Optimization of Dynamic Reactive Power Sources Using Mesh Adaptive Direct Search

Transcription

1 Acceted by IET Generation, Transission & Distribution on 6/2/207 Otiization of Dynaic Reactive Power Sources Using Mesh Adative Direct Search Weihong Huang, Kai Sun,*, Junjian Qi 2, Jiaxin Ning 3 Electrical Engineer and Couter Science Deartent, University of Tennessee, Knoxville, TN, USA 2 Energy Systes Division, Argonne National Laboratory, Argonne, IL, USA 3 Doinion Virginia Power, Richond, VA, USA * kaisun@utk.edu Abstract: Dynaic reactive ower sources can be used to effectively itigate the fault-induced delayed voltage recovery (FIDVR) and transient voltage instability issues. When any var sources need to be installed at lanned locations, otiization of their sizes is a colicated nonlinear otiization roble due to its non-convexity and the deendence of the constraint on tie-series trajectories of ost-fault voltages. Solving this otial sizing roble needs to utilize both a nonlinear otiization solver and a ower syste differential-algebraic equation solver. This aer rooses a new aroach for solving this roble under both a single contingency and ultile contingencies by using the Mesh Adative Direct Search algorith interfaced with a ower syste siulator. The roosed aroach is validated by case studies on a North Aerican Eastern Interconnection odel to otiize the sizes of lanned STATCOMs against critical contingencies. Index Ters: Dynaic var sources, FIDVR, Mesh Adative Direct Search, reactive ower control Noenclature Variables are listed below in the order fro caital font, sall font, and Greek, alhabetically. D M l P l Q k Rj a atrix reresenting a fixed set of directions the set of trial oints on a esh at iteration l the set oll trial oints at iteration l the colun vector of the sizes of dynaic var sources in a ower syste () t the ercentage voltage deviation on bus j at tie t for the contingency k U l V c i c β l l the set of evaluated oints at the beginning of iteration l the vector of all voltage agnitudes in a ower syste the cost coefficient for Q i the enalty of voltage violation oll center esh size at iteration l oll size at iteration l

2 Acceted by IET Generation, Transission & Distribution on 6/2/207. Introduction In recent years, with the increasing integration of renewable energy resources and ower electronic devices in ower transission systes, there have been increased researches on reactive ower anageent, esecially the otial allocation of reactive ower sources. However, any studies were based steady state analysis in [-2]. With the growth of ower electronic devices and single-hase induction otors on the load side, dynaic voltage security robles, esecially fault-induced delayed voltage recovery (FIDVR) issues, on transission systes have drawn ore attentions by ower syste engineers with electric utilities. Steady state analysis on reactive ower sources cannot cature the dynaics of a ower syste under shortter or transient voltage stability robles. If the FIDVR issues are not addressed roerly, they ay lead to fast voltage collase or even cascading failures. To guarantee ower syste reliability and itigate FIDVR issues in both effective and econoical anners, it is iortant to study the otial sizing and siting of dynaic var sources such as static var coensators (SVCs) and STATCOMs. Several aers have studied the otial sizing roble of dynaic var sources addressing FIDVR issue [3-]. Paer [3] solves the otial locations and caacities of SVCs by the fuzzy clustering ethod and interior oint ethod on the IEEE 9-bus syste taking into account transient stability constraints. In [4], a ixed integer nonlinear rograing roble is forulated and the solution is obtained by interfacing the branch-and-bound and ulti-start scatter algoriths with ower syste tie-doain siulation software. A siilar roble is solved in [5] by interfacing an otiization tool KNITRO with ower syste tie-doain siulation software. In [6] and [7], the sizing of dynaic var sources is obtained by heuristic linear rograing and Voronoi diagra, resectively. The aroach in [8] eloys the ean-variance aing otiization in cobination with an integrated ix-integer search strategy and two intervention schees. The ethod in [9] alies a ulti-objective evolutionary algorith to otiize the sizes of dynaic var sources. Paer [0] solves the otial locations and sizes of static and dynaic var sources as a sequence of ixed integer rograing robles. In [], the article swar otiization is alied for the installation of dynaic var sources and wind turbines. The ain difficulties of the otial sizing roble of dynaic var sources lie in two asects: first, the roble is a nonlinear, non-convex otiization roble as indicated by the geoetric characteristics of its solution sace [2]; second, checking the constraints requires the ost-fault ower syste trajectories, which can only be obtained by solving an accurate ower syste differential-algebraic equation (DAE) odel, i.e. ower syste tie-doain siulation. Thus, the resolution of this roble requires both a nonlinear otiization solver and a ower syste DAE solver, which should be integrated by eans of an efficient interfacing algorith. Most of the existing ethods can achieve global otia only when the 2

3 Acceted by IET Generation, Transission & Distribution on 6/2/207 nuber of var sources is sall. For large-scale ower systes, these ethods ay either suffer fro heavy coutational burdens or can only converge to local otia. One way to tackle such a roble at a large scale is the blackbox otiization regarding all tasks on ower syste siulation and security constraint checking as a blackbox, which rovides an otiized outut, i.e. the sizes of dynaic var sources, for given inuts, i.e. the objective function and security constraints on ost-fault voltage recovery. As one of effective blackbox otiization ethods under general nonlinear constraints, the Mesh Adative Direct Search (MADS) algorith [3], [4] erfors well on otiizations only based on the results fro exensive couter siulations, in which the derivatives are available, containated noise can exist, or feasible solutions are not easy to find. In [5] and [6], the MADS algorith is alied to two otial laceent robles in ower systes: the otial PMU laceent for ower syste dynaic state estiation based on eirical observability graian and the otial laceent of dynaic var sources based on eirical controllability covariance. Those studies deonstrate the effectiveness of the MADS algorith in solving ractical otiization robles on large-scale ower systes. In this aer, an MADS-based ethod is roosed to otiize the sizes of dynaic var sources at redeterined locations. An interfacing rogra is develoed to wra ower syste siulation and the checking of ost-fault voltage recovery constraints as a blackbox so as to counicate with an MADS solver. The roosed aroach is validated on an Eastern Interconnection ower syste odel fro the Multiregional Model Working Grou (MMWG) reresenting an anticiated 209 suer eak load condition of the interconnection. The sizes of seven STATCOMs lanned in the Doinion Virginia Power (DVP) region are otiized for itigating FIDVR issues under given critical single and ultile contingencies. The roosed aroach can quickly converge to an otial solution after a sall nuber of iterative evaluations. 2. Proble forulation In ower systes, deterining the otial sizes of dynaic var sources is a nonlinear and non-convex otiization roble. Verification of constraints requires the ost-fault voltage trajectories, which are usually obtained fro tie-doain siulation. For instance, under a contingency, the syste s ostcontingency resonse with a given dynaic var injection should be siulated and checked based on the ost-fault voltage erforance criteria. For an N-bus ower syste subject to K critical contingencies, this otiization roble can be forulated as ()(5). 3

4 Acceted by IET Generation, Transission & Distribution on 6/2/207 Subject to I J c Q c Z K () i i k i k L Q Q Q U, i ~ I (2) i i i x f ( x, V, Q ) (3) 0 g( x, V, Q ) (4) S, S2,, SM (5) where Q=[Q,, QI] T is a colun vector of the sizes of dynaic var sources at I candidate buses with uer liits Qi L and lower liits Qi U (i=,, I), ci is the cost coefficient for Qi, c is the enalty of voltage violations, and Zk is a binary variable equal to 0 if the voltages of all buses eet the criteria for contingency k, or, otherwise. Equations (3)(4) reresent the DAEs of ower syste dynaics, x is the state vector, and V is a vector of all voltage agnitudes. S to SM in equation (5) reresent the given M criteria about ost-fault voltage recovery erforances for tie-doain siulation results. The roble is to find the otial Q iniizing the objective function J satisfying all constraints in (2) to (5). This aer rooses using the blackbox otiization aroach to solve this roble as illustrated in Fig., in which the blackbox utilizes a ower syste siulator (a DAE solver) to obtain the ost-fault voltage trajectories V under contingency k and a verification odule to check all criteria for V. The inut of the blackbox is Q, and the oututs are Zk and the objective function value J(Q). Based on the oututs fro the blackbox, the otiization algorith iteratively udates the inut Q of the blackbox until a stoing criterion is satisfied. Q Blackbox Otiization algorith J(Q) Z k Q L Q Q U Fig.. Scheatic rocedure of blackbox otiization for the otial sizing roble. In the criteria verification odule of the blackbox, the criteria on ost-fault voltage recovery erforance can be selected according to the WECC/NERC lanning standards [7], [8]. Assue that K contingencies ay otentially cause FIDVR issues. Define the ercentage voltage deviation on bus j at tie t for the contingency k as: 4

5 Bus Voltage Magnitude Acceted by IET Generation, Transission & Distribution on 6/2/207 init Vj() t V k j Rj ( t) 00%, j N, k K (6) init V j where N is the nuber of buses and Vj init is the re-fault initial voltage agnitude. We denote by tcl the fault clearing tie, ts the ost transient tie, and IL and IG resectively the sets of load and generator buses. Without losing the generality, consider these four criteria SS4 on the ost fault voltage trajectories: k S: R ( t) 30%, t t t, j I, k K j cl s G k S2: R ( t) 25%, t t t, j I, k K j cl s L k S3: R ( t) 5%, t t, j N, k K j s S4: Duration R 20% 20 cycles, tcl t ts, j IL. j For a load bus, the ost-fault voltage trajectory and criteria S2S4 are illustrated in Fig. 2, where criterion S2 liits the axiu transient voltage overshoot and di, S3 liits voltage derivations in the ost-transient eriod, and S4 liits the duration of a transient voltage di. Initial voltage Maxiu transient voltage overshoot (Violating S 2 ).25V init.2v init Checking S 3 criteria.05v init V init Tie duration 20 cycles Checking S 4 constraint 0.75V init 5V init Post transient eriod Transient eriod t cl Fault cleared Fig. 2. Post-fault voltage erforance criteria for a load bus. Tie/s t s The MADS algorith is selected as the otiization algorith in this aer. It is a useful frae based ethod for blackbox otiization under general nonlinear constraints. In the next two sections, we will 5

6 Acceted by IET Generation, Transission & Distribution on 6/2/207 introduce the MADS algorith and then roose an MADS based aroach to solve the otial sizing roble of dynaic var sources. 3. Introduction of the MADS algorith The MADS algorith iterates on a tower of underlying eshes in the searching sace by controlling the refineent of discretization of the sace of variables [9]. It ais at handling general nonlinear constraints in a coutationally effective way. Constraints ay be of several tyes, including blackboxes, nonlinear inequalities, and yes/no or hidden constraints. Variables ay be integer, binary, or categorical. α 3 α 6 α β l β l+ α 4 α 5 α 2 (a) α 7 α 8 (b) α 9 00 α β l+2 β l+3 0 α α 3 (c) (d) Fig. 3. MADS algorith based blackbox otiization a =, = l b l l =/4, l =/2 l =/4, l2 =/2 l =/6, l3 =/4 c 2 d 3 Fig. 3 illustrates how a two-diensional non-convex Goldstein-Price function is otiized by MADS algorith without utilizing any gradient inforation on the function. The Goldstein-Price function has four local inia and is often used as a benchark roble for testing the erforances of otiization 6

7 Acceted by IET Generation, Transission & Distribution on 6/2/207 algoriths [20]. In Fig. 3, the value of the function in the searching sace is colored as a contour a. The colder the color, the saller the value of the objective function. The global otiu solution is arked by a star. Blackbox functions are evaluated at soe trial oints on a esh whose discrete structure at iteration l is defined by: where M n { Dz : z D } (7) l U l l l R + is the esh size, Ul is the set of oints where the objective function and constraints have been evaluated at the beginning of iteration l, and D is a n nd atrix reresenting a fixed finite set of nd directions in R n. In the Fig. 3(a), D is equal to In other words, Ml defines esh through the lattices sanned by the coluns of D, centered around β which belongs to the set Ul. Defining the esh in this way ensures all revious visited oints Ul lie on the esh, and new trial oints can be selected around any of the using the directions in D. Each iteration is coosed of three stes: oll, search, and udate. The search ste is flexible and allows the creation of trial oints anywhere on the esh while the oll is ore rigidly defined to exlore the esh near the current iterate βl with the following set of oll trial oints: P { d : d D } M (8) l l l l l where Dl is the set of oll directions. Each colun of Dl is an integer cobination of the coluns of D. In the Fig. 3(a), Dl is the subset of D which is st, 3 rd, and 6 th coluns of D and oints of Pl are α, α2 and α3. Points of Pl s distance to the oll center βl is constrained by the oll size satisfy ) l is always saller than l ; and 2) oints are generated on the esh at distance ossible choices for trial oints gradually increases. l is reduced faster than l. The esh and oll size ust l after failures. The oll trial l fro the oll center, which ilies that the nuber of The oll and search stes generate trial oints on the esh. The blackbox functions are evaluated at these oints. At the end of iteration l, an udate ste deterines the iteration status and then the next iterate βl+ is chosen. The esh size araeter is udated with w l (9) l l where τ > is a fixed rational nuber and wl is a finite integer, ositive or null if iteration l is a success, or strictly negative if the iteration fails. 7

8 Acceted by IET Generation, Transission & Distribution on 6/2/207 The relation between the esh and oll size araeters is illustrated in Fig. 3, in which thin lines reresent the esh of size l is equal to l and thick lines refer to the oints at distance l fro βl. As in the Fig. 3 (a), l and the trial oints α, α2, and α3 lie at the intersection of the thick and thin lines in n+ (here 2+=3) rando directions. When the three trial oints fail to find a better solution, l is reduced to /4 and the trial oints α4, α5, and α6 are chosen in three rando directions at distance l =/2. These trial oints successfully find a better solution at Fig. 3(b) search at α5, so l2 and l2 kee still, and the βl+2 ove to the α5 then evaluate the neighboring oints and find better solution. Three trial oints fail to find a better solution at α7, α8, α9, so l3 is reduced to /6 which is faster than the decrease of l3 =/4, and the nuber of candidate locations increases shown as Fig. 3(d). Three trial oints successfully find a better solution at α, the oll and search size kee still in the next ste. Iteratively, do the search and oll, the solution toward the otial until eet the sto criterial, such as total search ste is less than certain nuber, or oll size is less than a certain value. Rearks: Frist, regarding the selection of araeters in the MADS algorith, it is suggested that the axiu nuber of iterations be 0 4 and the iniu of the esh size in each diension be 0-7 ultilied by the range of search [2] in order for the global otial solution to be found at a high robability. However, that ay increase the tie cost of otiization. Second, regarding the selection of the initial guess and erforance of otiization, the MADS algorith erfors well in finding global otia of sallscale non-convex otiization robles or the best solutions known so far for soe difficult large-scale robles whose global otia have not yet been verified. In [22], the MADS algorith is tested on three real-life difficult robles including the Styrene roduction siulation [2], the ultidiscilinary design otiization [23], and the well laceent roble [24]. The use of the Variable Neighborhood Search ethod fro [2] can ake the otiization be indeendent of the selection of an initial guess. As deonstrated in [22], the MADS algorith is run starting fro both infeasible and feasible solutions as initial guesses and the otiizations all lead to the best-known solutions given so far in literature. The MADS algorith illustrated above can be alied to blackbox otiization for the sizes of dynaic var sources. In the case study section, the roosed MADS-based aroach is tested on the Eastern Interconnection syste odel. In the following, we first illustrate the iterations of blackbox otiization on the sae roble as [3] on the WSCC 9-bus syste for the otial sizes of two SVCs. In Fig. 4, the entire solution sace is colored as a contour a based on actual objective function values at different oints calculated by the exhaustive search ethod. The actual global otiu solution 8

9 Acceted by IET Generation, Transission & Distribution on 6/2/207 is arked by star. Fig. 4(a) to Fig. 4(d) illustrate how MADS finally identifies the global otiu by four iterations of searches, olls and udates. The tentative otiu starts fro βl in Fig. 4(a). Due to the failure to find a better solution in the search and oll ste, tentative otiu βl+ and βl+2 stay at βl, and the eshes and olls in Fig. 4(b)(c) shrink to generate ore trial oints. When finding a better solution in Fig. 4(c), then a tentative otiu βl+3 jus to α8 and the esh and oll size kee constant in Fig. 4(d), and finally a tentative otiu goes to α, which becoes closer to the global otiu. Such a rocedure will gradually aroach the global otiu and does not exlicitly require any gradient inforation regarding the objective function. α 3 α α 6 β α 7 l β l+ α 4 α 5 α 9 β l+2 α α 2 0 α β l+3 8 α α 2 (a) (b) (c) (d) x k+ Fig. 4. MADS algorith based blackbox otiization a =, = l b l l =/4, l =/2 l =/6, l2 =/4 l =/6, l3 =/4 c 2 d 3 4. Proosed MADS-based aroach The roosed MADS-based aroach for otiizing the sizes of dynaic var resources ackages ower syste siulation and criteria checking odules in one blackbox. Fig. 5 illustrates the flowchart of the roosed aroach, in which the MADS algorith is interfaced with a ower syste DAE solver (e.g. Sieens PTI PSS/E). DAEs (3)(4) are solved to obtain the ost-fault voltage resonses. The results on the checking of voltage criteria are sent to the MADS algorith through a data interface (e.g. Python). Secifically, it includes the following stes: 9

10 Acceted by IET Generation, Transission & Distribution on 6/2/207 Ste. Predeterine the locations of dynaic var sources in the ower syste. A widely adoted aroach is to calculate a voltage sensitivity index for each candidate location and select locations that have the largest overall average voltage iroveent on voltage trajectories under the ost severe contingency [3]. Another one is the eirical controllability covariance based ethod roosed in [6], which has less deendency on the selection of contingencies. Ste 2. Set initial values for dynaic var sources Q as well as their uer and lower liits. The initial values could coe fro either linear analysis result as done in [2] or rando sale oints. Ste 3. Perfor the search and oll stes of the MADS algorith, and ass the Q to the blackbox. Ste 4. In the blackbox, the data interface receives the Q fro the MADS algorith and uses Q as new caacities of STATCOMs in the ower syste DAE odel to be solved. Once receiving new results fro the DAE solver, the data interface checks all voltage trajectories V with the lanning standard S and feeds back the result of Zk and J(Q). Ste 5. Evaluate objective function J to find an iroved esh oint βl+ on the esh Ml; coarsen the esh or stay with revious βl+= βl. Then refine the esh and udate l, l, and Ul+. Ste 6. Check the stoing criterion: the total nuber of iterations l is ore than a certain nuber or l is less than a certain value indicating a convergence of the tentative otiu. If it is et, sto the rocedure; otherwise, go back to Ste 3. Ste 4. Detailed load odel, STATCOMs odel, Wind turbine Model, Generators, etc Alication Progra Interface Q Solve the DAE of the Power syste Black box V Checking Voltage with NERC/WECC Criteria J(Q) Z k Q L Q Q U Ste. Predeterined Locations of Dynaic Var Suorts Ste 2. Start with Initial Value MADS Algorith Q Ste 3. Perfor the search and oll stes No Ste 6. Does it eet search stoing criteria? Yes Ste 5. Find a better solution and coarsen the esh or refine the esh Final solution on var locations and aounts Fig. 5. Flow chart for ileenting the roosed aroach 0

11 Acceted by IET Generation, Transission & Distribution on 6/2/ Case Study In this section, the roosed MADS-based aroach is alied to otiize the sizes of seven STATCOMs laced in the DVP region of an Eastern Interconnection syste odel that reresents an anticiated 209 Suer Peak Load condition. The entire odel has buses and 8847 generators including 237 wind turbine generators. The DVP region has 93 generators and 066 buses and this eak load condition has totally MW active load and 544 Mvar reactive load. Several 500 kv level N- contingencies are siulated to validate the roosed ethod. Each contingency corresonds to a three-hase fault at one terinal of a 500 kv line which is tried after 5 cycles. The total siulation eriod is 30 seconds, the tie for ost transient voltage liits S3 checking is last 3 seconds of the eriod, and the tie for transient eriod criteria S, S2, and S4 checking is between the fault cleared and ost transient eriod. We assue c=$,000/mvar and c=$,360,000; i.e. all STATCOMs have an identical cost and all buses have an identical enalty for violating the voltage criteria. In this case, the objective is equivalent to iniizing the total aount of dynaic var suorts to eet the voltage criteria. The ower syste DAE solver adots Sieens PTI PSS/E 32, and the MADS algorith is ileented by the NOMAD (Nonlinear Otiization by MADS in MATLAB) solver [4], which counicate through an interfacing rogra develoed in Python. 5.. Load and STATCOM odelling In the tie-doain siulation, a user defined coosite load odel CMLDBL and a STATCOMS odel SVSMO3U rovided by PSS/E [25] are alied to siulate FIDVR issues. When a STATCOM reaches its liit, it behaves as a constant current source, or in other words, its reactive ower is roortional to the voltage. It can also regulate voltage by controlling echanically switched shunts (MSS), such as echanically switch caacitors (MSC) or reactors (MSR) available in the syste. The SVSMO3U odel is a voltage source converter (VSC) based generic user-defined static var syste (SVS). It is reresented in the ower flow case as a FACTS device. The CMLDBL load odel, as shown in Fig. 6, is used for ower syste lanning and oeration studies in PSS/E. It consists of three-hase otors, a single-hase air conditioner otor, electronic loads and static loads. It is connected to a low-voltage load bus whose dynaic resonse is reflected at the high voltage syste bus. The araeters of the coonents used in this study are listed in Table I and the user can also define other araeters, such as stator resistance, otor breakdown value, etc.

12 Static Electronic Motor D--hase Motor C-3-hase Motor B-3-hase Motor A-3-hase Acceted by IET Generation, Transission & Distribution on 6/2/207 High voltage syste bus jxxf Substation Low-side :T Transforer Bss Substation shunt Rfdr+jXfdr FbBfdr Distribution Feeder Equivalent (-Fb)Bfdr Far end load bus M M M AC Load Model Coonents Fig. 6. User defined coosite load odel. Table I Paraeters for user-written coosite load odel Paraeter Value 3-hase otor A, % 20 3-hase otor B, % 6 3-hase otor C, % 6 Single hase AC otor, % 20 Electronic load, % 3 Static load, % DVP Syste An encoded one-line diagra reresenting a local DVP transission network is shown in Fig. 7. The laceents of seven STATCOMs are redeterined at the 500 kv, 230 kv, and 5 kv levels (indicated by red circles). The uer liits of STATCOM to STATCOM 7 are 50 Mvar, 50 Mvar, 60 Mvar, 50 Mvar, 250 Mvar, 250 Mvar, and 250 Mvar, resectively. STATCOMs, 2 and 3 are connected to bus 09, whose sizes are denoted by Q, Q2, and Q3, and the forer two are at the 230 kv level and the last one is at 2

13 Acceted by IET Generation, Transission & Distribution on 6/2/207 the 5 kv level. STATCOM 4 with size Q4 is connected to bus 27 at 230 kv level and STATCOMs 5, 6, and 7 with sizes of Q5, Q6, and Q7 are connected to bus 7, bus 3, and bus 30 at the 500 kv level, resectively Q 5 02 Q Q Q 2 Q Q 6 Q Fig. 7. Part of encoded Doinion syste (soe iortant buses are highlighted in the following discussion) Case : Single contingency This test otiizes the sizes of STATCOMS for the ost severe N- contingency, which is a threehase fault on bus 27 and cleared by triing line 2728 after 5 cycles. The FIDVR issue is observed in the siulation by checking criteria at ste 4 as shown in Fig. 8(a). Ten bus voltage rofiles with the ost severe FIDVR issue are selected fro Fig. 8(a) and shown in Fig. 8(b), where one can see that the voltage agnitudes dro to below 75% of the initial voltage at the clearance of the fault violating S2 and reain under 80% of initial voltage for ore than 30 cycles violating S4. 3

14 Post transient Voltage/u Voltage/u Post transient Voltage/u Voltage/u Acceted by IET Generation, Transission & Distribution on 6/2/207 V U =.25V init V U =.25V init.2 <20 cycles.2v init.2.2v init..05v init V init Duration R 20% V init V init >30 cycles <20 cycles V L =0.75V init 0.7 V L =0.75V init Violating S 2 : Transient eriod R 25% Tie/s Tie/s (a) (b) Fig. 8. Voltage resonses of the case contingency without STATCOMs a All buses voltage resonses of the case contingency without STATCOMs coensation b 0 lowest voltage resonses of the case contingency without STATCOMs coensation.2 V U =.25V init <20 cycles.2v init.2 V U =.25V init.2v init..05v init. 5V init V init <20 cycles V L =0.75V init 0.7 V init V L =0.75V init Transient eriod Tie/s Tie/s (a) (b) Fig. 9. Post-fault voltage resonse with otiized STATCOMs for contingency. a All buses voltage resonses of the case contingency with otiized STATCOMs b 0 lowest voltage resonses of the case contingency with otiized STATCOMs 30 Using one deskto couter with Intel i rocessor, the roosed aroach takes 4 hours in total to give the final otial solution. The rocess runs PSS/E to conduct 240 to 250 tie-doain siulations on the contingency based on ultile individual tests on the aroach. Each siulation takes about 0 inutes due to the details and size of the syste. The final otial solution has Q=50 Mvar, 4

15 Post transient Post transient Post transient Acceted by IET Generation, Transission & Distribution on 6/2/207 Q2=50 Mvar, Q3=20 Mvar, Q4=50 Mvar, Q5= 70Mvar, Q6=250 Mvar, and Q7=0 Mvar with the objective function equal to $,00,000. Fig. 9 shows the voltage trajectories with the otiized dynaic var suort, checking with criteria at ste 4, which reain above 75% of initial voltage and have durations between 75% and 80% of initial voltage less than 20 cycles. In the steady stage, the lowest voltage stays above 95% of its initial value as circled in Fig. 9(b) Case 2: Multile contingencies This case assues three severe N- contingencies: triing line 2324 due to a three-hase fault on bus 23, triing line 2327 due to a three-hase fault on bus 23, and triing line 0927 due to a three-hase fault on bus 09. All three faults last for 5 cycles. The dynaic var sizing roble needs to find the sizes of seven STATCOMs to iniize the total cost while eeting the voltage criteria for all three N- contingencies. The roosed aroach totally takes 23 hours to find the final otial solution on the deskto couter. The whole rocess conducts 240 to 250 tie-doain siulations on each of the three contingencies and each siulation takes about 0 inutes. Siulations on the ultile contingencies can be arallelized to reduce the total tie cost. The final objective reaches $940,000 at 30 Mvar, 40 Mvar, 50 Mvar, 60 Mvar, 230 Mvar, 230Mvar, and 0 Mvar in the searching sace for Q to Q7, resectively. Figs. 0 illustrate the bus voltage rofiles in tie doain without and with the otiized dynaic var suorts Violating S 4 criteria Violating S 4 criteria Violating S 4 criteria Duration R 20% =43 cycles >20 cycles Duration R 20% =65 cycles >20 cycles Duration R 20% =55 cycles >20 cycles Violating S 2 criteria: R 25% Transient eriod Transient eriod Transient eriod (a) (b) (c) Fig. 0. Voltage resonses of the case 2 contingencies without STATCOMs a Voltage resonse of the line outage of line 2324 by a three-hase-to-ground fault on bus 23 b Voltage resonse of the line outage of line 2327 by a three-hase-to-ground fault on bus 23 c Voltage resonse of the line outage of line 0927by a three-hase-to-ground fault on bus 09 5

16 Post transient Post transient Post transient Acceted by IET Generation, Transission & Distribution on 6/2/ Duration R 20% <7 cycles<20 cycles Duration R 20% <7 cycles <20 cycles Duration R 20% <7 cycles<20 cycles Transient eriod 0.7 Transient eriod 0.7 Transient eriod (a) (b) (c) Fig.. Post-fault voltage resonse with otiized STATCOMs for case 2 contingencies. a Voltage resonse of the line outage of line 2324 by a three-hase-to-ground fault on bus 23 b Voltage resonse of the line outage of line 2327 by a three-hase-to-ground fault on bus 23 c Voltage resonse of the line outage of line 0927by a three-hase-to-ground fault on bus 09 For the line 2324 contingency, the syste resonses without any var suort and with otiized dynaic var suorts are shown in Fig. 0(a) and Fig. (a) for coarison. In Fig. 0(a), soe bus voltages dro to the zone of 75% to 80% of initial voltage for 43 cycles, which violates S4 the 20 cycles liit. In addition, soe bus voltages dro to less than 75% of initial voltage after the fault is cleared violating S2. However, with the otiized var suorts, Fig. (a) shows that soe voltages stay in the zone of 75% to 80% of initial voltage for less than 7 cycles and never dro below 75% of initial voltage. In other words, all voltage trajectories eet the voltage criteria S-S4 with STATCOMs of the otiized sizes. For the line 2327 contingency, soe bus voltages enter the zone of 75% to 80% of initial voltage for ore than 65 cycles if STATCOMs are not added as shown in Fig. 0(b). This is violating the voltage criteria S2 by checking voltage criteria at ste 4. With otiized STATCOMs, that tie duration reduces to less than 7 cycles and eets the criteria S-S4 as shown in Fig. (b). Finally, for the line 0927 contingency, soe bus voltages violate the criteria S2 due to entering the 75% to 80% of initial voltage zone for 55 cycles if STATCOMs are not added as shown in Fig. 0(c). With the otiized STATCOMs, all voltage trajectories eet the voltage criteria S-S4 as shown in Fig. (c) Sensitivity to changes of load This section studies the sensitivity of the results fro the roosed aroach to changes of load. The DVP area load is first decreased by 5% fro the eak load condition to 2688 MW and 569 Mvar. But the single contingency case and the ultile contingencies case are siulated using the sizes of STATCOMs otiized for the base loading condition. No FIDVR issue is caused. 6

17 Acceted by IET Generation, Transission & Distribution on 6/2/207 Although the otiization with the roosed aroach is actually erfored on the suer eak load condition of the syste, in order to test the sensitivity of the obtained otial solution to a change of the load, the load of the DVP area is further increased by 5% fro the eak load condition to 2397 MW and 573 Mvar. Both the single contingency and ultile contingencies case are tested. The single contingency case still has FIDVR issue due to violations of S2 and S4. All of voltage trajectories in the syste are illustrated in Fig. 2(a) with details in the tie window fro.083 s to.7 s as shown in Fig. 2(b). Voltages at soe buses dro to 0.75 u, which violate the criterion S2. The lowest bus voltage trajectory violates criterion S4, i.e. the duration of voltage between 0.75 u and u exceeding 27 cycles. For the case 2 with ultile contingencies, only one of the three contingencies, i.e. triing lines with three-hase fault on bus 23, still causes a FIDVR issue as shown in Fig. 3(a) while the other two contingencies do not bring any FIDVR issue. Fig. 3(b) enlarges Fig. 3(a) for the tie eriod fro.083 s to.6 s. Soe buses violate criterion S2 and the lowest voltage trajectory violates the criterion S4. Therefore, a 5% load increase on the eak load condition ay cause the otial solution not to guarantee satisfactions to all criteria. These results atch well with the conclusion in [2]: the otial solution for a heavy-load condition can reain the feasibility under a light-load condition. Therefore, an advisable strategy is to consider the eak load condition with a sufficient ercentage of otor loads in siulation and otiization of dynaic var sources as this case study does with the MMWG 209 suer eak load condition and large ercentage of otor load Violating S 4 criteria Duration R 20% >27 cycles >20 cycles Violating S 2 criteria: R 25% (a) Fig. 2. Voltage resonses of the case contingency with a 5% load increase in DVP a Voltage resonses of all buses b Detailed voltage resonses in.083 s to.700 s (b) 7

18 Acceted by IET Generation, Transission & Distribution on 6/2/ Violating S 4 criteria Duration R 20% >22 cycles >20 cycles Violating S 2 criteria: R 25% (a) Fig. 3. Voltage resonses following the line outage due to the fault on bus 23 of case 2 contingencies with a 5% load increase in DVP a Voltage resonses of all buses b Detailed voltage resonses in.083s to.600 s (b) 6. Conclusions This aer rooses an MADS-based blackbox otiization aroach for the otial sizing roble of the dynaic var sources, which efficiently interfaces a ower syste DAE solver with the MADS algorith. The roosed aroach can assist ower syste lanning engineers in otiizing the sizes of dynaic var sources at redeterined locations. Otial allocation of dynaic var sources such as SVCs and STATCOMs is an iortant and tie-consuing task at any electric utilities that have otential FIDVR issues or short-ter voltage stability issues based on the NERC/WECC reliability criteria on dynaic voltage erforance. The roosed aroach has been successfully validated on a detailed North Aerican Eastern Interconnection odel having buses and 8847 generators. Although the test focuses on the Doinion Virginia Power region to otiize seven STATCOMs, the tie-doain siulations were erfored on the whole syste odel to obtain realistic syste resonses and rovide useful inforation to syste lanning engineers. Coutation of the whole aroach for a single contingency can be finished in tens of hours on one deskto couter. With high-erforance couters, the aroach ay also have otentials to suort ower syste oerations such as otiizing the settings of installed dynaic var sources hourly or even every 0-5 inutes to be reared for anticiated contingencies based on the current syste condition. That will be studied by our future research. 8

19 Acceted by IET Generation, Transission & Distribution on 6/2/ Acknowledgents 8. Biograhies 9. References [] Fang, X., Li, F., Wei, Y., Azi, R.: Reactive ower lanning with high enetration of wind energy using Benders decoosition, IET Gener. Trans. Distrib., 205, 9, (4), [2] Duan, C., Fang, W., Jiang, L., Niu, S.: FACTS devices allocation via sarse otiization, IEEE Trans. Power Syst., 206, 3, (2),. 308 [3] Wang, Y., Chen, H., Li, F., Y.: Reactive ower lanning with transient rocess stability constraint, IEEE PES General Meeting. Denvor, CO, 205 [4] Tiwari, A., Ajjarau, V.: Otial allocation of dynaic VAR suort using ixed integer dynaic otiization, IEEE Trans. Power Syst., 20, 26, (), [5] Paraasiva, M., et al.: Dynaic otiization based reactive ower lanning to itigate slow voltage recovery and short ter voltage instability, IEEE Trans. Power Syst., 203, 28, (4), [6] Huang, W., Sun, K., Qi, J., Xu, Y.: A new aroach to otiization of dynaic reactive ower sources addressing FIDVR issues, IEEE PES General Meeting, National Harbor, MD, 204 [7] Huang, W., Sun, K., Qi, J., Xu, Y.: Voronoi diagra based otiization of dynaic reactive ower sources, IEEE PES General Meeting. Denvor, CO, 205 [8] Wildenhues, S., et al.: Otial allocation and sizing of dynaic var sources using heuristic otiization, IEEE Trans. Power Syst., 205, 30, (5), [9] Y. Xu, et al.: Dynaic otiization based reactive ower lanning to itigate slow voltage recovery and short ter voltage instability, IEEE Trans. Power Syst., 204, 29, (6), [0] Liu, H., Krishnan, V., McCalley, J. D., Chowdhury, A.: Otial lanning of static and dynaic reactive ower resources, IET Gener. Trans. Distrib., 204, 8, (2), [] Hussain, Z., Chen, Z., Thogersen, P., Lund, P.: Dynaic reactive ower coensation of large-scale wind integrated ower syste, IEEE Trans. Power Syst., 205, 30, (5), [2] Huang, W., Sun, K., Qi, J., Ning, J.: Otial allocation of dynaic var sources using Voronoi diagra ethod integrating linear rograing, IEEE Trans. Power Systes, acceted [3] Digabel, S. Le.: Algorith 909: NOMAD: Nonlinear otiization with the MADS algorith, ACM Trans. Matheatical Software., 20, 37, (4) [4] Audet, C., Dennis, J. E.: Mesh adative direct search algoriths for constrained otiizaiton, SIAM J. Oti., 2006, 7, (), [5] Qi, J., Sun, K., Kang, W.: Otial PMU laceent for ower syste dynaic state estiation by using eirical observability graian, IEEE Trans. Power Systes, 205, 30, (4),

20 Acceted by IET Generation, Transission & Distribution on 6/2/207 [6] Qi, J., Huang, W., Sun, K., Kang, W.: Otial laceent of dynaic var sources by using eirical controllability covariance, IEEE Trans. Power Systes, 207, 30, (), [7] Shou, D. J., Paserba, J. J., Taylor, C. W.: A survey of current ractices for transient voltage di/sag criteria related to ower syste stability, IEEE PES PSCE., Oct [8] NERC/WECC Planning Standards, WECC, Ar [9] Audet, C., Custόdio, A. L., Dennis, J. E., Erratu, Jr.: Mesh adative direct search algoriths for constrained otiizaiton, SIAM J. Oti., 2008, 8, (4), [20] Ali, M. M., Khoatraorn, C., Zabinsky, Z. B.: A nuerical evaluation of several stochastic algoriths on selected continuous global otiization test robles, J. Global Oti., 2005, 3, [2] Audet, C., Béchard, V., Le Digabel, S.: Nonsooth otiization through esh adative direct search and variable neighborhood search, J. Glob. Oti., 2008, 4(2), [22] Audet, C., Dennis, J.: Globalization strategies for Mesh Adative Direct Search, Cout. Oti. Al., 200, 46, [23] Sobieszczanski-Sobieski, J., Agte, J.S., Sandusky, R.R. Jr.: Bi-level integrated syste synthesis (BLISS), Technical Reort NASA/TM , NASA, Langley Research Center, Aug.998 [24] Fowler, K.R., Reese, J.P., Kees, C.E., Dennis, J.E. Jr., Kelley, C.T., Miller, C.T., Audet, C., Booker, A.J., Couture, G., Darwin, R.W., Farthing, M.W., Finkel, D.E., Gablonsky, J.M., Gray, G., Kolda, T.G.: Coarison of derivativefree otiization ethods for groundwater suly and hydraulic cature counity robles, Adv. Water Resour., 2008, 3(5), [25] Sieens PTI Power Technologies Inc., PSS/E 32, Progra Alication Guide, 200, V 20