2624 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6 .
2624IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6, NOVEMBER 2016Optimal Operation Mode Selectionfor a DC MicrogridWann-Jiun Ma, Member, IEEE, Jianhui Wang, Senior Member, IEEE, Xiaonan Lu, Member, IEEE,and Vijay Gupta, Member, IEEEAbstract—This paper considers an optimal control problemto improve dc microgrid stability while minimizing its operation cost. A dc microgrid consists of various components,such as renewable energy sources, loads, and power lines.Every component may change its role during operation byswitching to a different mode in real time. A switched systemapproach is employed to ensure the stability of a dc microgrid with a rich array of operation modes. Meanwhile, anoptimal control algorithm is designed to improve the systemperformance by appropriately selecting the operation modes.A typical dc microgrid with three source buses and one loadbus is implemented. The effectiveness of the algorithms is verified by MATLAB/Simulink time-domain tests and numericalstudies.Index Terms—DC, microgrids, switched system, optimalcontrol.I. I NTRODUCTIONICROGRIDS was proposed several years ago and havebeen intensively studied these years [1]–[3]. They bringenergy sources close to electricity users by supplying energyfrom distributed generation sources. A microgrid can operatein a grid-connected mode or in a stand-alone fashion. When apower outage occurs, a microgrid can automatically disconnectwith the main grid. With the supply from the distributed generation sources, it can operate in a stand-alone mode withoutinterruptions.Most microgrids are designed based on an alternating current (AC) system. Thus, the issues that exist in an ACdistribution system such as reactive power control, and frequency control may degrade the system performance of anAC microgrid. Recently, a direct current (DC) microgridhas been proposed to increase power distribution qualityMManuscript received December 10, 2014; revised May 24, 2015,September 20, 2015, and December 19, 2015; accepted December 23, 2015.Date of publication January 25, 2016; date of current versionOctober 19, 2016. The work of W.-J. Ma and V. Gupta was supportedby the National Science Foundation under Grant 1239224. The work of J.Wang and X. Lu was supported by the Office of Electricity of the U.S.Department of Energy. Paper no. TSG-01210-2014.W.-J. Ma is with the Department of Mechanical and Materials Science,Duke University, Durham, NC 27708 USA (e-mail: wann.jiun.ma@duke.edu).J. Wang and X. Lu are with the Energy Systems Division, ArgonneNational Laboratory, Lemont, IL 60439 USA (e-mail: jianhui.wang@anl.gov;xiaonan.lu@anl.gov).V. Gupta is with the Department of Electrical Engineering, University ofNotre Dame, Notre Dame, IN 46556 USA (e-mail: vgupta2@nd.edu).Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSG.2016.2516566(see, e.g., [4]–[9]). A DC microgrid has a natural interface with renewable energy sources (RESs) and loads withDC coupling, which enhances the system performance during operation by shortening the power conversion chain andavoiding using most of the DC-AC and AC-DC conversionstages [10]–[13].This paper focuses on optimizing DC microgrid operation.A DC microgrid consists of various components such as RESs,loads, and power lines connecting sources and loads. From thesupply side, photovoltaic (PV) power generation is consideredto be one of the typical RESs in a DC microgrid. The highvariability of PV power generation complicates the control ofDC microgrid operation. For example, the effect of passingclouds on the PV arrays results in unpredictable variation ofthe power generation. As the interface circuits, PV inverteris operated in maximum power point tracking (MPPT) modeto maximize the renewable power generation [14]. By executing MPPT, a maximum amount of power is extracted fromPV panels. However, it is not always appropriate to let thePV power source operate in the MPPT mode because a maximum power generation may introduce an excessive amountof energy which may affect the stability of an islanded microgrid [14]. If this is the case, the PV power source can beswitched to an output voltage control mode to maintain thevoltage stability of the microgrid [14]. A battery is anotherexample of component in a microgrid which has different operation modes. In order to achieve optimal life time, a batteryshould be charged or discharged in a regulated fashion [14].Also, a battery can act as a control unit to maintain the voltage stability of a microgrid in both charging and dischargingmodes. Therefore, from both the supply and demand sides, different components may play different roles in the microgridduring operation, and the components may change their rolesdepending on the actual situation in real-time, which resultsin a rich array of operation modes.In this paper, a switched system approach is adopted andan optimal control algorithm is designed to regulate microgrid operation with various operation modes. A switchedsystem framework has gained tremendous popularity in thecontrol community; see, e.g., [15] and references therein. Thisframework generalizes linear and nonlinear systems. It models a dynamical system by allowing it to switch from onemode to other modes in real-time. Each mode representsa particular linear or nonlinear system. The switched system framework characterizes a dynamical system with variousoperation modes. Here, this modeling framework is adopted toc 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.1949-3053 See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
MA et al.: OPTIMAL OPERATION MODE SELECTION FOR A DC MICROGRIDthe DC microgrid with different operation modes and a suitable controller is designed to improve the performance of themicrogrid based on this framework.Of particular interest to our work is the supervisory energymanagement system proposed in [14]. The supervisory systemcontrols the operation modes in a coordinated manner suchthat the charging and discharging of batteries are in a regulated fashion. The output power of a PV source is extractedappropriately while maintaining the voltage stability of themicrogrid. However, the control law in [14] is based on aheuristic rule. Despite the fact that the simulation results verify the effectiveness of the algorithm, there is no analyticalperformance guarantee about the algorithm. In this paper, aswitched system framework is adopted to analyze the stabilityof the microgrid with various operation modes and to provide a performance guarantee for the proposed controller byoptimally selecting the operation modes.The remainder of the paper is organized as follows.Section II presents the model of the DC microgrid. Section IIIanalyzes the stability of the microgrid. The optimal controllerdesign for the microgrid is presented in Section IV. Simulationresults can be found in Section V. Section VI concludes thepaper.II. M ODELING OF DC M ICROGRIDA microgrid consists of different components such as renewable sources (e.g., solar PV source), loads (e.g., electricappliance, battery, etc.), and power lines. Figure 1 showsthe structure of a node (bus) in the microgrid. This sectionformulates each component in detail and obtain a dynamical model by combining all components together. Based onthis dynamical model, an optimal control algorithm for microgrids is proposed to maintain the stability and to improve theperformance of the microgrid.A. Microgrid DynamicsA microgrid is characterized as a graph. Specifically, consider a connected and undirected graph G (Ng , E), whereNg denotes the set of vertices of the graph and E Ng Ngdenotes the set of edges in the graph. The edge (i, j) connects vertices i and j. For our setup, the set of verticesNg : {1, ., Ng } denotes the set of loads and sources in themicrogrid, where Ng N 0 is the total number of loads andsources. Suppose that Nl out of Ng vertices are loads and letthe set Nl : {1, ., Nl } be the set of loads. Similarly, suppose that there are Ns out of Ng vertices are sources and letthe set Ns : {1, ., Ns } be the set of sources. The edge (i, j)denotes a power line connecting vertices i and j. Each powerline in the microgrid has an associated power capacity constraint which restricts the magnitude of the current flowingthrough the power line. Now, each component is discussed indetail.Source: There are two types of sources considered herein the microgrid. One of the types is a droop-controlled voltagesource. The other type is a constant power source (CPS).Suppose that Nd out of Ns sources are droop-controlled voltage sources, and let a set Nd : {1, ., Nd } be the set of2625Fig. 1.Small signal model of node j.droop-controlled voltage sources. This type of sources can bemodeled by a voltage source in series with a virtual resistance.For the inverter controller, the bandwidth of the inner loop ismuch higher than the outer loop, i.e., droop control loop. Sincethis paper focuses on the higher level operation, i.e., microgrid operation with multiple interface converters, the innerloop dynamics of the inverters are neglected. Following [16],the dynamics of voltage/current inner loops are ignored whichleads to a voltage and power relationship as follows.For each droop-controlled voltage source j Nd ,Vj Vj0 dj PDj ,(1)where Vj is the output voltage of the droop controller, Vj0is the nominal voltage value of the droop controller, PDj isthe corresponding output power of the droop controller, anddj is the droop gain of the droop controller. The referencedirection of PDj is toward the outside of the droop controlledvoltage source, i.e., the positive PDj represents the positive output power of the droop-controlled voltage source. The droopcontroller is further linearized around an operating point whichresults in a small-signal model as dj V̄jiDj ,(2)vj 1 dj Ījwhere vj is the corresponding small-signal perturbation of thevoltage, V̄j is the steady-state operating point of DC voltage, iDj is the corresponding small-signal perturbation of thecurrent, and Īj is the steady-state operating point of DC current.The individual small-signal models is aggregated whichleads tov d̃iD ,(3)where v : [v1 , v2 , · · · , vNd ], iD : [i1 , i2 , · · · , iNd ], and d̃ is a dj V̄j , j 1, ., Nd .diagonal matrix with diagonal elements 1 dj ĪjFor example, a battery can be modeled as a droop-controlledvoltage source [14]. Also when a PV generator is operated inthe voltage source converter mode [14], it can be modeled asa droop-controlled voltage source as well.Suppose that NCPS out of Ns sources are CPSs, and let theset NCPS : {1, ., NCPS } be the set of CPSs. By linearizing
2626IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6, NOVEMBER 2016around an operating point, a CPS can be modeled as a currentsource in parallel with a conductance as follows [16], [17].For each CPS j NCPS ,iCPSj gCPSj vj ĩCPSj ,(4)where iCPSj is the corresponding small-signal output current ofthe CPS and vj is the corresponding small-signal output voltage of the CPS. The values of the corresponding conductancegCPSj and current source ĩCPSj are determined by the powerand voltage at an operating point asgCPSj ICPSj PCPSj2VDCj,2PCPSjVDCj,iCPS gCPS vCPS ĩCPS .iCPSvCPSĩCPS(6)(7)and gCPS is a diagonal matrix with diagonal elementsgCPSj , j 1, ., NCPS .For example, when a PV panel is operated in the MPPTmode, it can be modeled as a CPS [14].Load: Following [16], loads in the microgrid are modeled as constant power loads (CPLs). There are other typesof loads, e.g., constant current loads and constant impedanceloads, which are all possible models for loads. Essentially,CPL mainly affects the stability of a DC microgrid. Sincemaintaining the stability of a DC microgrid is one of the maingoals of the proposed approach, in the paper, CPLs are mainlyconsidered in the system.Similar to CPS, a CPL is modeled as a negative conductancein parallel to a current sink as follows.For each CPL j NCPL ,iCPLj gCPLj vj ĩCPLj ,(8)where iCPLj is the corresponding small-signal output currentof the CPL and vj is the corresponding small-signal outputvoltage of the CPL.The values of the corresponding conductance gCPLj and current sink ĩCPLj are determined by the power and voltage at anoperating point asgCPLj ICPLjwhere,2VDCj2PCPLj ,VDCj(9)(10)(11) iCPL : iCPL1 , iCPL2 , · · · , iCPLNCPL , vCPL v1 , v2 , · · · , vNCPL , ĩCPL ĩCPL1 , ĩCPL2 , · · · , ĩCPLNCPL ,and gCPL is a diagonal matrix with diagonal elementsgCPLj , j 1, ., NCPLFor example, when a battery is operated in a regulatedcharging mode, it can be modeled as a CPL [14].Power Line: Suppose that there are Nb power lines in themicrogrid. Denote the set of power lines by Nb . The powerlines connecting the adjacent vertices (sources and loads) aremodeled as follows.For each power line j Nb ,dibj r b j ib j ,(12)dtwhere vbj and ibj are the corresponding power line voltageand current, respectively. lbj is the power line inductance, andrbj is the power line resistance. The line capacitance can beaggregated with CPS or CPL. Our model accommodates bothCPS and CPL. Even if the line capacitance is not aggregatedwith CPS or CPL, due to the low voltage operation condition,the line capacitance can be neglected. Here, the line resistance and inductance are considered. The reason is that theline impedances have directly interactions with the impedancesin the source/load equivalent circuit models. For example, theoutput impedance in the equivalent circuit model of droopcontrolled voltage source is directly in series with the lineimpedance, which indicates highly coupled dynamics.All the individual small-signal models are aggregatedleading tovbj lbj : iCPS1 , iCPS2 , · · · , iCPSNCPS , v1 , v2 , · · · , vNCPS , ĩCPS1 , ĩCPS2 , · · · , ĩCPSNCPS ,PCPLjiCPL gCPL vCPL ĩCPL .(5)where PCPSj is the output power at the operating point andVDCj is the DC (nominal) voltage at the operating point of theCPS. Note that ICPSj is the nominal current source value andĩCPSj represents the small-signal perturbation of the currentsource. All the individual small-signal models are aggregatedleading towherewhere PCPLj is the output power at the operating point andVDCj is the DC (nominal) voltage at an operating pointof the CPL.All the individual small-signal models are aggregated leading tovb lbdib rb ib ,dt(13)where vb : vb1 , vb2 , · · · , vbNb , ib ib1 , ib2 , · · · , ibNb ,(14)lb is a diagonal matrix with diagonal elements lbj , j 1, ., Nb , and rb is a diagonal matrix with diagonal elementsrbj , j 1, ., Nb .Microgrid Dynamics[16]: By applying Kirchhoff’s voltage law (KVL),vb Mv,(15)where M RNb Ng . M consists of { 1, 0, 1}. The j-th columnin each row corresponding to branch ( j, k) is equal to 1 andthe k-th column is equal to 1. The remaining entries of Mare zeros.
MA et al.: OPTIMAL OPERATION MODE SELECTION FOR A DC MICROGRIDBy applying Kirchhoff’s current law (KCL),iD iCP MT ib ,(16)where iCP is a shorthand notation which contains the elementsof iCPS and iCPL based on the configuration of the system.By combining (3), (7), (11), (13), (15), and (16), themicrogrid dynamics has the formi̇b Aib BĩCP ,where A l 1M d̃ 1 gCPb 1B l 1 gCPb M d̃ 1 1B. A Switched System MT rb ,.(18)j̃ Nd̃ d̃j j̃where Ndj is the set of droop-controlled voltage sources atbus j. Similarly, if there are multiple CPLs/CPSs at bus j Ng ,an aggregate model can be used:1,1j̃ NCPj rCPICPj j̃ICPj̃ ,(20)j̃ NCPjwhere NCPj is the set of CPLs/CPSs at bus j.The control dynamics in the converter-based-interface ateach source bus can be modeled as CPS at the correspondingbus. To model a pure resistance load at bus j Ng , the corresponding resistance and current source of the CPL or CPS areset to infinity and zero, respectively. Then, the virtual resistance of the droop-controlled voltage source is replaced withthe corresponding resistance of the resistor.Remark 1: The sufficient conditions for the stability of thelinear system (17) can be derived from the eigenvalues of thesystem matrix A. Following [16], the sufficient conditions forstability ared̃j 1,gCPjwhere the set NCP : NCPS NCPL . Notice that the conditions (21) impose upper bounds on the output powers of CPLsand CPSs at operating points. In order to maintain the stabilityof the microgrid operation, for a given nominal voltage, thedroop gains and the output powers of CPLs and CPSs needto satisfy the constraints (21) to maintain the stability of themicrogrid. The stability is measured by the magnitude of thepower line current state ib .(17)The models in this paper for droop controlled voltage sourceand CPS/CPL are equivalent circuit representations of thecorresponding devices. The dynamics involved by the circuitcomponents, e.g., terminal capacitance, mainly impacts theintermediate voltage control loop and inner current controlloop, which feature much higher control bandwidth comparedto the outer loop.If there is no droop-controlled voltage source at bus j Ng ,the corresponding virtual resistance is set as infinity, i.e., d̃j to model an open-loop connection. Similarly, if there is noCPL or CPS at bus j Ng , the corresponding resistance andcurrent source can be set as infinity and zero, respectively. Onthe other hand, if there are multiple droop-controlled voltagesources at the same bus, an aggregate model as shown belowcan be used:1d̃j ,(19)1rCPj 2627 j NCP ,(21)The controller can select different operation modes depending on the needs in an actual situation. For example, dueto the uncertainties of solar power sources, a PV panel isdesirable to operate in the MPPT mode to inject maximumavailable power. However, it is required to switch to thedroop-controlled voltage source mode for the purpose of theconservation of common voltage [14]. On the other hand, abattery operates in the regulated charging mode (modeled asa CPL), and is required to switch to the droop-control voltagesource mode as needed.Depending on the different operation modes, a microgridmay be modeled by various linear dynamics. In the following,a switched linear dynamics approach is adopted to model thesedifferent linear systems and the switching behavior betweendifferent modes. Consider that there are M operation modesand denote the set of modes by M : {1, ., M}. For a givenmode σ M, the microgrid is modeled asi̇b Aσ ib Bσ ĩCPσ ,(22)where 1Aσ l 1bσ Mσ d̃σ gCPσ 1Bσ l 1bσ Mσ d̃σ gCPσ 1 1 MTσ rbσ ,.(23)The model (23) accommodates various operation modes.For example, each mode may specify the operating powerrequirement. By switching to different modes, the operatingpoint is switched; or the switch may represent the aforementioned change between the voltage-controlled source andCPS/CPL.III. S TABILITY OF DC M ICROGRID O PERATIONIn this section, the stability of the microgrid model as shownin (22) is analyzed. To simplify the notation, the microgrid isrepresented byẋ(t) Aσ x(t) bσ ,σ M,(24)where x Rnx is the system state, σ M is the index ofthe operation mode, Aσ Rnx nx is the system matrix formode σ , and bσ Rnb is the input for mode σ . The systemstate x consists of the branch current ib . The input bσ is definedas bσ : Bσ ĩCPσ , for any σ M.
2628IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6, NOVEMBER 2016Following [18], given a desired equilibrium point xr , thetracking error dynamics is modeled as:ė(t) Aσ e(t) kσ ,kσ bσ Aσ xr ,e(t) x(t) xr ,σ M,σ M,σ (x(t)) arg max {vσ (e(t))},σ Mσ M(26)where vσ is an auxiliary function associated with each mode,and V(σ (t), σ ) is the Lyapunov function of the switched affinesystem. The system performance is characterized byv̇σ (e(t)) ασ vσ (e(t)) 0,(27)where vσ (e(t)) 0, for all σ M and e(t) 0. ασ , σ Mare given positive constants which characterize the decreasing rate of the state x measured by the Lyapunov functionV(e(t), σ ) in mode σ M.Considervσ (e(t)) e(t)T Pσ e(t) 2e(t)T Sσ ,(28)where Pσ PTσ Rnx nx , and Sσ Rnx .Similarly to [18], the stability can be characterized by linearmatrix inequality (LMI). For a complete discussion, the readeris referred to [18].Theorem 1: If there exist Pσ PTσ Rnx nx , and Sσ Rnxsuch that T T Pσ Sσkσ Pσ SσT AσPσ Aσ ATσ Pσ 0, ασSσT0kσT Pσ SσT Aσ2SσT kσσ M,MPσ 0, σ,Sσ 0.(29)σ 1The switching signal σ is selected according toσ (x(t)) arg max {vσ (e(t))}.σ MThis section focuses on the design of the optimal controllerthat is applied to the microgrid (17) with different operationmodes. The controller optimally selects which operation modeis active or inactive.(25)where e Rnx is the error, xr is the tracking reference, σ Mis the index of mode, Aσ is the system matrix for the modeσ , and kσ Rnb is the input for mode σ .This study focuses on the stability of (22) and the smallsignal tracking reference xr is set to zero.The switching signal σ is selected according to:V(e(t), σ ) : max {vσ (e(t))},IV. O PTIMAL C ONTROL FOR DC M ICROGRID O PERATION(30)Then the state x will converge to zero (xr 0) as t andthe system performance (27) is satisfied. The pair (Pσ , Sσ )can be found by solving LMI (29). The LMI solver providedin Matlab can be used to perform the computation.Note that the state represents the small-signal perturbationof the branch current. The results provide a switching rule (30)to ensure the current stability of the DC microgrid with variousoperation modes. The convergence rate of mode σ M iscontrolled by the parameter ασ in (27). A larger ασ results ina faster convergence speed.A. Optimal Control of Switched Affine SystemThe optimal controller conducts a switch pattern planningand schedules the operation modes {σk , k K}. The indices{σk , k K} are the decision variables. If σk 1, it indicates that mode σ is activated at time k. On the other hand,if σk 0, it indicates that mode σ is inactivated at time k.The available modes for selection are obtained from the higherlevel energy management algorithm, e.g., the optimal powerflow algorithm. The higher level algorithm decides the steadystate operating points (voltage, power, etc.) of the componentsin the DC microgrid and the costs associated with the operating modes. Given these operation modes, a planning processof the mode switching is carried out to optimize the transientbehavior and to minimize the operating cost of the system. Thecontroller is implemented in an offline manner. The optimalcontrol algorithm is executed at the lower level. The higherlevel algorithm is executed on a less frequent basis.The objective is to minimize the perturbation of the smallsignal branch current and the operation cost while satisfyingthe microgrid dynamics, which leads to the following optimization problem:c(x(k)) fσ (γσ (k))minimize{σ M}k KMzσ (k) x(k),subject tok Kσ 1Mγσ (k) 1,k K,σ 1zσ (0) x(0)γσ (0), σ M,zσ (k 1) (Aσ x(k) Bσ )γσ (k 1),σ M, k K,x(k) x(k) x(k),k K,(31)where zσ (k) Rnx , γσ (k) {0, 1}, for all σ M andk K. zσ (k) is an auxiliary variable associated with modeσ, σ M. If the variable γσ is selected as γσ (k) 1, itindicates that mode σ is active at time k. The term c(x(k))represents the cost function of the state x(k). For example, ifthe cost function c(x(k)) is selected as c(x(k)) x((k)) , themagnitude of the state x(k) is minimized. The term fσ (γσ (k))is a linear or quadratic function of γσ (k) which encodes thecost of activating mode σ at time k, where σ M, k K.If mode σ is not an optimal operating mode obtained fromthe higher level energy management algorithm, then the termfσ (γσ ) is adopted to penalize being activated in mode σ .The switching signals derived from (31) are different fromthe ones derived from (30). The current stability of the optimized mode can be determined by including the constraintsx(k) x(k) x(k), k K in (31).
MA et al.: OPTIMAL OPERATION MODE SELECTION FOR A DC MICROGRID2629By solving the optimization problem (31), we are able tofind the optimal switching signals such that the stability ofthe small signal branch current perturbation is ensured and theoperating cost is minimized. Note that the available modes forselection and the associated operating costs are derived fromthe higher level energy management algorithm.Problem (31) can be transformed to a mixed-integerquadratic program (MIQP) as follows [19]:zσ (0) x(0) λ(1 γσ (0)), σ M, zσ (0) x(0) λ(1 γσ (0)), σ M,zσ (0) λγσ (0), σ M,zσ (0) λγσ (0), σ M,Fig. 2.Configuration of the test system.(32)A. Time-Domain Testandzσ (k 1) (Aσ x(k) Bσ ) λ(1 γσ (k 1)), σ M, zσ (k 1) (Aσ x(k) Bσ ) λ(1 γσ (k 1)), σ M,zσ (k 1) λγσ (k 1), σ M,zσ (k 1) λγσ (k 1),σ M,(33)where λ Rnx is a sufficiently large constant. The resultingMIQP isc(x(k)) fσ (γσ (k))minimize{γσ (k) {0,1},σ M,k K}k KMzσ (k) x(k),subject tok Kσ 1Mγσ (k) 1,k K,σ 1x(k) x(k) x(k),(32), (33).k K,(34)V. S IMULATION V ERIFICATIONA DC microgrid with four buses was implemented, asshown in Figure 2, to test the effectiveness of the optimal control formulation (31) presented in Section IV. In particular,three buses in the DC microgrid, i.e., Buses 1-3, are connectedto the source converter and the fourth bus, i.e., Bus 0, is connected to the load. The nominal voltage is set to 400 V. Thepower line resistance (per kilometer) and the inductance (perkilometer) are 0.05 /km and 102.5 μH/km„ respectively. Thedroop gain dj is set to 0.2 10 3 . On the source side, thereare two types of source buses, i.e., droop-controlled voltagesource and constant power source. Two operation modes arestudied, as shown below:Mode 1: Bus 1: droop-controlled voltage source, Bus 2:CPS, Bus 3: CPS, Bus 0: load. Mode 2: Bus 1: CPS, Bus 2:droop-controlled voltage source, Bus 3: droop-controlled voltage source, Bus 0: load.The simulation verification is separated into two parts. Thefirst part is the time-domain MATLAB/Simulink test to verifythe effectiveness of mode transition between different operation modes, and the second part is the numerical demonstrationto test the optimal control formulation.For the mode transition from mode 1 to mode 2, the waveforms of the output power for each interface converter is shownin Figure 3. Before the mode transition, in mode 1, the output power references for the sources connected at Bus 2 andBus 3 are set to 20 kW and 15 kW, respectively. Meanwhile,droop control is used for the source at Bus 1. Since the loadpower of 40 kW is selected, the output power of source 1 isautomatically regulated to 5 kW. After the mode transition, thepower references of source 1 is set to 15 kW. Since droopcontrolled voltage sources are connected at Bus 2 and 3, theoutput power of source 2 and 3 are gradually equalized. Thewaveform of common load bus voltage is shown in Figure 4. Itcan be seen that after a short time transient adjustment, the DCbus voltage turns back around the nominal voltage of 400 V.The transient voltage overshoot is about 12 V, which is 3% tothe rated voltage.For the mode transition from mode 2 to mode 1, the waveforms of the output power for each interface converter is shownin Figure 5. Before the mode transition, in mode 2, the outputpower reference of source 1 is set to 15 kW, while droopcontrollers are used in source 2 and 3. Hence, the outputpower of source 2 and 3 are equalized. Since the load poweris selected as 40 kW, the output power of source 2 and 3are 12.5 kW, respectively. After the mode transition, in mode1, the references of the output power of source 2 and 3 areset to 20 kW and 15 kW, while droop controller is used insource 1. Hence, the output power of source 1 is changed to5 kW. Meanwhile, the common load bus voltage waveformis shown in Figure 6. It can be seen that the common loadbus voltage can be regulated to around the nominal voltage of400 V after a short period of time. Meanwhile, the overshootof the voltage is around 13 V, which is around 3% of the ratedvoltage.B. Numerical VerificationAfter testing the mode transitions in the time domain simulation, numerical test is conducted to verify the effectivenessof the proposed optimal control formulation. The same asthe cases in the time-domain simulation, mode 1 and 2 areselected in the numerical study. The sources with corresponding rated power are used to feed the load. Here, the rated loadpower is set to 100 kW in order to highlight the controlled
2630Fig. 3. Mode transition from mode 1 to mode 2. Output power of eachinterface converter.IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6, NOVEMBER 2016Fig. 6. Mode transition from mode 2 to mode 1. Output voltage of eachinterface converter.Fig. 4. Mode transition from mode 1 to mode 2. Output voltage of eachinterface converter.Fig. 7. The switching pattern between Mode 1 (stable, suboptimal mode)and Mode 2 (marginal unstable, optimal mode).transient behavior while minimizing the operation cost. Thecost function is selected asK x(k) 2 ργ1 (k).(35)k 0Fig. 5. Mode transition from mode 2 to mode 1. Output power of eachinterface converter.mode transitions. Based on the small signal branch currentdynamics (22), mode 2 has one positive eigenvalue (close tothe imaginary axis) and is a marginal unstable mode. Fromthe higher level energy management prospective, mode 2 is theoptimal operation mode. However, for the lower level control,adopting mode 2 all the time will certainly lead to an unstablesystem. If the system instead adopts the suboptimal but stablemode 1, the cost will be high. The optimal controller helpsthe microgrid to select the switching patterns between the twomodes. In particular, the optimal control formulat
2624 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6, NOVEMBER 2016 Optimal Operation Mode Selection for a DC Microgrid Wann-Jiun Ma, Member, IEEE, Jianhui Wang, Senior Member, IEEE, Xiaonan Lu, Member, IEEE, and Vijay Gupta, Member, IEEE Abstract—This paper considers an optimal control problem to
IEEE 3 Park Avenue New York, NY 10016-5997 USA 28 December 2012 IEEE Power and Energy Society IEEE Std 81 -2012 (Revision of IEEE Std 81-1983) Authorized licensed use limited to: Australian National University. Downloaded on July 27,2018 at 14:57:43 UTC from IEEE Xplore. Restrictions apply.File Size: 2MBPage Count: 86Explore furtherIEEE 81-2012 - IEEE Guide for Measuring Earth Resistivity .standards.ieee.org81-2012 - IEEE Guide for Measuring Earth Resistivity .ieeexplore.ieee.orgAn Overview Of The IEEE Standard 81 Fall-Of-Potential .www.agiusa.com(PDF) IEEE Std 80-2000 IEEE Guide for Safety in AC .www.academia.eduTesting and Evaluation of Grounding . - IEEE Web Hostingwww.ewh.ieee.orgRecommended to you b
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English Language Arts: Grade 3 READING Guiding Principle: Students read a wide range of fiction, nonfiction, classic, and contemporary works, to build an understanding of texts, of themselves, and of the cultures of the United States and the world; to acquire new information; to respond to the needs and demands of society and the workplace .
