Safety during Transient Response in Direct Current Microgrids using Control Barrier Functions
SSafety during Transient Response in Direct CurrentMicrogrids using Control Barrier Functions
K. C. Kosaraju, S. Sivaranjani, and V. Gupta
Abstract —We consider the problem of guaranteeing that thetransient voltages and currents stay within prescribed bounds inDirect Current (DC) microgrids, when the controller does nothave access to accurate system dynamics due to the load beingunknown and/or time-varying. To achieve this, we propose anoptimization based controller design using control barrier func-tions. We show that the proposed controller has a decentralizedstructure and is robust with respect to the uncertainty in theprecise values of the system parameters, such as the load.
I. I
NTRODUCTION
The design and operation of microgrids, which are intercon-nected clusters of Distributed Generation Units (DGUs), loadsand energy storage units interacting with each other throughdistribution lines, has been widely studied in the literature.This paper focuses on Direct Current (DC) microgrids [1]–[4].The main control objective in DC microgrids is to ensure thatthe load voltage is stabilized to a desired value [5]. To achievethis objective, several control methods have been proposed,e.g., droop control [6], plug-and-play control [7], sliding modecontrol [8], passivity-based control [9], output regulation [10]and input-to-state stability based control [11].However, there is limited literature studying the problemof maintaining the voltages and currents in the system withinsome prescribed bounds during transient operation in DC mi-crogrids. Violating such constraints could lead to deteriorationof equipment, ultimately leading to its failure. In this paper,we consider this problem. There are at least two challengeshere. First, the control strategy should be decentralized (orat least distributed), for reasons of scalability and robustness.Second, since the knowledge of system parameters (such asthe load values) is always imperfect, the controller should notrely on precise values of such parameters being available.As a first step towards solving the more general problem,we design a decentralized control algorithm for voltage andcontrol regulation during transient operation for an islandedDC microgrid with purely resistive lines that does not requireaccurate information of load values.Our controller design relies on casting the problem as one ofsafety and utilizing Control Barrier Functions (CBFs) to ensurethat the system trajectories remain in a desired safe set. CBFsare now a widely accepted tool for designing safety basedcontrollers [12]–[14]. CBFs guarantee the existence of controlinputs under which a super-level set of a function (typically
K.C. Kosaraju, V. Gupta are with the Department of Electrical Engineering,University of Notre Dame, Notre Dame, IN 46556, USA (email: { kkosaraj,vgupta2 } @nd.edu). S. Sivaranjani is with the Department of Electrical andComputer Engineering, Texas A&M University, College Station, TX 77843,USA (email: [email protected]). representing a specification such as safety) is forward invariantunder a given dynamics. However, they do not seem tohave been widely explored in the context of DC microgrids.Our main contribution over the existing literature is that weutilize CBFs to design an optimization based controller forDC microgrids that guarantees constraint satisfaction duringtransient response, while being decentralized and not requiringa precise knowledge of the system parameters.The paper is organized as follows. In Section II, we presentthe model of the DC microgrid and provide the problemformulation. In Section III, we propose a new control designusing control barrier functions that solves the problem. InSection IV, we corroborate the proposed control design insimulations using a DC microgrid with four buck converters.Finally, in Section V, we conclude and provide some directionsfor future work. For completeness, a short review of controlbarrier functions is provided in Appendix A. Notation: R n denotes the space of n -dimensional realvectors and R denotes the space of real numbers. denotes thevector of all ones with the dimension clear from the context.For a vector v ∈ R n , v T denotes its transpose, (cid:107) v (cid:107) denotes its2-norm, v i denotes its i -th element, and [ v ] denotes a diagonalmatrix with v i as the i th diagonal entry, ∀ i ∈ { , . . . , n } . Fortwo vectors u, v ∈ R n , the inequality u ≤ v is interpretedelement wise. A graph G = ( V , E ) is defined by a node set V and an edge set E . For a set S , |S| denotes its cardinality.A function h : R n → R is said to be of class C k if the first k derivatives exist and are continuous. A continuous function h : [0 , a ) → [0 , ∞ ) , a > is said to belong to class K function if it is strictly increasing and h (0) = 0 . It is said tobelong to class K ∞ if a = ∞ and h ( r ) → ∞ as r → ∞ .Given f ( x ) : R n → R n and h ( x ) ∈ C , L f h ( x ) denotes theLie derivative of h ( x ) along the direction of f ( x ) . A function f : A → B is Lipschitz if there exists a constant L satisfying (cid:107) f ( b ) − f ( a ) (cid:107) ≤ L (cid:107) b − a (cid:107) , for all a, b ∈ A and class C if it is continuously differentiable.II. DC M ICROGRID M ODEL AND P ROBLEM F ORMULATION
In this section, we introduce the DC microgrid model andformulate the problem of guaranteeing bounds on voltage andcurrent during transient operation.
A. DC microgrid model
We model a DC microgrid by a connected undirected graph G = ( V , E ) , where each node i ∈ V represents a distributedgenerating unit (DGU) containing a voltage source, buckconverter to step-down the voltage, and a load, and eachedge e ∈ E represents a transmission line interconnecting the a r X i v : . [ ee ss . S Y ] F e b s,i Switch i L i I i u i V i G i C i R ij DGU i Line ij Fig. 1: Electrical scheme of DGU i ∈ V and line k ∼ { i, j } ∈E , j ∈ N i , where N i is the set of the DGUs connected toDGU i .corresponding DGUs. Let |V| = n and |E| = m. For eachedge ( u, v ) ∈ E , arbitrarily choose one of the ends and assignit the value +1 ; similarly, assign the value − to the otherend. Define the incidence matrix of the graph G by B ∈ R n × m through the relation: B ik = +1 if ( i, k ) ∈ E and +1 was assigned to i − if ( i, k ) ∈ E and − was assigned to i otherwise . Note that, by construction, B (cid:62) . (1)We assume that each transmission line is purely resistive [15].A schematic of each DGU is provided in Figure 1 andthe variables used to specify the dynamic evolution of themicrogrid are summarized in Table I.TABLE I: A summary of the used variables States and input Parameters I i Generated current L i Filter inductance V i Load voltage C i Shunt capacitor u i Control input G i Load impedance G li , G hi Bounds for load impedance V s,i Voltage source R tk Line resistance
The state of the DC microgrid is specified via the voltageat every node and current through every edge. At each node i ,denote the current through the inductor L i by I i , and denotethe voltage across the load impedance G i by V i . Stack thesevariables for all the nodes to form the current and voltagevectors I, V : R + → R n . The control input at the i -th nodeis given by the duty-ratio of the buck converter of the i -thDGU, which is denoted by u i ∈ (0 , . V s,i ∈ R + denotes thevoltage source connected to the i -th node. Stack the controlinputs and the voltage sources for all the nodes to define thecontrol input and the voltage source for the entire system as u : R + → (0 , n and V s ∈ R n + , respectively. Finally, definepositive definite and diagonal matrices L, C, G ∈ R n × n and R t ∈ R m × m obtained by stacking and diagonalizingthe resulting vector of DGU inductances, capacitances, loadimpedances and line resistances. With these quantities defined, the dynamics governing theDC microgrid are given by the relations − L ˙ I = V − V s uC ˙ V = I − GV − B R − t B (cid:62) V. (2)For future reference, denote G p (cid:44) G + B R − t B (cid:62) . (3)We make the following assumptions in the paper. Assumption 1.
The current I i and the voltage V i are availableby direct measurements at each DGU i ∈ V . Assumption 2.
While the exact value of the load G isunknown, upper and lower bounds are known as G l ≤ G ≤ G h , (4) where G l , G h > and the inequalities are interpretedelementwise.B. Problem formulation The control objective in the microgrid defined by theequations (2) is to design the control input u such that thevoltage V across the load G is stabilized to a desired value.In our formulation, we have assumed that the exact value of theload G is unknown. Thus, the traditional objective of ensuringa specified value for the voltage may be too stringent. Instead,we assume that each load has a safe operating region in termsof a permitted lower bound v l ∈ R and upper bound v h ∈ R for the voltage across the load G i . Consequently, we definethe first control objective as follows: Objective 1 (Safe voltage regulation) . The voltage across theload G must satisfy v l ≤ V ( t ) ≤ v h , t ≥ . (5)The second objective is to prevent the over or under drawingthe current from the source. Thus, we define the second controlobjective as follows: Objective 2 (Safe current regulation) . The current through theload must satisfy the bounds I l ≤ I ( t ) ≤ I h , t ≥ , (6) where I l , I h ∈ R are lower and upper bounds for theallowable values of the current. The problem we consider can now be stated as follows.
Problem Statement:
Given the DC microgrid as described in(2), we would like to design decentralized control inputs u i ateach node i ∈ V to achieve Objectives 1 and 2, with each u i being computed using only local information about the statevariables I i , V i at node i .Since the desired voltage across the load should be less thanthe supply voltage V s , a trivial bound for v h in (5) is givenby the source vector V s . To see this more formally, note thatfor a given constant input u ∈ R n , the corresponding steadystate solution ( I, V ) of (2) satisfies − V + V s u = 0 , (7) I − GV − B R − t B (cid:62) V = 0 . (8)hus, the set of all feasible forced equilibria are given bythe tuples ( I, V , u ) that satisfy (7) and 8. Since < u ≤ , the equation (7) implies that V ≤ V s . We formalize thisthrough the following assumption needed for the feasibility ofthe problem. Assumption 3.
We assume that v l ≤ v h < V s . (9)Finally, we assume that the problem is feasible. Assumption 4.
The problem stated above is feasible in thesense that there exists at least one control sequence achievingObjectives 1 and 2. In particular, the initial conditions for thevoltages V and the currents I are in the sets defined in (5)and (6). III. P
ROPOSED S OLUTION
The problem formulated above poses two difficulties. Thefirst is that of guaranteeing the safety objectives 1 and 2. Thesecond is the limited knowledge of the load G . For pedagogicalease, we tackle these issues sequentially and present oursolution to the problem formulated above in three steps: • We first show how Objective 1 can be guaranteed whenthe load G is known. • We then extend the solution to the case when only a lowerand an upper bound on the load G are known. • Finally, we extend the solution to also include Objec-tive 2.The proofs of all the results are provided in the Appendices.
A. Guaranteeing Objective 1 with known load
We begin by designing a controller that guarantees objec-tive 1 when the load G is known. We utilize Control BarrierFunctions (CBFs) for the purpose. A brief introduction toCBFs is provided in Appendix A for the interested reader.Specifically, for all nodes i ∈ V , we propose the controllerobtained by solving the following optimization problem: u opt i, = arg min a i ∈ R (cid:107) a i (cid:107) (10)s.t. a i V s,i − V i + η l,i ( I i − Gv l ) ≥ − a i V s,i + V i − η h,i ( I i − Gv h ) ≥ ≤ a i ≤ , where ≤ η l,i , η h,i ≤ are tuning parameters.We now show that this controller stabilizes the system tothe safe set (1). Theorem 1.
Consider the problem formulated in Section IIwith the load G being known. The controller u opt i, designed in (10) ensures that the system (2) satisfies Objective 1.Proof. See Appendix B.Note that the last constraint in the QP (10) models thephysical limits of the control input.
Remark 1 (Decentralized controller) . In the proposed opti-mization based controller, computing u opt i, requires only localinformation of the states ( V i , I i ) . Thus, the controller has adecentralized structure as required. The proposed controller is independent of the exact valuesof the parameters L and C . Thus, even if we have limitedknowledge of these parameters, the proposed controller pro-vides some robustness with respect to variation in the valuesof these parameters. B. Guaranteeing Objective 1 with unknown load
While the decentralized controller presented in Proposition1 ensures that the voltage V ( t ) lies with in the prescribedsafe set, it requires an accurate knowledge of the load G .In practice, the exact value of the load is unknown andusually tends to change slowly with time. We now extend thecontroller presented above to not require this knowledge andhence be robust with respect to the precise value of the load G .Specifically, we consider the controller obtained by solvingthe following optimization problem: u opt i, = arg min a i ∈ R (cid:107) a i (cid:107) (11)s.t. a i V s,i − V i + η l,i ( I i − G l v l ) ≥ − a i V s,i + V i − η h,i ( I i − G h v h ) ≥ ≤ a i ≤ , where ≤ η l,i , η h,i ≤ are tuning parameters. We can thenstate the following result. Theorem 2.
Consider the problem formulated in Section IIwith only the bounds G l and G h on the load G being known.The controller u opt i, calculated as proposed in (11) ensuresthat the system (2) satisfies Objective 1.Proof. See Appendix C.
C. Guaranteeing both Objectives 1 and 2
To satisfy Objective 2, we can proceed in a similar fashionas above. However, we note that the voltages and currents ineach DGU are not independent, and hence, their constraintsneed to be studied jointly. To this end, we note that if wedefine ˜ I l (cid:44) max { v l G l , I l } ˜ I h (cid:44) min { v h G h , I h } , (12)where the operations max and min are defined elementwise,then satisfying the conditions ˜ I l ≥ and ˜ I h ≥ will ensurethat both Objectives 1 and 2 are met.We consider the controller obtained by solving the followingoptimization problem: u opt i, = arg min a i ∈ R (cid:107) a i (cid:107) (13)s.t. a i V s,i − V i + η l,i ( I i − ˜ I l,i ) ≥ − a i V s,i + V i − η h,i ( I i − ˜ I h,i ) ≥ ≤ a i ≤ , where ≤ η l,i , η h,i ≤ are tuning parameters and state thefollowing result. Theorem 3.
Consider the problem formulated in Section II.The controller u opt i, calculated as proposed in (13) ensuresthat the system (2) satisfies both Objectives 1 and 2.roof. The proof is analogous to that of Theorems 1 and 2and is omitted.
Remark 2.
Although for notational and pedagogical ease,we have assumed in the above development that the lowerbound I l and the upper bound I h for all the DGUs are thesame, all the arguments above can be extended to considerheterogeneous bounds. In fact, in the simulation study below,we demonstrate the heterogeneous case. IV. S
IMULATION R ESULTS I l I l I l I l Fig. 2: DC microgrid considered with four buck converters.In this section, we illustrate the proposed controller througha simulation study. We consider a Kron reduced DC microgridconsisting of four DGUs connected as shown in Figure 2.The parameters of the DGU and distribution lines are reportedin Table II and Table III, respectively. These parameters aresimilar to those used in [1], [9] and [16] for simulationand experimentation respectively. We assume that the desirednominal voltage for each DGU is . , with the safe regionbetween . and . . Moreover, the precise valueof the load parameter G is unknown and only the bounds G l = 0 . × G and G h = 1 . × G are known. Finally, foroptimal working of the voltage source, we assume that thegenerating currents are bounded as shown in in Table II.At time t = 0 , we start the system with a feasible voltagevalue. However, to test if our method can handle slight viola-tions of the feasibility Assumption 4, the generating currentsare initialized outside the prescribed safe region. Since theproposed controller assumes initialization within the feasibleset, we use a numerical heuristic to bring the desired valuesquickly within the feasible region. Specifically, we modify theoptimization problem (13) to u opt i = arg min a i ,(cid:15) l,i ,(cid:15) h,i ∈ R a i + P l,i (cid:15) l,i + P h,i (cid:15) h,i (14)s.t. a i V s,i − V i + η l,i ( I i − ˜ I l,i ) + (cid:15) l,i ≥ − a i V s,i + V i − η h,i ( I i − ˜ I h,i ) + (cid:15) h,i ≥ ≤ a i ≤ , for large values of constants P l,i and P h,i . Once the currentsand voltages enter the feasible set, we switch to the controller(13). The simulation is conducted for . . The tuningparameters of the controller are given in Table IV. After .
25 sec the load at all the nodes is increased by . Thevoltage and current signals of the resulting simulation areplotted in Figure 5 and Figure 4, respectively. The control input Time (sec) D u t y - r a t i o u u u u Fig. 3: Control input generated by the proposed controller.
Time (sec) C u rr e n t ( A ) I I l ,1 I h ,1 Time (sec) C u rr e n t ( A ) I I l ,2 I h ,2 Time (sec) C u rr e n t ( A ) I I l ,3 I h ,4 Time (sec) C u rr e n t ( A ) I I l ,4 I h ,4 Fig. 4: Currents from the voltage source in each DGU.generated by the optimization problem is plotted in Figure3. We observe that the controller ensures satisfaction of bothObjectives 1 and 2 (modulo the infeasbility of the specifiedinitial condition) despite the abrupt load change.V. C
ONCLUSIONS AND FUTURE WORK
We presented a new decentralized optimization based con-troller for DC microgrids. We used Control Barrier Functionsto ensure that the voltage and current signals lie within thepermitted safety bounds despite the load being unknown andpotentially time-varying. Future research directions includeextending this technique for ZIP loads and towards achievesafe current sharing in DC microgrids.R
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ABLE II: Microgrid Parameters
DGU 1 2 3 4 L i ( mH ) . . . . C i ( mF ) . . . . V (cid:63)i ( V ) . . . . V l,i ( V ) . . . . V h,i ( V ) . . . . G i ( Ω ) / . / . / . / . I l,i ( A ) . . . . I h,i ( A ) . . . . TABLE III: Line Parameters
Line 1 2 3 4 R k ( mΩ )
70 50 80 60
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Time (sec) V o l t a g e ( V ) V V l ,1 V h ,1 Time (sec) V o l t a g e ( V ) V V l ,2 V h ,2 Time (sec) V o l t a g e ( V ) V V l ,3 V h ,4 Time (sec) V o l t a g e ( V ) V V l ,4 V h ,4 Fig. 5: Voltage across the load of each DGU, considering aload variation of at time t = 0 . seconds.VI. A PPENDIX
A: C
ONTROL B ARRIER F UNCTIONS
Consider the affine nonlinear system ˙ x = f ( x ) + g ( x ) u (15)with f and g are locally Lipschitz continuous, x ∈ R n and u ∈ U ⊂ R m . Let Ω be a zero super level-set of a continuouslydifferentiable function h : R n → R , i.e., Ω = { x ∈ R n | h ( x ) ≥ } . (16)The function h ( x ) is called a zeroing control barrier function(ZCBF), if there exists a locally Lipschitz extended class K function α such that sup u ∈ R m { L f h ( x ) + L g h ( x ) u + α ( h ( x )) } ≥ ∀ x ∈ R n (17)where L f h ( x ) and L g h ( x ) denote the Lie derivative of h ( x ) in the direction of f and g , respectively. Given a ZCBF h ( x ) ,define the set for all x ∈ Ω K zcbf ( x ) = { u ∈ U : L f h ( x ) + L g h ( x ) u + α ( h ( x )) ≥ } . (18) Theorem 4 (See [12], [13]) . Consider system (15) , with x ∈ R n and u ∈ U denoting the state and the input, respectively, fand g are locally Lipschitz. Let Ω be a zero super level-set of continuously differentiable function h ( x ) , defined in (16) . Ifthe relative-degree of (15) w.r.t h ( x ) is (i.e., L g h ( x ) (cid:54) = 0 ),then any Lipschitz continuous controller u ∈ K zcbf ( x ) rendersthe set Ω forward invariant and asymptotically stable. For many physical systems, including the one consideredin the paper, the relative degree r is greater than one. In thiscase, as shown in [14], one can recursively define the controlbarrier functions h i ( x ) = L f h i − ( x ) + α I ( h i − ( x )) and theirzero super level sets Ω i = (cid:8) x ∈ R n | h i − ( x ) ≥ (cid:9) , where i ∈ { . . . r − } , α , · · · , α r − are all class K functions and h ( x ) = h ( x ) . This is the construction used in the paper.A PPENDIX
B: P
ROOF OF T HEOREM b l , b h : R n → R n as b l (cid:44) V − v l b h (cid:44) v h − V, (19)and their corresponding zero super level-set of (19) as Ω b (cid:44) (cid:8) ( I, V ) ∈ R n | b l ≥ , b h ≥ (cid:9) . (20)Note that for all ( I, V ) ∈ Ω b , the voltage V satisfies Objective1. Since from Assumption 4, the initial condition is in this set,our proof below will show that the controller in (10) rendersthe set Ω b forward invariant and asymptotically stable for thesystem (2).To this end, consider b l as the candidate zeroing controlbarrier function. By choosing C − G p as the class K function,the condition (17) for a zeroing control barrier function statedin the Appendix A simplifies to ˙ b l + C − G p b l ≥ . (21)Multiplying C on both the sides yields that (21) is equivalentto the condition C ˙ V + G p ( V − v l ) ≥ . (22)Using the relation C ˙ V + G p V = I from (2) and (3), we canrewrite (22) as I − v l G p ≥ . (23)Finally, since B (cid:62) , we can simplify (23) to the relation I − Gv l ≥ . (24)Similarly, by considering b h as the zeroing control barrierfunction, we can obtain the condition − I + Gv h ≥ . (25)Since the zero super level set of b l and b h is given by Ω b , Theorem 4 implies that if (24) and (25) hold, then Objective 1is met.However, there is no control input appearing in (24)and (25). This is a consequence of the fact that the relativedegree of the system (2) with respect to either b l or b h is . Following [14], we can overcome this issue by recursivelydefining zeroing control barrier functions. To this end, definefunctions B l , B h : R n → R n , given by B l (cid:44) I − v l G ,B h (cid:44) − I + v h G , (26) with the zero super level set Ω B (cid:44) (cid:8) ( I, V ) ∈ R n | B l ≥ , B h ≥ (cid:9) . (27)To enforce (24), consider B l as the candidate zeroing controlbarrier function. By using L − [ η l ] as the class K function, wecan write the condition in (17) as ˙ B l + L − [ η l ] B ≥ . (28)Multiplying both sides of (28) with L and simplifying byusing (2), we can rewrite this condition as − V + [ V s ] u + [ η l ]( I − Gv l ) ≥ . (29)Similarly, by considering B h as the candidate zeroing controlbarrier function, we can obtain the condition V − [ V s ] u − [ η h ]( I − Gv h ) ≥ . (30)Thus, we have shown that enforcing (29) and (30) ensuresthat the set Ω B is forward invariant, which further enforces that Ω b is forward invariant and asymptotically stable and hencethat Objective 1 is met. But the first two constraints in theoptimization problem (10) are equivalent to (29) and (30).Thus, we conclude that the controller u opt i, designed in (10)ensures that the system (2) satisfies Objective 1.A PPENDIX
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