) (12) represent the constrants of node voltage, generator output and lnes transmsson lmts. Apart from (1) (5), the other constrants for natural gas system are presented n (13) (15). k s the dual varables for the constrants. s m;mn s m s m;max k 2 m ; k3 m ð13þ p m;mn p m p m;max k 4 m,k5 m ð14þ H mn;mn H mn H mn;max k 6 m ; k7 m ð15þ Equatons (13) and (14) are respectvely the gas supply and gas pressure lmts. 2.2 Robust optmal operaton model for reslent energy systems As the attacks on ntegrated energy system are random and unpredctable, the complex nature of ths problem makes robust optmzaton be the most sutable method that can take account of nherent randomness and uncertantes [11]. Ths robust optmzaton problem ams to fnd the optmal enhancement and dspatch plans accordng to the worst case caused by attackers. As descrbed n the frst secton, reslency analyss for ntegrated energy system can be formulated n three stages, whch s also known as defender-attacker-defender game model. The frst stage s the hardenng of transmsson lnes by defender to mnmze the damage caused by contngences. In the second stage, attackers dsrupt the energy system to produce a most damagng attack. In the thrd stage, defender wll respond to the damage caused by attacks to optmze the operaton of ntegrated energy system. Therefore, the three-stage problem can be formulated as a mn-max-mn problem whch s shown as follows: mn max mn ðf CðrÞþf LC ðrþþ ð16þ h2h a2a r2rðh;aþ where h and a are respectvely the hardenng and attackng scenaros; H and A are respectvely the feasble set for lne hardenng plans and the uncertanty set for attackng plans; r represents the optmal electrcty and gas flow varables; RðH; AÞ denotes the energy system response set based on H and A; f LC ðrþ s the total economc loss due to load curtalment; f C ðrþ s the costs for optmal operaton of energy system. In ths secton, we analyze the model from nner mnmum problem to outer mnmum problem Defender response model In Secton 2.1, we have dscussed the optmal operaton plannng for electrcty and gas network n normal stuaton. After an attack, the topology of electrcty and gas network may change, whch wll result n nevtable load curtalment. In defender response model, reactve power s gnored when calculatng power flow n electrcty network. Therefore, constrants for power system are modfed n (17) (21). P j x j ¼ h h j ð17þ P þ j2j P j þ LC elec ¼ P L ð18þ P j;max P j P j;max ð19þ P ;mn P P ;max k 8 ; k9 ð20þ h ;max h h ;max k 10 ; k 11 ð21þ where LC elec s the load curtalments for electrcty network. The constrant for natural gas network n (1) s modfed as:

5 1070 Hao CON et al. s m ¼ f Hmn ðh mn Þþd m þ e m ðp m Þ ðm;nþ2n g f mn LCm gas ðm;nþ2n CP where LC gas m ð22þ s the load curtalment for natural gas network. The defender response model (DRM) can be smply descrbed n (23). mn CðrÞþf LC ðrþþ ðp ;h;p;sþ2r ð23þ f C ðrþ ¼fc elec ðp ; hþþfc gas ðp; sþ ð24þ f LC ðrþ ¼ 2I s elec LC elec þ s gas m LCgas m m2m ð25þ subject to (2) (4) and (16) (21). where I and M represent the sets of load curtalment nodes for electrcty and gas network respectvely; s elec and s gas m denote the coeffcents of economc loss for electrcty and gas network respectvely. Equaton (24) s the smplfed objectve functon of (5) Attacker nterdcton model There are varous of attacks that may result n system blackout, whch can be generally categorzed nto two knds: terrorst attack and natural dsaster. However, attackers wll destroy network elements, such as generators, compressors or transmsson lnes, regardless of the type of attacks. As energy transmsson falure occurs when network elements are attacked, we assume that attackers only attack transmsson lnes for smplcty. We ntroduce a elec t;j and a gas t;mn to represent the attackng states for electrcty and gas transmsson lnes respectvely. Then we set: a elec t;j ¼ 1 f electrcty transmsson lne -j s attacked 0 otherwse a gas 1 f gas transmsson lne m-n s attacked t;mn ¼ 0 otherwse ð26þ ð27þ The lnes constrants (2), (3), (17) (19) and (22) must be modfed due to the ntroduced varables a elec t;j and a gas t;mn. s m ¼ a gas t;mn f H mn ðh mn Þþd m þ e m ðp m Þ ðm;nþ2n CP a gas t;mn f ð28þ mn LCm gas ðm;nþ2n g f mn ¼ a gas t;mn sgnðp m p n ÞC mn qffffffffffffffffffffffffffffffffffff p 2 m p2 n ð29þ fmn cp ¼ agas t;mn sgnðp m p n ÞH mn h a ð30þ maxðp k mn2 k m ;p n Þ mn1 mnðp m ;p n Þ P j x j ¼ a gas t;mn ðh h j Þ k 15 P þ P j þ LC elec ¼ P L k 16 j2j P j a elec t;j P j;max k 17 ; k 18 ð31þ ð32þ ð33þ As some of generators n electrcty system are customers of natural gas system, attacks on gas transmsson lnes may cause blackouts of generators, compressors and gas loads, whch wll result n more damage to entre energy system. Therefore, attackers plan for ntegrated energy system wll be qute dfferent from the plan n power system only. Attackers wll fnd the optmal attackng plan whch maxmzes the economc loss caused by electrcty and gas load curtalment. The attacker nterdcton problem (AIP) s formulated n (34) (36). max ðf CðrÞþf LC ðrþþ ð34þ ði;mþ2a where A ¼ ðlc; I; MÞj mn ðf CðrÞþf LC ðrþþ ðp ;h;p;sþ2r subject to (4), (13) (15), (20), (21), (28) (33), and ð35þ a elec t;j ; agas t;mn 2 f0; 1g ð36þ It s obvous that ths model s a two-stage optmzaton problem Defender renforcement model Lnes renforcement s an effectve measure that defenders use to harden networks and mprove reslence of systems. We assume that hardened lnes cannot be attacked. Smlar to attacker nterdcton model dscussed n Secton 2.2.2, we also ntroduce h elec l;j to represent the hardenng states for electrcty and gas transmsson lnes respectvely. We set: h elec 1 f electrcty transmsson lne -j s hardened l;j ¼ 0 otherwse and h gas l;mn ð37þ h gas 1 f gas transmsson lne m-n s hardened l;mn ¼ 0 otherwse ð38þ In consderaton of economc budget, the number of hardened lnes s lmted, whch s gven by (39).

6 Robust optmzaton for mprovng reslence of ntegrated energy systems wth electrcty elec j ;j2n where 1 elec j h elec l;j þ m;n2n S 1 gas mn hgas l;mn B h ð39þ and 1 gas mn are unt cost for lnes renforcement; B h s the budget for renforcement. Defenders wll fnd the optmal renforcement plan to mnmze the economc loss caused by load curtalment. The defender renforcement problem (DRP) s formulated as follows: mn AIPðhÞ h¼ðh elec ;h gas l;j l;mn Þ2H subject to (30) ð40þ h elec l;j ; h gas l;mn 2 f0; 1g ð41þ where AIP s presented n Secton It s obvous that ths model s a nested two-stage optmzaton problem. 3 Soluton methodology The formulated model n Secton 2.2 s a nested twostage robust optmzaton problem whch s shown n Secton We apply a nested Benders decomposton algorthm to solve ths problem. 3.1 Master problem The master problem of the two-stage model s the mnmzaton of economc loss for load curtalment under lnes renforcement. mn z ð42þ subject to z f C ðrþþf LC ðrþ 1 elec j ;j2n h elec l;j ; h gas l;mn 2 H h elec l;j þ m;n2n S 1 gas mn hgas l;mn B h ð43þ ð44þ ð45þ Solvng the master problem needs a set of network renforcement plan H and the worst case under attack whch can be derved from the subproblem. 3.2 Subproblem The subproblem s amng to fnd the worst case under attack wth gven renforcement plan, whch s the maxmzaton of economc loss for load curtalment. For a gven attackng plan, DRM wll formulate an optmal operaton plan for ntegrated system. The subproblem s actually a max-mn optmzaton problem whch s formulated n Secton as AIP. max ði;mþ2a mn ðf CðrÞþf LC ðrþþ ð46þ ðp ;h;p;sþ2r subject to a elec t;j ; agas t;mn 2 A ð47þ and (4), (13) (15), (20), (21), (28) (33). As the objectve and constrants n (4), (28) (30) and (46) are nonlnear, we wll fnd ways to lnearze them. Varables a mn n (4), a n (5) and a m n (28) are generally qute small, so we take the dervatve of the second order terms for lnearzaton. Equatons (4), (5), (28) are modfed as: f Hmn ðh mn Þ¼ð2a mn þb mn ÞH mn þ c mn k 1 m ð48þ ( ) mn ½ð2a þ b ÞP þ c Šþ K m s m ð49þ 2N m2n S s m ¼ a gas t;mn f H mn ðh mn Þþd m þð2a m þ b m ÞP m ðm;nþ2n CP þ c m a gas t;mn f mn LCm gas k 12 m ðm;nþ2n g ð50þ As for the constrants n (29) and (30), we apply Taylor seres expanson to lnearze them by omttng the hgher order terms [6]. Natural gas flow expresson (29) s expanded around the neghbor pont ðp m0 ; p n0 Þ as: f mn f mn ðp m0 ; p n0 Þþ of mn op m ðp m p m0 Þþ of mn op n ðp n p n0 Þ agas t;mnc mn ¼ pffffffffffffffffffffffffffffffffffffff p 2 m0 p2 n0 ðp m0 p m p n0 p n Þ¼a gas t;mn ðu mnp m V mn p n Þ k 13 m ð51þ where we assume that p m [ p n. Smlarly, gas flow through compressors n (30) can be expanded n the neghbor pont ðp m0 ; p n0 ; H mn0 Þ as: f cp mn f cp mn ðp m0; p n0 ; H mn0 Þþ of cp mn þ ðh mn H mn0 Þ¼ oh mn þ k 1 H mn p n0 p m0 a k2 ¼ a gas of cp mn op m ðp m p m0 Þþ k 1 t;mn ðucp p n0 p m0 of cp mn ðp n p n0 Þ op n ah mn0 k 1 p a n0 p m a k2 2 p n p m0 p n0 p a m0 mn p m Vmn CP p n þ Wmn CP H mnþ k 14 m ð52þ As mentoned n [6], the doman of f mn s dvded nto (NL) 2 grds. NL s the number of segments whch s decded by the accuracy requrement. Selecton of neghbor ponts and approxmaton error analyss are also gven n [6], whch wll not be repeatedly explaned here. To solve ths max-mn model, the nner mnmzaton problem s transformed nto ts dual problem. Dual varables for the constrants are gven n the end of each

7 1072 Hao CON et al. formulaton, whch are k 1 k 18. The dual problem s a maxmzaton program shown n (53) (67). max m2n g C mn k 1 m þ m2n S ðs m;max k 2 m s m;mnk 3 m Þ þ ðh mn;max k 6 m H mn;mnk 7 m Þ ðm;nþ2n CP þ t;j P jmax ðk 17 k 18 Þþ ðp m;max k 4 m p m;mnk 5 m Þ m2n g ð;jþ2n el a elec þ ðp ;max k 8 P ;mn k 9 Þþ d m k 12 m þ P L k 16 2N g m2n g 2N e worst case. Set UB! MnðUB; obj max SP Þ; n c! UB LB; K ¼ K þ 1: 3) Step 3: Solve the master problem wth attackng plan A and optmal system operaton varables R derved from Step 2. et the mnmum economc loss obj mn MP. Set LB! MaxðLB; obj mn MP Þ; n c! UB LB. Update lnes renforcement plan H. 4) Step 4: If n c satsfes convergence condton, stop the process. Otherwse, return to Step 2. þ h max ðk 10 k 11 Þ 2N e ð53þ 4 Case study subject to a elec t;j ; agas t;mn 2 A ð54þ k 2 m þ k3 m þ k12 m ¼ K m m 2 N s ð55þ k 4 m þ k5 m þ agas t;mn ðu mnk 13 m V mnk 13 n Þ þ a gas t;mn ðucp mn k14 m VCP mn k14 n Þ¼0 ðm; nþ 2N g ð56þ ð2a mn þ b mn Þk 1 m þ k6 m þ k7 m þ WCP mn k14 m ¼ 0 ðm; nþ 2N CP ð57þ k 8 þ k 9 þð2a m þ b m Þk 12 m þ k16 ¼ 2a þ b 2 N ; m 2 N g ð58þ k 10 þ k 11 þ a elec t;j ðk 15 k 15 j Þ¼0 ; j 2 N e ð59þ k 17 þ k 18 þ x j k 15 þ k 16 ¼ 0 2 N e ð60þ a gas t;mn k12 m þ k13 m ¼ 0 ðm; nþ 2N el ð61þ a gas t;mn k12 m þ k14 m ¼ 0 ðm; nþ 2N CP ð62þ k 12 m k 16 ¼ sgas m m 2 N g ð63þ ¼ s elec 2 N e ð64þ k 2 m ; k4 m ; k6 m ; k8 ; k10 ; k 17 0 ð65þ k 3 m ; k5 m ; k7 m ; k9 ; k11 ; k 18 0 ð66þ k 1 m ; k12 m k16 unlmted ð67þ The dual maxmzaton problem s lnear and can be solved by an MILP solver. 3.3 Soluton step for nested two-stage robust optmzaton algorthm 1) Step 1: Intalzaton of varables. Set teratons K! 0; LB! 1; UB!1: 2) Step 2: Solve the subproblem wth gven renforcement plan H. et the objectve maxmum economc loss obj max SP and the optmal attackng plan A for the To show the performance of our proposed optmzaton model for ntegrated electrcty and natural gas energy system, we apply two testng systems whch are IEEE 30-bus power system wth 7-node natural gas system and IEEE 118-bus power system wth 14-node natural gas system. 4.1 IEEE 30-bus system Ths case s based on the modfed IEEE 30-bus power network and 7-node natural gas system. The modfed IEEE-30 bus power system shown n Fg. 2 s consst of 6 thermal unts ncludng 3 fossl unts and 3 natural gas-fred unts, 41 branches, 4 transformers and 20 demand sdes. Data for buses, branches and load demands are from [21]. Characterstcs for generator locatons, costs and lmts are taken from [22]. The 7-node natural gas system depcted n Fg. 3 has 2 natural gas supplers, 5 ppelnes, 1 compressor and 5 natural gas loads Fg. 2 IEEE 30-bus power system

8 Robust optmzaton for mprovng reslence of ntegrated energy systems wth electrcty 1073 S1 S Fg. 3 7-node natural gas system ncludng 3 demand loads for gas-fred unts. Parameters for natural gas network consults [5]. In normal condtons where no transmsson lnes are hardened and no elements are damaged, the ntegrated energy system wll operate n a safe and economc manner. Some of the optmal dspatch plan results for the system n a certan tme perod are gven n Tables 1 3. L1, L3 and L5 are varable gas loads for natural gas-fred unts 2, 3 and 4, whch depends on the outputs and loads of electrcty network. In normal operaton condtons, no congestons happen and no loads are curtaled. 3 L5 Table 1 Unts outputs for electrcty network Unt Node No. P (MW) Table 2 as outputs for natural gas network Suppler Node No. Output (kcf) S S Table 3 as loads for natural gas network Loads Node No. L gas (kcf) L L L L L L3 L4 L1 L2 In reslent operaton condtons, lnes renforcement and load curtalment are consdered. As both of defenders and attackers have operatng budgets, defenders cannot renforce all transmsson lnes and attackers cannot damage all elements n the system. Therefore, n ths case, we assume that attackers have budgets for damagng up to 5 lnes n electrcty network and 1 lne n natural gas network. Then we analyze the renforcement plan for defenders of whch budgets can support from 0 to 5 lnes renforcement for electrcty network and 0 to 1 lne renforcement for natural gas network. The lnes renforcement plans for ntegrated system under dfferent budgets are gven n Table 4. In Table 4, HL elec and HL gas are hardened lnes for electrcty and natural gas networks respectvely. When no lne s hardened, attackers wll cause damage up to MW electrcty load curtalments and 4764 kcf natural gas load curtalments. By hardenng 5 lnes n electrcty network and 1 lne n gas network, the worst case only causes 57.5 MW electrcty load curtalments and 2351 kcf natural gas load curtalments, whch s a reducton of nearly 50%. Congestons caused by attackng for 6 dfferent defendng plans are shown n Table 5. B cg s the branch that congeston happens. S j and S lm are apparent power and lmts for transmsson lnes. It s obvous that more hardened lnes wll allevate congestons of transmsson lnes. Therefore, reslence of ntegrated energy system s mproved greatly. Load curtalments for power system and natural gas system are shown n Fgs. 4 and 5 respectvely. Curtaled loads decrease wth the ncrease of hardened transmsson lnes. However, the slope for the reducton of load curtalments tend to be gentle. It means that more money wll be cost to reduce the same quanttes of load curtalments wth the ncrease of hardened lnes. Defenders must decde the optmal number of transmsson lnes to be hardened to gan more proft despte unpredctable attacks. 4.2 IEEE 118-bus system Ths case s studed to test the valdty of proposed method n large-scaled systems. It s based on a modfed IEEE 118-bus power network and 14-node natural gas system. The modfed IEEE 118-bus power system depcted n Fg. 6 s conssts of 54 thermal unts ncludng 42 fossl unts and 12 natural gas-fred unts, 186 branches, 9 transformers and 99 loads. Data for generators, buses, branches and load demands are from [23]. The 14-node natural gas system depcted n Fg. 7 has 3 natural gas supplers, 12 ppelnes, 2 compressors and 16 natural gas loads ncludng 8 demand loads for gas-fred unts. Parameters for natural gas network consults [5].

9 1074 Hao CON et al. Table 4 Renforcement plans for ntegrated energy system Lnes HL elec HL gas Worst case Load curtalments Electrcty as Electrcty (MW) as (kcf) 0 lne None None 2 4, 6 8, 15 18, 10 22, lne None 6 9, 8 28, 10 22, 15 23, lnes 10 22, None 4 6, 8 28, 19 20, 15 23, lnes 15 23, 10 22, None 1 3, 3 4, 12 13, 12 14, lnes 1 3, 15 23, 10 22, None 3 4, 4 6, 12 13, 12 14, lnes 1 3, 3 4, 15 23, 10 22, None 1 3, 3 4, 15 23, 10 22, Table 5 Congestons caused by attacks for 6 renforcemet plans Plan No. B cg S j (MW) S lm (MW) , 6 28, , 42.62, , 32, , 15 23, , 23.14, , 16, , , , , 6 28, , 37.54, , 32, , 22 24, , 26.35, , 16, , 6 28, , 37.29, , 32, , 22 24, , 25.89, , 16, Fg. 4 Load curtalment for IEEE 30-bus power system We assume that attackers can damage up to 10 transmsson lnes n electrcty network and 3 transmsson lnes n natural gas network. Defender has budget for hardenng 0 10 power transmsson lnes and 0 3 gas Fg. 5 Load curtalments for 7-node natural gas system transmsson lnes. Fgures 8 and 9 depcts load curtalments n power system and natural gas network respectvely. As 12 of the generators n power system are drven by gas fuel, natural gas network s qute mportant to the safe and stable operatng of electrcty system. It s obvous from the followng fgures that gas lnes hardenng plays a sgnfcant role n mprovng reslence of ntegrated energy system. From the results shown n Fgs. 8 and 9, we can see that two gas ppelnes hardened and three gas ppelnes hardened has nearly the same gas and power load curtalments. However, compared wth none gas lne hardened and one gas lne hardened, load curtalments obvously decrease. Smlarly, 7 10 power transmsson lnes hardened also have nearly the same results. As the costs wll ncrease f more lnes are hardened, 7 power transmsson lnes and 2 gas ppelnes hardened s the most economc and effectve plan to protect ntegrated energy system from attacks.

10 Robust optmzaton for mprovng reslence of ntegrated energy systems wth electrcty Fg. 6 IEEE 118-bus power system L6, L9, L14 L7 L8, L11, L16 S S L5, L10, L15 L1, L12 S3 11 L2, L13 Fg node natural gas system 9 From the results for ths large-scale system, we can also draw that our proposed model and method s feasble to mprove and analyze reslence of ntegrated energy 8 L3 L4 7 Fg. 8 Load curtalments for IEEE 118-bus power system systems. Optmal plan for hardenng power and gas lnes can be derved from smulaton results, whch breaks the tradtonal vew that reslence of system wll mprove as long as the number of hardened lnes ncreases. When appled for large-scale system or more defense and attack budgets, our proposed method s more tme savng. Compared wth column and constrant generaton

11 1076 Hao CON et al. Table 7 Computaton tme for C&C n [16] and our proposed method n the case of IEEE 118-bus system Number of hardened lnes Computaton tme (s) C&C Proposed method ,210 15,647 Fg. 9 Load curtalments for 14-node natural gas system algorthm used n [16], computaton tme s much shorter as computatonal complexty ncreases. As stated n [24], C&C algorthm has dffcultes n operatng n reasonable tme when dealng wth large-scale real problems. Therefore, algorthms based on benders decomposton or L-shape method should be developed [24]. The computaton tme s shown n Tables 6 and Impact of energy storage system The concern on energy storage technologes s rapdly ncreasng due to hgh penetraton of renewable and ntermttent energy plug-n. As an effectve manner to mprove energy utlzaton effcency, energy storage systems can solve the problem of msmatch between energy supply and demand sde n tme and space [25]. The system dscussed n ths paper ntegrates electrcty and natural gas. Therefore, energy storage systems may nclude both electrcty and gas storage systems. In power system, storage system s an effectve manner to adjust peak. Excess power s stored nto storage devces at offpeak perod and stored energy wll be transported back to grd when load demand s at peak. Moreover, as the unpredctable and ntermttent nature of wnd, solar and other renewable energy generaton have great nfluence on reslence of power system, storage devces are usually nstalled nearby for trackng load changes [25]. However, electrcty has the feature of easy to transport but dffcult to store. Therefore, large-scale electrcty storage technology s stll a challenge. Compared wth electrcty, gas has the feature of easy to store but dffcult to transport. Natural gas transportaton costs manly depend on the volume of gas supply and transport dstance [26]. In fact, natural gas consumers are usually far away from gas sources and the cost for transportng s qute expansve, as a result of whch the storage facltes for natural gas s of vtal mportance. Ether n power system or natural gas system, energy storage s an effectve way to mprove the reslence for the ntegrated system. We place 3 electrcty storage devces at buses 21, 50, 96 n IEEE 118-bus power system and 2 gas storage devces at nodes 3, 5 n natural gas system. The capacty of each electrcty storage devces s 50 MW and the capacty of each gas storage devces s 1000 kcf. The results consderng energy storage systems s shown n Fgs. 10 and 11. Compared wth Fgs. 8 and 9, load curtalments n both electrcty and natural system decrease apparently. Wth 7 10 electrcty transmsson lnes and 2 3 gas transmsson lnes hardened, gas load curtalments are prevented thoroughly and electrcty load curtalments reduced by Table 6 Computaton tme for C&C n [16] and the proposed method n the case of IEEE 30-bus system Number of hardened lnes Computaton tme (s) C&C Proposed method Fg. 10 Load curtalments for IEEE 118-bus power system wth energy storage devces

12 Robust optmzaton for mprovng reslence of ntegrated energy systems wth electrcty 1077 lnk to the Creatve Commons lcense, and ndcate f changes were made. References Fg. 11 Load curtalments for 14-node gas system wth energy storage devces nearly 80%. The energy storage system plays a sgnfcant role n mprovng reslence of ntegrated energy system. In Secton 2.1, we have stated that we assume natural gas system operates n steady state and lne pack s gnored. Actually, large amounts of gas stored n ppelnes, such as lne-pack, can provde addtonal gas supply after the occurrence of contngences. In ths regard, the load sheddng results obtaned from steady-state model are conservatve. Certan amounts of load sheddng can be compensated by lne-pack and hence can be avoded. 5 Concluson Ths paper proposes a robust optmzaton model for reslent operaton of ntegrated energy system wth electrcty and natural gas nfrastructures. The proposed model s formulated as a three-stage optmzaton problem whch consders network renforcement, damage caused by attackers and defenders response of both electrcty and natural gas systems. Lnearzaton technologes and decomposton algorthms are appled to reformulate and solve ths defender-attacker-defender problem. Numercal results valdate the effectveness of our proposed model. Studes also pont to the mportance of energy storage systems n mprovng reslence of ntegrated energy system aganst contngences. In future works, we wll focus on the reslence of energy systems whch ntegrate other forms of energes and demand response management. And dstrbuton networks or mcogrds wll also be concerned n future works. Acknowledgements Ths work was supported by Natonal Natural Scence Foundaton of Chna (No ). Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( 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 [1] abbar HA, Bower L, Pandya D et al (2014) Reslent mcro energy grds wth gas-power and renewable technologes. In: Proceedngs of the 2nd IEEE conference on power engneerng and renewable energy, Bal, Indonesa, 9 11 Dec 2014, 6 pp [2] u Y, Ma Y, Zhang et al (2013) The optmzaton of the operaton program for natural gas ppelne transmsson and end segment storng gas. In: Proceedngs 2013 nternatonal conference on mechatronc scences, electrc engneerng and computer, Shengyang, Chna, December 2013, 5 pp [3] Unshuay C, Lma JWM, Souza ACZD (2007) Modelng the ntegrated natural gas and electrcty optmal power flow. In: Proceedngs of 2007 IEEE power engneerng socety general meetng, Tampa, USA, June 2007, 7 pp [4] edl M, Andersson (2007) Optmal power flow of multple energy carrers. IEEE Trans Power Syst 22(1): [5] Lu C, Shahdehpour M, Fu Y et al (2009) Securty-constraned unt commtment wth natural gas transmsson constrants. IEEE Trans Power Syst 24(3): [6] Shao C, Wang, Shahdehpour M et al (2017) An MILP-based optmal power flow n multcarrer energy systems. IEEE Trans Sustan Energy 8(1): [7] Nezamoddn N, Mousavan S, Erol-Kantarc M (2017) A rsk optmzaton model for enhanced power grd reslence aganst physcal attacks. Electr Power Syst Res 143: [8] The ReslNets. php/defntons#reslence [9] Khodae A (2014) Reslency-orented mcrogrd optmal schedulng. IEEE Trans Smart rd 5(4): [10] Chen J, Zhu Q (2017) A game-theoretc framework for reslent and dstrbuted generaton control of renewable energes n mcrogrds. IEEE Trans Smart rd 8(1): [11] Yuan W, Wang J, Qu F et al (2016) Robust optmzaton-based reslent dstrbuton network plannng aganst natural dsasters. IEEE Trans Smart rd 7(6): [12] Chen C, Wang J, Qu F et al (2016) Reslent dstrbuton system by mcrogrds formaton after natural dsasters. IEEE Trans smart grd 7(2): [13] Yao Y, Edmunds T, Papageorgou D et al (2007) Trlevel optmzaton n power network defense. IEEE Trans Syst Man Cybern Part C (Appl Rev) 37(4): [14] Zeng B, Zhao L (2013) Solvng two-stage robust optmzaton problems usng a column-and-constrant generaton method. Oper Res Lett 41(5): [15] Manshad SD, Khodayar ME (2015) Reslent operaton of multple energy carrer mcrogrds. IEEE Trans Smart rd 6(5): [16] Wang C, We W, Wang J et al (2017) Robust defense strategy for gas-electrc systems aganst malcous attacks. IEEE Trans Power Syst 32(4): [17] Correa-Posada CM, Sanchez-Martn P (2014) Securty-constraned optmal power and natural-gas flow. IEEE Trans Power Syst 29(4): [18] Chen S, We Z, Sun et al (2017) Identfyng optmal energy flow solvablty n electrcty-gas ntegrated energy systems. IEEE Trans Sustan Energy 8(2): [19] Mercado RR (2002) Natural gas ppelne optmzaton. Handbook of Appled Optmzaton. Oxford Unversty Press, Oxford

13 1078 Hao CON et al. [20] Correa-Posada CM, Sánchez-Martín P (2015) Integrated power and natural gas model for energy adequacy n short-term operaton. IEEE Trans Power Syst 30(6): [21] Alsac O, Stott B (1974) Optmal load flow wth steady-state securty. IEEE Trans Power Appar Syst PSA 93(3): [22] Ferrero RW, Shahdehpour SM, Ramesh VC (1997) Transacton analyss n deregulated power systems usng game theory. IEEE Trans Power Syst 12(3): [23] Power system test case archve. research/pstca/ [24] Zhao L, Zeng B (2012) An exact algorthm for two-stage robust optmzaton wth mxed nteger recourse problems. doc88.com/p html [25] Ma Y, Yang P, Zhou et al (2016) Research revew on energy storage technology. In: Proceedngs of 2016 IEEE nternatonal conference on mechatroncs and automaton, Harbn, Chna, 7 10 August 2016, 6 pp [26] Sh, Jng Y, Zhang et al (2009) Prospects of natural gas storage and transportaton usng hydrate technology n Chna. In: Proceedngs of th IEEE conference on ndustral electroncs and applcatons, an, Chna, May 2009, 5pp Hao CON receved the B.S. degree n the Department of Electrcal Engneerng from Shangha Jao Tong Unversty, Shangha, Chna, n 2014, where he s currently pursung the Ph.D. degree. Hs research nterests nclude Energy Internet, electrcty market, and power system optmzaton. Yang HE works at the State rd Henan Electrc Power Company. Hs research nterests nclude power system optmzaton and dstrbuton network operaton. u WAN receved the B.S. degree n Electrcal Engneerng from Southeast Unversty, Nanjng, Chna, n 2010 and the Ph.D. degree n Electrcal Engneerng from Shangha Jao Tong Unversty, Shangha, Chna, n Currently, he s wth the School of Electronc Informaton and Electrcal Engneerng, Shangha Jao Tong Unversty, Shangha, Chna. He s now a vstng postdoctoral researcher at Illnos Insttute of Technology, Chcago, IL, USA. Hs research nterests nclude optmal dspatch of power system wth large-scale renewable energes and optmzaton for smart grds. Chuanwen JIAN receved the M.S. and Ph.D. degrees from the Huazhong Unversty of Scence and Technology, Wuhan, Chna, n 1996 and 2000, respectvely. He was nvolved n post-doctoral research wth the Department of Electrcal Engneerng, Shangha Jao Tong Unversty, Shangha, Chna. He s currently a Professor wth the Department of Electrcal Engneerng, Shangha Jao Tong Unversty. He s currently nvolved n the research of reservor dspatch, power system analyss, electrcty markets, and power system economcs and optmzaton.