Optimal Operation Mode Selection for a DC Microgrid - University of ...

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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 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 by switching to a different mode in real time. A switched system approach is employed to ensure the stability of a dc microgrid with a rich array of operation modes. Meanwhile, an optimal control algorithm is designed to improve the system performance by appropriately selecting the operation modes. A typical dc microgrid with three source buses and one load bus is implemented. The effectiveness of the algorithms is verified by MATLAB/Simulink time-domain tests and numerical studies. Index Terms—DC, microgrids, switched system, optimal control.

I. I NTRODUCTION ICROGRIDS was proposed several years ago and have been intensively studied these years [1]–[3]. They bring energy sources close to electricity users by supplying energy from distributed generation sources. A microgrid can operate in a grid-connected mode or in a stand-alone fashion. When a power outage occurs, a microgrid can automatically disconnect with the main grid. With the supply from the distributed generation sources, it can operate in a stand-alone mode without interruptions. Most microgrids are designed based on an alternating current (AC) system. Thus, the issues that exist in an AC distribution system such as reactive power control, and frequency control may degrade the system performance of an AC microgrid. Recently, a direct current (DC) microgrid has been proposed to increase power distribution quality

M

Manuscript 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 version October 19, 2016. The work of W.-J. Ma and V. Gupta was supported by 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: [email protected]). J. Wang and X. Lu are with the Energy Systems Division, Argonne National Laboratory, Lemont, IL 60439 USA (e-mail: [email protected]; [email protected]). V. Gupta is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online 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 with DC coupling, which enhances the system performance during operation by shortening the power conversion chain and avoiding using most of the DC-AC and AC-DC conversion stages [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 the supply side, photovoltaic (PV) power generation is considered to be one of the typical RESs in a DC microgrid. The high variability of PV power generation complicates the control of DC microgrid operation. For example, the effect of passing clouds on the PV arrays results in unpredictable variation of the power generation. As the interface circuits, PV inverter is operated in maximum power point tracking (MPPT) mode to maximize the renewable power generation [14]. By executing MPPT, a maximum amount of power is extracted from PV panels. However, it is not always appropriate to let the PV power source operate in the MPPT mode because a maximum power generation may introduce an excessive amount of energy which may affect the stability of an islanded microgrid [14]. If this is the case, the PV power source can be switched to an output voltage control mode to maintain the voltage stability of the microgrid [14]. A battery is another example of component in a microgrid which has different operation modes. In order to achieve optimal life time, a battery should 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 discharging modes. Therefore, from both the supply and demand sides, different components may play different roles in the microgrid during operation, and the components may change their roles depending on the actual situation in real-time, which results in a rich array of operation modes. In this paper, a switched system approach is adopted and an optimal control algorithm is designed to regulate microgrid operation with various operation modes. A switched system framework has gained tremendous popularity in the control community; see, e.g., [15] and references therein. This framework generalizes linear and nonlinear systems. It models a dynamical system by allowing it to switch from one mode to other modes in real-time. Each mode represents a particular linear or nonlinear system. The switched system framework characterizes a dynamical system with various operation modes. Here, this modeling framework is adopted to

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MA et al.: OPTIMAL OPERATION MODE SELECTION FOR A DC MICROGRID

the DC microgrid with different operation modes and a suitable controller is designed to improve the performance of the microgrid based on this framework. Of particular interest to our work is the supervisory energy management system proposed in [14]. The supervisory system controls the operation modes in a coordinated manner such that the charging and discharging of batteries are in a regulated fashion. The output power of a PV source is extracted appropriately while maintaining the voltage stability of the microgrid. However, the control law in [14] is based on a heuristic rule. Despite the fact that the simulation results verify the effectiveness of the algorithm, there is no analytical performance guarantee about the algorithm. In this paper, a switched system framework is adopted to analyze the stability of the microgrid with various operation modes and to provide a performance guarantee for the proposed controller by optimally selecting the operation modes. The remainder of the paper is organized as follows. Section II presents the model of the DC microgrid. Section III analyzes the stability of the microgrid. The optimal controller design for the microgrid is presented in Section IV. Simulation results can be found in Section V. Section VI concludes the paper. II. M ODELING OF DC M ICROGRID A microgrid consists of different components such as renewable sources (e.g., solar PV source), loads (e.g., electric appliance, battery, etc.), and power lines. Figure 1 shows the structure of a node (bus) in the microgrid. This section formulates each component in detail and obtain a dynamical model by combining all components together. Based on this dynamical model, an optimal control algorithm for microgrids is proposed to maintain the stability and to improve the performance of the microgrid. A. Microgrid Dynamics A microgrid is characterized as a graph. Specifically, consider a connected and undirected graph G = (Ng , E), where Ng denotes the set of vertices of the graph and E ⊂ Ng × Ng denotes the set of edges in the graph. The edge (i, j) connects vertices i and j. For our setup, the set of vertices Ng := {1, ..., Ng } denotes the set of loads and sources in the microgrid, where Ng ∈ N>0 is the total number of loads and sources. Suppose that Nl out of Ng vertices are loads and let the set Nl := {1, ..., Nl } be the set of loads. Similarly, suppose that there are Ns out of Ng vertices are sources and let the set Ns := {1, ..., Ns } be the set of sources. The edge (i, j) denotes a power line connecting vertices i and j. Each power line in the microgrid has an associated power capacity constraint which restricts the magnitude of the current flowing through the power line. Now, each component is discussed in detail. Source: There are two types of sources considered here in the microgrid. One of the types is a droop-controlled voltage source. 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 of

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Fig. 1.

Small signal model of node j.

droop-controlled voltage sources. This type of sources can be modeled by a voltage source in series with a virtual resistance. For the inverter controller, the bandwidth of the inner loop is much higher than the outer loop, i.e., droop control loop. Since this paper focuses on the higher level operation, i.e., microgrid operation with multiple interface converters, the inner loop dynamics of the inverters are neglected. Following [16], the dynamics of voltage/current inner loops are ignored which leads 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, Vj0 is the nominal voltage value of the droop controller, PDj is the corresponding output power of the droop controller, and dj is the droop gain of the droop controller. The reference direction of PDj is toward the outside of the droop controlled voltage source, i.e., the positive PDj represents the positive output power of the droop-controlled voltage source. The droop controller is further linearized around an operating point which results in a small-signal model as   dj V¯ j iDj , (2) vj = − 1 + dj I¯j where vj is the corresponding small-signal perturbation of the voltage, V¯ j is the steady-state operating point of DC voltage, iDj is the corresponding small-signal perturbation of the current, and I¯j is the steady-state operating point of DC current. The individual small-signal models is aggregated which leads to ˜ D, v = −di

(3)

where v := [v1 , v2 , · · · , vNd ], iD := [i1 , i2 , · · · , iNd ], and d˜ is a  dj V¯ j  , j = 1, ..., Nd . diagonal matrix with diagonal elements 1+d j I¯j For example, a battery can be modeled as a droop-controlled voltage source [14]. Also when a PV generator is operated in the voltage source converter mode [14], it can be modeled as a droop-controlled voltage source as well. Suppose that NCPS out of Ns sources are CPSs, and let the set NCPS := {1, ..., NCPS } be the set of CPSs. By linearizing

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around an operating point, a CPS can be modeled as a current source in parallel with a conductance as follows [16], [17]. For each CPS j ∈ NCPS , iCPSj = gCPSj vj + ˜iCPSj ,

(4)

where iCPSj is the corresponding small-signal output current of the CPS and vj is the corresponding small-signal output voltage of the CPS. The values of the corresponding conductance gCPSj and current source ˜iCPSj are determined by the power and voltage at an operating point as gCPSj = ICPSj =

PCPSj 2 VDC j

,

2PCPSj VDCj

,

iCPS = gCPS vCPS + ˜iCPS .

iCPS vCPS ˜iCPS

(6)

(7)

and gCPS is a diagonal matrix with diagonal elements gCPSj , j = 1, ..., NCPS . For example, when a PV panel is operated in the MPPT mode, 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 types of loads, e.g., constant current loads and constant impedance loads, which are all possible models for loads. Essentially, CPL mainly affects the stability of a DC microgrid. Since maintaining the stability of a DC microgrid is one of the main goals of the proposed approach, in the paper, CPLs are mainly considered in the system. Similar to CPS, a CPL is modeled as a negative conductance in parallel to a current sink as follows. For each CPL j ∈ NCPL , iCPLj = −gCPLj vj + ˜iCPLj ,

(8)

where iCPLj is the corresponding small-signal output current of the CPL and vj is the corresponding small-signal output voltage of the CPL. The values of the corresponding conductance gCPLj and current sink ˜iCPLj are determined by the power and voltage at an operating point as gCPLj = ICPLj

where

, 2 VDCj 2PCPLj = , VDCj

(9) (10)

(11)

  iCPL := iCPL1 , iCPL2 , · · · , iCPLNCPL ,  vCPL = v1 , v2 , · · · , vNCPL ,   ˜iCPL = ˜iCPL , ˜iCPL , · · · , ˜iCPLN , 1 2 CPL

and gCPL is a diagonal matrix with diagonal elements gCPLj , j = 1, ..., NCPL For example, when a battery is operated in a regulated charging mode, it can be modeled as a CPL [14]. Power Line: Suppose that there are Nb power lines in the microgrid. Denote the set of power lines by Nb . The power lines connecting the adjacent vertices (sources and loads) are modeled as follows. For each power line j ∈ Nb , dibj

+ r b j ib j , (12) dt where vbj and ibj are the corresponding power line voltage and current, respectively. lbj is the power line inductance, and rbj is the power line resistance. The line capacitance can be aggregated with CPS or CPL. Our model accommodates both CPS and CPL. Even if the line capacitance is not aggregated with 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 the line impedances have directly interactions with the impedances in the source/load equivalent circuit models. For example, the output impedance in the equivalent circuit model of droopcontrolled voltage source is directly in series with the line impedance, which indicates highly coupled dynamics. All the individual small-signal models are aggregated leading to vbj = lbj

  := iCPS1 , iCPS2 , · · · , iCPSNCPS ,  = v1 , v2 , · · · , vNCPS ,   = ˜iCPS1 , ˜iCPS2 , · · · , ˜iCPSNCPS ,

PCPLj

iCPL = −gCPL vCPL + ˜iCPL .

(5)

where PCPSj is the output power at the operating point and VDCj is the DC (nominal) voltage at the operating point of the CPS. Note that ICPSj is the nominal current source value and ˜iCPSj represents the small-signal perturbation of the current source. All the individual small-signal models are aggregated leading to

where

where PCPLj is the output power at the operating point and VDCj is the DC (nominal) voltage at an operating point of the CPL. All the individual small-signal models are aggregated leading to

vb = lb

dib + 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 elements rbj , 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 column in each row corresponding to branch ( j, k) is equal to 1 and the k-th column is equal to −1. The remaining entries of M are zeros.

MA et al.: OPTIMAL OPERATION MODE SELECTION FOR A DC MICROGRID

By applying Kirchhoff’s current law (KCL), iD − iCP = MT ib ,

(16)

where iCP is a shorthand notation which contains the elements of iCPS and iCPL based on the configuration of the system. By combining (3), (7), (11), (13), (15), and (16), the microgrid dynamics has the form ˙ib = Aib + B˜iCP , where

−1

˜ −1 − gCP B = −l−1 M . d b

B. A Switched System

(18)

The models in this paper for droop controlled voltage source and CPS/CPL are equivalent circuit representations of the corresponding devices. The dynamics involved by the circuit components, e.g., terminal capacitance, mainly impacts the intermediate voltage control loop and inner current control loop, which feature much higher control bandwidth compared to 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., d˜ j = ∞ to model an open-loop connection. Similarly, if there is no CPL or CPS at bus j ∈ Ng , the corresponding resistance and current source can be set as infinity and zero, respectively. On the other hand, if there are multiple droop-controlled voltage sources at the same bus, an aggregate model as shown below can be used: 1 d˜ j = , (19) 1 ˜j∈N ˜ d˜ dj ˜j

where Ndj is the set of droop-controlled voltage sources at bus j. Similarly, if there are multiple CPLs/CPSs at bus j ∈ Ng , an aggregate model can be used: 1 1

˜j∈NCP rCP j ˜

ICPj =

,

j

ICP˜j ,

(20)

˜j∈NCP j

where NCPj is the set of CPLs/CPSs at bus j. The control dynamics in the converter-based-interface at each source bus can be modeled as CPS at the corresponding bus. To model a pure resistance load at bus j ∈ Ng , the corresponding resistance and current source of the CPL or CPS are set to infinity and zero, respectively. Then, the virtual resistance of the droop-controlled voltage source is replaced with the corresponding resistance of the resistor. Remark 1: The sufficient conditions for the stability of the linear system (17) can be derived from the eigenvalues of the system matrix A. Following [16], the sufficient conditions for stability are d˜ j ≤

1 , gCPj

where the set NCP := NCPS ∪ NCPL . Notice that the conditions (21) impose upper bounds on the output powers of CPLs and CPSs at operating points. In order to maintain the stability of the microgrid operation, for a given nominal voltage, the droop gains and the output powers of CPLs and CPSs need to satisfy the constraints (21) to maintain the stability of the microgrid. The stability is measured by the magnitude of the power line current state ib .

(17)



 −1 T ˜ −1 − gCP A = −l−1 M M + r d b , b

rCPj =

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∀j ∈ NCP ,

(21)

The controller can select different operation modes depending on the needs in an actual situation. For example, due to the uncertainties of solar power sources, a PV panel is desirable to operate in the MPPT mode to inject maximum available power. However, it is required to switch to the droop-controlled voltage source mode for the purpose of the conservation of common voltage [14]. On the other hand, a battery operates in the regulated charging mode (modeled as a CPL), and is required to switch to the droop-control voltage source mode as needed. Depending on the different operation modes, a microgrid may be modeled by various linear dynamics. In the following, a switched linear dynamics approach is adopted to model these different linear systems and the switching behavior between different modes. Consider that there are M operation modes and denote the set of modes by M := {1, ..., M}. For a given mode σ ∈ M, the microgrid is modeled as ˙ib = Aσ ib + Bσ ˜iCPσ ,

(22)

where 

 −1 −1 T ˜ Aσ = −l−1 − g M + r M d σ CPσ bσ , σ σ bσ

−1 ˜ −1 M − g . Bσ = −l−1 d σ CP σ σ bσ

(23)

The model (23) accommodates various operation modes. For example, each mode may specify the operating power requirement. By switching to different modes, the operating point is switched; or the switch may represent the aforementioned change between the voltage-controlled source and CPS/CPL.

III. S TABILITY OF DC M ICROGRID O PERATION In this section, the stability of the microgrid model as shown in (22) is analyzed. To simplify the notation, the microgrid is represented by x˙ (t) = Aσ x(t) + bσ ,

σ ∈ M,

(24)

where x ∈ Rnx is the system state, σ ∈ M is the index of the operation mode, Aσ ∈ Rnx ×nx is the system matrix for mode σ , and bσ ∈ Rnb is the input for mode σ . The system state x consists of the branch current ib . The input bσ is defined as bσ := Bσ ˜iCPσ , for any σ ∈ M.

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Following [18], given a desired equilibrium point xr , the tracking error dynamics is modeled as: e˙ (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 ∈M

σ ∈M

(26)

where vσ is an auxiliary function associated with each mode, and V(σ (t), σ ) is the Lyapunov function of the switched affine system. The system performance is characterized by v˙ σ (e(t)) + ασ vσ (e(t)) < 0,

(27)

where vσ (e(t)) > 0, for all σ ∈ M and e(t) = 0. ασ , σ ∈ M are given positive constants which characterize the decreasing rate of the state x measured by the Lyapunov function V(e(t), σ ) in mode σ ∈ M. Consider vσ (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 linear matrix inequality (LMI). For a complete discussion, the reader is referred to [18]. Theorem 1: If there exist Pσ = PTσ ∈ Rnx ×nx , and Sσ ∈ Rnx such that     T T  Pσ Sσ kσ Pσ + SσT Aσ Pσ Aσ + ATσ Pσ < 0, + α σ SσT 0 kσT Pσ + SσT Aσ 2SσT kσ σ ∈ M, M

Pσ > 0, ∀σ, Sσ = 0. (29) σ =1

The switching signal σ is selected according to σ (x(t)) = arg max {vσ (e(t))}. σ ∈M

This section focuses on the design of the optimal controller that is applied to the microgrid (17) with different operation modes. The controller optimally selects which operation mode is active or inactive.

(25)

where e ∈ Rnx is the error, xr is the tracking reference, σ ∈ M is the index of mode, Aσ is the system matrix for the mode σ , 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 → ∞ and the system performance (27) is satisfied. The pair (Pσ , Sσ ) can be found by solving LMI (29). The LMI solver provided in Matlab can be used to perform the computation. Note that the state represents the small-signal perturbation of the branch current. The results provide a switching rule (30) to ensure the current stability of the DC microgrid with various operation modes. The convergence rate of mode σ ∈ M is controlled by the parameter ασ in (27). A larger ασ results in a faster convergence speed.

A. Optimal Control of Switched Affine System The optimal controller conducts a switch pattern planning and 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 higher level energy management algorithm, e.g., the optimal power flow algorithm. The higher level algorithm decides the steady state operating points (voltage, power, etc.) of the components in the DC microgrid and the costs associated with the operating modes. Given these operation modes, a planning process of the mode switching is carried out to optimize the transient behavior and to minimize the operating cost of the system. The controller is implemented in an offline manner. The optimal control algorithm is executed at the lower level. The higher level 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 satisfying the microgrid dynamics, which leads to the following optimization problem:

c(x(k)) + fσ (γσ (k)) minimize {σ ∈M}

subject to

k∈K M

σ =1 M

zσ (k) = x(k), γσ (k) = 1,

k∈K k ∈ K,

σ =1

zσ (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 and k ∈ K. zσ (k) is an auxiliary variable associated with mode σ, σ ∈ M. If the variable γσ is selected as γσ (k) = 1, it indicates that mode σ is active at time k. The term c(x(k)) represents the cost function of the state x(k). For example, if the cost function c(x(k)) is selected as c(x(k)) = x((k)), the magnitude of the state x(k) is minimized. The term fσ (γσ (k)) is a linear or quadratic function of γσ (k) which encodes the cost of activating mode σ at time k, where σ ∈ M, k ∈ K. If mode σ is not an optimal operating mode obtained from the higher level energy management algorithm, then the term fσ (γσ ) is adopted to penalize being activated in mode σ . The switching signals derived from (31) are different from the ones derived from (30). The current stability of the optimized mode can be determined by including the constraints x(k) ≤ x(k) ≤ x(k), k ∈ K in (31).

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By solving the optimization problem (31), we are able to find the optimal switching signals such that the stability of the small signal branch current perturbation is ensured and the operating cost is minimized. Note that the available modes for selection and the associated operating costs are derived from the higher level energy management algorithm. Problem (31) can be transformed to a mixed-integer quadratic 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 Test

and zσ (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 resulting MIQP is

c(x(k)) + fσ (γσ (k)) minimize {γσ (k)∈{0,1},σ ∈M,k∈K}

subject to

k∈K M

σ =1 M

zσ (k) = x(k), γσ (k) = 1,

k∈K k ∈ K,

σ =1

x(k) ≤ x(k) ≤ x(k), (32), (33).

k ∈ K, (34)

V. S IMULATION V ERIFICATION A DC microgrid with four buses was implemented, as shown 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 connected to the source converter and the fourth bus, i.e., Bus 0, is connected to the load. The nominal voltage is set to 400 V. The power line resistance (per kilometer) and the inductance (per kilometer) are 0.05 /km and 102.5 μH/km„ respectively. The droop gain dj is set to 0.2 × 10−3 . On the source side, there are two types of source buses, i.e., droop-controlled voltage source and constant power source. Two operation modes are studied, 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. The first part is the time-domain MATLAB/Simulink test to verify the effectiveness of mode transition between different operation modes, and the second part is the numerical demonstration to 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 shown in Figure 3. Before the mode transition, in mode 1, the output power references for the sources connected at Bus 2 and Bus 3 are set to 20 kW and 15 kW, respectively. Meanwhile, droop control is used for the source at Bus 1. Since the load power of 40 kW is selected, the output power of source 1 is automatically regulated to 5 kW. After the mode transition, the power references of source 1 is set to 15 kW. Since droopcontrolled voltage sources are connected at Bus 2 and 3, the output power of source 2 and 3 are gradually equalized. The waveform of common load bus voltage is shown in Figure 4. It can be seen that after a short time transient adjustment, the DC bus voltage turns back around the nominal voltage of 400 V. The transient voltage overshoot is about 12 V, which is 3% to the rated voltage. For the mode transition from mode 2 to mode 1, the waveforms of the output power for each interface converter is shown in Figure 5. Before the mode transition, in mode 2, the output power reference of source 1 is set to 15 kW, while droop controllers are used in source 2 and 3. Hence, the output power of source 2 and 3 are equalized. Since the load power is selected as 40 kW, the output power of source 2 and 3 are 12.5 kW, respectively. After the mode transition, in mode 1, the references of the output power of source 2 and 3 are set to 20 kW and 15 kW, while droop controller is used in source 1. Hence, the output power of source 1 is changed to 5 kW. Meanwhile, the common load bus voltage waveform is shown in Figure 6. It can be seen that the common load bus voltage can be regulated to around the nominal voltage of 400 V after a short period of time. Meanwhile, the overshoot of the voltage is around 13 V, which is around 3% of the rated voltage. B. Numerical Verification After testing the mode transitions in the time domain simulation, numerical test is conducted to verify the effectiveness of the proposed optimal control formulation. The same as the cases in the time-domain simulation, mode 1 and 2 are selected in the numerical study. The sources with corresponding rated power are used to feed the load. Here, the rated load power is set to 100 kW in order to highlight the controlled

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Fig. 3. Mode transition from mode 1 to mode 2. Output power of each interface converter.

IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6, NOVEMBER 2016

Fig. 6. Mode transition from mode 2 to mode 1. Output voltage of each interface converter.

Fig. 4. Mode transition from mode 1 to mode 2. Output voltage of each interface 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. The cost function is selected as K

||x(k)||2 + ργ1 (k).

(35)

k=0

Fig. 5. Mode transition from mode 2 to mode 1. Output power of each interface converter.

mode transitions. Based on the small signal branch current dynamics (22), mode 2 has one positive eigenvalue (close to the imaginary axis) and is a marginal unstable mode. From the higher level energy management prospective, mode 2 is the optimal operation mode. However, for the lower level control, adopting mode 2 all the time will certainly lead to an unstable system. If the system instead adopts the suboptimal but stable mode 1, the cost will be high. The optimal controller helps the microgrid to select the switching patterns between the two modes. In particular, the optimal control formulation (31) is adopted to schedule the mode switching and to optimize the

The planning process is repeated every 10 seconds and the switching is only allowed once every 10 seconds. Consider total 200 seconds. Between the first 100 seconds, the penalty term ρ is set to 0.5; during the final 100 seconds, the penalty term is changed to 20. In the cost function (35), the first term x(k)2 is used to penalize large branch current perturbation, and the second term is used to penalize suboptimal operation mode (mode 1). Since mode 2 is the optimal mode derived from the higher level energy management algorithm, mode 1 is penalized by the penalty term ρ. Mode 2 is set to be the initial mode, i.e., γ2 (0) = 1 and is also activated when the penalty changes. The cost function value (35) obtained from the optimal control algorithm (31) is 37874. If only considering mode 1 all the time, i.e., γ2 (k) = 1 for all k, the cost function value is 39981. The optimal control algorithm (31) improves the cost function value. The optimal control algorithm optimally selects the modes and generates a smaller cost function value than the one using mode 1 all the time. Figure 7 shows the switching pattern between mode 1 and mode 2. Figure 8 shows the norm of the small-signal branch current

MA et al.: OPTIMAL OPERATION MODE SELECTION FOR A DC MICROGRID

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to mode 1) happens around 8 seconds. Figure 10 shows the norm of the small signal. In Figure 9, as the penalty term ρ decreases (changed from 1 to 0.01), during the first 10 seconds, the switching (form mode 2 to mode 1) happens between 2 to 3 seconds. Figure 10 shows the norm of the small signal. By tuning the penalty term ρ, a different trade-off between the transient stability and the cost function value is obtained. VI. C ONCLUSION

Fig. 8. The norm of the small-signal branch current trajectory from the switching pattern in Figure 7.

A DC microgrid with a rich array of operation modes based on a switched system framework is modeled in this paper. Meanwhile, the stability of the DC microgrid has been analyzed. An optimal control algorithm to select the operation modes of the DC microgrid is proposed. The algorithm stabilizes and improves the performance of the DC microgrid with various operation modes. R EFERENCES

Fig. 9. The switching pattern between mode 1 (stable, suboptimal mode) and mode 2 (unstable, optimal mode) with ρ = 1, 0.01.

Fig. 10. The norm of the small-signal branch current trajectory from the switching pattern in Figure 9.

obtained from the optimal control formulation (31). The optimal control algorithm takes transient stability into account and optimally selects the operation modes. It is interesting to see how the penalty affects the switching pattern of the system. In Figure 9, the penalty term ρ is set to 1, during the first 10 seconds, the switching (form mode 2

[1] N. Hatziargyriou, H. Asano, R. Iravani, and C. Marnay, “Microgrids,” IEEE Power Energy Mag., vol. 5, no. 4, pp. 78–94, Jul./Aug. 2007. [2] H. Nikkhajoei and R. H. Lasseter, “Distributed generation interface to the CERTS microgrid,” IEEE Trans. Power Del., vol. 24, no. 3, pp. 1598–1608, Jul. 2009. [3] J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. de Vicuña, and M. Castilla, “Hierarchical control of droop-controlled AC and DC microgrids—A general approach toward standardization,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 158–172, Jan. 2011. [4] C. Liang and M. Shahidehpour, “DC microgrids: Economic operation and enhancement of resilience by hierarchical control,” IEEE Trans. Smart Grid, vol. 5, no. 5, pp. 2517–2526, Sep. 2014. [5] A. A. A. Radwan and Y. A.-R. I. Mohamed, “Linear active stabilization of converter-dominated DC microgrids,” IEEE Trans. Smart Grid, vol. 3, no. 1, pp. 203–216, Mar. 2012. [6] H. Kakigano, Y. Miura, and T. Ise, “Distribution voltage control for DC microgrids using fuzzy control and gain-scheduling technique,” IEEE Trans. Power Electron., vol. 28, no. 5, pp. 2246–2258, May 2013. [7] A. Kwasinski and C. N. Onwuchekwa, “Dynamic behavior and stabilization of dc microgrids with instantaneous constant-power loads,” IEEE Trans. Power Electron., vol. 26, no. 3, pp. 822–834, Mar. 2011. [8] Y. Gu, X. Xiang, W. Li, and X. He, “Mode-adaptive decentralized control for renewable DC microgrid with enhanced reliability and flexibility,” IEEE Trans. Power Electron., vol. 29, no. 9, pp. 5072–5080, Sep. 2014. [9] L. Xu and D. Chen, “Control and operation of a DC microgrid with variable generation and energy storage,” IEEE Trans. Power Del., vol. 26, no. 4, pp. 2513–2522, Oct. 2011. [10] T. Dragicevic, X. Lu, J. C. Vasquez, and J. M. Guerrero, “DC microgrids—Part I: A review of control strategies and stabilization techniques,” IEEE Trans. Power Electron., to be published. [11] J. M. Guerrero, P. C. Loh, T.-L. Lee, and M. Chandorkar, “Advanced control architectures for intelligent microgrids—Part II: Power quality, energy storage, and AC/DC microgrids,” IEEE Trans. Ind. Electron., vol. 60, no. 4, pp. 1263–1270, Apr. 2013. [12] H. Kakigano, Y. Miura, and T. Ise, “Low-voltage bipolar-type DC microgrid for super high quality distribution,” IEEE Trans. Power Electron., vol. 25, no. 12, pp. 3066–3075, Dec. 2010. [13] A. Maknouninejad, Z. Qu, F. L. Lewis, and A. Davoudi, “Optimal, nonlinear, and distributed designs of droop controls for DC microgrids,” IEEE Trans. Smart Grid, vol. 5, no. 5, pp. 2508–2516, Sep. 2014. [14] T. Dragicevic, J. M. Guerrero, J. C. Vasquez, and D. Skrlec, “Supervisory control of an adaptive-droop regulated DC microgrid with battery management capability,” IEEE Trans. Power Electron., vol. 29, no. 2, pp. 695–706, Feb. 2014. [15] H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear systems: A survey of recent results,” IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 308–322, Feb. 2009. [16] S. Anand and B. G. Fernandes, “Reduced-order model and stability analysis of low-voltage DC microgrid,” IEEE Trans. Ind. Electron., vol. 60, no. 11, pp. 5040–5049, Nov. 2013.

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Wann-Jiun Ma (S’11–M’15) received the B.S. degree in mechanical engineering from National Taiwan University, Taipei City, Taiwan; the M.S. degree in electrical and computer engineering from the University of Maryland at College Park, College Park, MD, USA; and the Ph.D. degree in electrical engineering from the University of Notre Dame, Notre Dame, IN, USA. He is currently a Postdoctoral Associate with the Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA. His research interests include distributed and scalable machine learning and optimization, and reliable and efficient design of large-scale systems.

Jianhui Wang (M’07–SM’12) received the Ph.D. degree in electrical engineering from the Illinois Institute of Technology, Chicago, IL, USA, in 2007. He is currently the Section Lead of Advanced Power Grid Modeling with the Energy Systems Division, Argonne National Laboratory, Lemont, IL, USA. Dr. Wang is an Affiliate Professor with Auburn University and an Adjunct Professor with the University of Notre Dame. He has held visiting positions in Europe, Australia, and Hong Kong, including a VELUX Visiting Professorship with the Technical University of Denmark. He was a recipient of the IEEE Power and Energy Society (PES) Power System Operation Committee Prize Paper Award in 2015. He is the Secretary of the IEEE PES Power System Operations Committee. He is an Associate Editor of the Journal of Energy Engineering and an Editorial Board Member of Applied Energy. He is the Editor-in-Chief of the IEEE T RANSACTIONS ON S MART G RID and is an IEEE PES Distinguished Lecturer.

IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 6, NOVEMBER 2016

Xiaonan Lu (S’11–M’14) was born in Tianjin, China, in 1985. He received the B.E. and Ph.D. degrees in electrical engineering from Tsinghua University, Beijing, China, in 2008 and 2013, respectively. From 2010 to 2011, he was a guest Ph.D. student with the Department of Energy Technology, Aalborg University, Denmark. From 2013 to 2014, he was a Postdoctoral Researcher with the Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville. In 2015, he joined the Energy Systems Division, Argonne National Laboratory, where he is currently a Postdoctoral Appointee. His research interests are modeling and control of power electronic converters in renewable energy systems and microgrids, hardware-in-the-loop real-time simulation, active distribution system, multilevel converters, and matrix converters. He was a recipient of the Outstanding Reviewer Award for the IEEE T RANSACTIONS ON P OWER E LECTRONICS in 2013 and IEEE T RANSACTIONS ON S MART G RID in 2015. He was the Co-Chair of the special session entitled DC Microgrids: Control, Operation, and Trends at the IEEE Energy Conversion Congress and Exposition in 2015. Dr. Lu is a Member of the IEEE PELS, IAS, and PES Societies.

Vijay Gupta received the B.Tech. degree from the Indian Institute of Technology, Delhi, and the M.S. and Ph.D. degrees from the California Institute of Technology, all in electrical engineering. He served as a Research Associate at the Institute for Systems Research, University of Maryland, College Park. He is an Associate Professor with the Department of Electrical Engineering, University of Notre Dame. His research interests include cyber-physical systems, distributed estimation, detection and control, and in general, the interaction of communication, computation, and control. He was a recipient of the NSF CAREER Award in 2009 and the Donald P. Eckman Award from the American Automatic Control Council in 2013.