Scheduling of Separable Mobile Energy Storage Systems with Mobile Generators and Fuel Tankers to Boost Distribution System Resilience
TThis work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. Abstract —Mobile energy resources (MERs) have been shown to boost DS resilience effectively in recent years. In this paper, we propose a novel idea, the separable mobile energy storage system (SMESS), as an attempt to further extend the flexibility of MER applications. “Separable” denotes that the carrier and the energy storage modules are treated as independent parts, which allows the carrier to carry multiple modules and scatter them independently throughout the DS. The constraints for scheduling SMESSs involving carriers and modules are derived based upon the interactive behavior among them and the DS. In addition, the fuel delivery issue of feeding mobile emergency generators (MEGs), which was usually bypassed in previous studies involving the scheduling of MEGs, is also considered and modeled. SMESSs, MEGs, and fuel tankers (FTs) are then jointly routed and scheduled, along with the dynamic DS reconfiguration, for DS service restoration by integrating them in a mixed-integer linear programming (MILP) model. Finally, the test is conducted on a modified IEEE 33-node test system, and results verify the effectiveness of the model in boosting DS resilience.
Index Terms —Mobile energy resources, separable mobile energy storage system, fuel tankers, distribution system resilience. N OMENCLATURE
Sets Set of time spans. / DP Set of nodes/fuel depots. M / S / G Set of nodes supporting the access of MERs/SMESSs/MEGs. / S / G / F Set of MERs/Carrs (of SMESSs)/MEGs/FTs. Set of Mods of SMESSs. Set of DS branches. FO / FC Set of nodes with faulted open/closed load switches. FO / FC Set of faulted open/closed branches.
Variables x j , i , t Binary variable; 1 if MER j is parked at node i during time span t , 0 otherwise. v j , i , t Binary variable; 1 if MER j is traveling to node i during time span t , 0 otherwise. S j , t Travel time to be consumed by MER j during time span t . R j , t Residual travel time of MER j during time span t . ω j , t Binary variable; 1 if MER j is traveling during time spans t − t . ζ k , i , t Binary variable; 1 if Mod k belongs to node i during time span t , 0 otherwise. γ k , j , t Binary variable; 1 if Mod k belongs to Carr j during time span t , 0 otherwise. α j , i , k , t Binary variable; 1 if Carr j carrying Mod k arrives at node i during time span t , 0 otherwise. c k , i , t / d k , i , t Binary variable; 1 if Mod k is charged/discharged at node i during time span t , 0 otherwise. P c.S k , i , t / P d.S k , i , t Active power output of Mod k charged/discharged at node i during time span t . Q S k , i , t Reactive power output of Mod k charged/discharged at node i during time span t . SOC k , t State of charge of Mod k at the end of time span t . P G m , i , t / Q G m , i , t Active/reactive power output of MEG m at node i during time span t . B m , i , t / B + m,i,t Gross/extra fuel demand for the generation of MEG m at node i during time span t . τ m , t , l Binary variable; 1 if the output of MER m during time span t is within the l th interval [ p m , l − , p m , l ], 0 otherwise. SOF m / h / i , t State of fuel of MEG m /FT h /node (depot) i at the end of time span t . b m , t Binary variable; 1 if B m , i , t ≤ SOF m , t · F m , 0 otherwise. ι m/h,i,t Binary variable; 1 if MEG m/FT h can exchange fuel with node (depot) i , 0 otherwise. G m , i , t Supplemented fuel of MEG m at node (or fuel depot) i during time span t . D h , i , t Fuel output of FT h at node (or fuel depot) i during time span t . f ix ii' , t Flow of commodity i x from nodes i to i' during time span t . λ ii' , t Binary variable; 1 if arc ( i , i' ) is included in the directed fictitious spanning tree during time span t , 0 otherwise. μ ii' , t Binary variable; 1 if branch ( i , i' ) is included in the fictitious spanning tree during time span t , 0 otherwise. κ ii' , t Binary variable; 1 if branch ( i , i' ) of the DS is closed during time span t , 0 otherwise. P i'i,t / Q i'i,t Active/reactive power flow on branch ( i' , i ) from node i' to node i during time span t . P IN i , t / Q IN i , t Active/reactive power injected by a Mod or MEG into node i during time span t . δ i , t Binary variable; 1 if the load at node i is picked up during time span t , 0 otherwise. V i , t Squared voltage magnitude at node i during time span t . r i'i /x i'i Resistance/reactance of branch ( i' , i ). η i , t Binary variable; 1 if node i is energized during time span t , 0 otherwise. ρ i , t Binary variable; 1 if at least one Mod or MEG is connected to node i during time span t , 0 otherwise. σ i , t Binary variable; 1 if at least one energized node is connected to node i during time span t , 0 otherwise. χ i' - ii' , t Binary variable; 1 if node i' is connected to node i and energized during time span t , 0 otherwise. Parameters Δ t Length of a single time span. M / ε A large/small positive number. T j , ii' Time spans spent traveling from node i to node i' for MER j . W k Capacity consumed by Mod k . A j Carrying capacity Carr j . P c.S k ,max / P d.S k ,max Maximum charging/discharging power of Mod k . S k ,Mod Rated apparent power of Mod k . E k Energy capacity of Mod k . e c k / e d k Charging/discharging efficiency of Mod k . SOC k ,min / SOC k ,max Lower/upper bound of allowable range of
SOC k , t . P G m ,max / Q G m ,max Maximum active/reactive power output of MEG m . S m ,MEG Rated apparent power of MEG m . B m ,max Maximum fuel consumption of MEG m at full load. Wei Wang,
Student Member, IEEE , Xiaofu Xiong,
Member, IEEE , Yufei He, Jian Hu, Hongzhou Chen.
Scheduling of Separable Mobile Energy Storage Systems with Mobile Generators and Fuel Tankers to Boost Distribution System Resilience p m , l Load level l to depict the fuel consumption characteristic of MEG m . l Î {1, 2, …, L }, where L is the number of those levels. y m , l , z m , l Coefficients of piecewise linearization for fuel consumption characteristic of MEG m within the interval [ p m , l − , p m , l ]. F m / h / i Available fuel storage capacity of MEG m /FT h /node (depot) i . υ h ,in / υ h ,out Maximum rate of fuel input/output of FT h . i r The substation node. P L i , t / Q L i , t Active/reactive load at node i during time span t . a i , a i Coefficients used to determine P IN i , t / Q IN i , t . V i ,min / V i ,max Lower/upper bound of the allowable voltage magnitude at node i . S i'i,max Apparent power capacity of branch ( i' , i ). w i Priority weight of the load at node i . φ travel / φ fuel Cost coefficient for MERs for traveling/fuel exchange. I. I NTRODUCTION
OBILE energy resources (MERs) are powerful tools that make distribution systems (DSs) respond to disasters in a rapid and flexible way [1]. The intelligent use of two main types of MERs, i.e. , mobile energy storage systems (MESSs) and mobile emergency generators (MEGs), has been highlighted in recent years due to their effectiveness in enhancing power system resilience and economics [2], [3]. To enhance DS resilience, in [4], MEGs are prepositioned prior to a disaster and repositioned to energize the loads after the disaster and resulting damage are known. In [5], the dynamic routing of MERs is further considered. In [6] and [7], MESSs are routed among several parts of a DS or several DSs to realize optimal energy allocation under emergency situations and the DS reconfiguration is involved, which means that the DS topology can be changed to coordinate with the connection of MESSs. In [8], repair crew dispatch is further considered along with the routing and scheduling of MERs and DS reconfiguration for DS restoration. In addition, MESSs also contribute to the improvement in power system economics, whether for bulk power systems [9] or DSs [10]. In [11], the investment decision is studied considering the operation of MESSs during both normal and emergency conditions. From a review of the relevant studies, MERs can truly act as effective “first-aid boxes” for DS service restoration. Compared to the common stationary energy resources, MERs bring higher flexibility to DS restoration, and thus endow DSs with more feasibility and a better response to disasters. Now, standing on the shoulders of the authors of these studies, a conjecture may arise: Can this flexibility be further extended? To answer this question, we focus our attention on the MESS. Note that in the present studies, the energy storage module (Mod) and carrier (Carr) of an MESS are “fastened” together to be scheduled as one whole, which implies that equivalently only one Mod is contained inside. This restriction creates a gap between the research and the practical fields, if we focus on the state of the art of energy storage and MESS. As a vital characteristic of battery-based energy storage applications and development, the modularity enables energy storage systems to be scalable within a wide range regarding the power or energy capacity [12], [13]. In brief, one can tailor the desired energy storage capacity by easily assembling multiple standard energy storage units, similar to building blocks [14], [15]. In addition, most MESS solutions currently on the market or in projects are containerized and towable [3], e.g. , the MESSs provided by RES [16], Consolidated Edison [17], and Aggreko [18]. Thus, these MESSs are inherently separable, i.e. , the Mod can be easily loaded on/unloaded from the Carr. Given the above, the idea of a separable MESS (SMESS) is conceived in this paper: We release the above restriction and “split” the only Mod of an MESS into multiple smaller Mods and then let these Mods be independently scattered by the Carr throughout the DS, similar to “one rocket launching multiple satellites”. By this action, a more decentralized allocation of energy storage as back-up sources can be realized. Note that the traditional MESS solution is included in the SMESS since we need only to make the Mods and the Carr of an SMESS always be together. Thus, the SMESS solution may be able to endow DS restoration with more flexibility and feasibility, which is attractive as it offers the possibility to push the MERs application to a higher level. Then, we move on to MEGs. Most of them are currently powered by fossil fuels. Once an MEG is deployed somewhere in the DS, there should be adequate fuel to ensure its operation for a period of time. Fuel delivery is an inevitable task in deploying MEGs for DS restoration [3], [19] and a critical issue related to supply resilience, as introduced by the EPRI [20]. However, apart from only a few studies involving the scheduling of MEGs, most related studies commonly bypassed the fuel delivery issue and left it out of the scheduling by assuming that, e.g. , the adequate fuel has been preallocated [5], [8]. Extreme weather striking DSs typically occurs suddenly and is hard to anticipate, so it is likely that such preallocation of fuel may not be fully accomplished and that this assumption is too ideal. In many circumstances, it is necessary to schedule fuel tankers (FTs) jointly with MEGs in a coordinated way to guarantee adequate fuel for MEGs operation. To the best of our knowledge, such joint scheduling has not yet been developed. To bridge the above gaps, in this paper, we propose a method for jointly scheduling SMESSs, MEGs, and FTs, along with the dynamic DS reconfiguration, for DS restoration. The main contributions of this paper are threefold: 1) A novel SMESS concept is proposed, and the constraints for scheduling SMESSs are derived. In contrast to a conventional MESS solution, an SMESS can carry multiple Mods and scatter them independently throughout the DS. 2) Fuel delivery coordinated with the scheduling of MEGs is considered, and the constraints for scheduling FTs are derived. 3) A joint scheduling model for DS restoration, involving the routing and scheduling of SMESSs, MEGs, and FTs and the scheduling of the dynamic DS reconfiguration, is developed in the form of mixed-integer linear programming (MILP) and its advantage in boosting the DS resilience is verified by tests. The rest of this paper is organized as follows: Section II derives the constraints for routing and scheduling of SMESSs, MEGs and FTs; Section III gives the constraints for dynamic DS reconfiguration and operation; Section IV summarizes the joint scheduling model for DS restoration; Section V provides the numerical studies; and Section VI concludes this paper. II. S CHEDULING OF
SMESS S , MEG
S AND F UEL D ELIVERY A. Evolution from MESS to SMESS
According to the investigation of [3], most of the MESS products at present are towable, i.e. , they are combinations of tractors and trailers. As depicted in Fig .1, the key evolutions from the scheduling of MESSs to SMESSs can be recognized as follows: 1) A Carr ( i.e. , a tractor) carries multiple detachable M B. Routing of MERs
In our prior work [21], we proposed a novel mobility model for the routing of MERs, which was shown to have a good computational efficiency. This model, as reviewed in (1), is used for the routing of all three kinds of MERs, i.e. , SMESSs (actually the Carrs of SMESSs), MEGs and FTs in this paper.
M M , , , ,
1, , j i t j i ti i x v t j ∈ ∈ + = ∀ ∈ ∈ ∑ ∑ (1a) ( ) ( ) ( ) ( )
M MM M , , 1 , , , , , , 1 , , , , 1, , 1 , , , , , , 1 , , , , 1M j i t j i t j i t j i t j i t j i ti ij i t j i t j i t j i t j i t j i ti i x x v v v vx x v v v vt j i + + +∈ ∈+ + +∈ ∈ ≥ + − + − − ≤ + − − − +∀ ∈ ∈ ∈ ∑ ∑∑ ∑ (1b) ( )
M M M , , , 1 , , , , , M, ,0, , j t j i t j ii j i t j ii j iii i ij t
S x T v T T iSt j ′ ′ ′ ′− ′ ′ ′∈ ∈ ∈ ≥ + − ∀ ∈ ≥∀ ∈ ∈ ∑ ∑ ∑ (1c) M , , 1 , , , 1 , , j t j t j t j i ti R R S v t j − −∈ = + − ∀ ∈ ∈ ∑ (1d) M , , , , , , j t j i t j ti R M v R t j ∈ ≤ ≤ ∀ ∈ ∈ ∑ (1e) ( ) ( ) M M , , , 1 , ,, , , , , 1 , M
21 1 ,, , j t j i t j i ti ij t j i t j i t j t v vv v it j ω εω ω −∈ ∈− ≥ + − +− − ≤ − ≤ − ∀ ∈∀ ∈ ∈ ∑ ∑ (1f) , ,0 ,0 ,0 ,0 j j i j j j x S R j ω= = = = ∀ ∈ , (1g) Specific explanation of (1a)-(1g) can be found in [21]. The above can realize the routing of MEGs and FTs by replacing with G and M with G DP , and realizes the routing of SMESSs by replacing with S and M with S . C. Scheduling of SMESSs
A Mod always belongs to either a node or a Carr, and the carrying capacity of Carrs cannot be exceeded. Thus, we have
S S , , , , k i t k j ti j t k ζ γ ∈ ∈ + = ∀ ∈ ∈ ∑ ∑ (2a) , , S , , k k j t jk
W A t j γ ∈ ≤ ∀ ∈ ∈ ∑ (2b) , ,0 k k i k ζ = ∀ ∈ (2c) In addition, Mod k is initially located at node i k , as denoted by (2c). All the possible interactive behaviors in the SMESS itself and between the SMESS and DS are enumerated in Fig. 2, upon which the scheduling model of the SMESS can be derived from the perspectives of both Carrs and nodes. Between Carrs and Mods Scenario 1 for Carrs . It is reasonable to assume that a node always dominate all the Mods located at it, as shown in Fig.2 (a)-(b). In other words, for a Carr, it does not own any Mod when it arrives or is parked at a node. Thus, we have S , , , , S k j t j i ti x t j k γ ∈ ≤ − ∀ ∈ ∈ ∈ ∑ (3a) Scenario 2 for Carrs . Then, when a Carr departs from a node, it can carry away some of the Mods owned by this node, as shown in Fig. 2 (c). We can formulate that as , , , , 1 , , , , 1 S S k j t k i t j i t j i t x x t j i k γ ζ − − − ≤ + − ∀ ∈ ∈ ∈ ∈ (3b)
Scenario 3 for Carrs . Other than the above scenarios, i.e. , when a Carr is traveling, as shown in Fig. 2 (d), the Carr should hold the Mods on it. This can be formulated as ( )
S SS S , , 1 , , , , , , 1, , 1 , , S , , , j i t j i t k j t k j ti ij i t j i ti i x xx x t j k γ γ − −∈ ∈−∈ ∈ − + ≤ − ≤+ ∀ ∈ ∈ ∈ ∑ ∑∑ ∑ (3c) Between Nodes and Mods Scenario 1 for nodes . If a Carr carrying a Mod arrives at a node, then this node must own this Mod, as shown in Fig. 2 (a).
Scenario 2 for nodes . Otherwise, ζ k , i , t ≤ ζ k , i , t − should be ensured, which means that node i cannot obtain Mod k when ζ k , i , t − =0 or may lose Mod k to some Carr when ζ k , i , t − =1. To distinguish the above two scenarios, a binary variable α j , i , k , t is defined to indicate whether Carr j carrying Mod k arrives at node i during time span t , i.e. , α j , i , k , t =(¬ x j , i , t − ) ˄ x j , i , t ˄ γ k , j , t − . According to [22] and [23], we can formulate it by , , , , , 1 , , , , ,, , , , , 1 , , , , , 1 , , , , 1S S j i k t j i t j i k t j i tj i k t k j t j i k t j i t j i t k j t x xx xt i j k α αα γ α γ −− − − ≤ − ≤ ≤ ≥ − + + −∀ ∈ ∈ ∈ ∈ (4a) Thus, we have S , , , , , S , , , k i t j i k tj t i k ζ α ∈ ≥ ∀ ∈ ∈ ∈ ∑ (4b) S , , , , 1 , , , S , , , k i t k i t j i k tj t i k ζ ζ α − ∈ − ≤ ∀ ∈ ∈ ∈ ∑ (4c) Constraints (4b) and (4c) represent Scenario 1 for nodes and
Scenario 2 for nodes , respectively.
Scenario 3 for nodes . In addition, if node i is “undisturbed”, i.e. , no Carr arrives at or departs from it, as shown in Fig. 2 (d), each Mod should maintain the status regarding it, i.e. , ζ k , i , t = ζ k , i , t − . This can be naturally realized by the above constraints. Suppose that node i is undisturbed during time span t , and we know from (4a) that Σ j Î S α j , i , k , t =0. When ζ k , i , t − =0, then ζ k , i , t =0 from (4c). On the other hand, when ζ k , i , t − =1, let us consider where this Mod k can go during the next time span t : i) For any other node i ( i ≠ i ), we know from (2a) that γ k , j , t − =0 for any Carr j and ζ k , i , t − =0; then, based on (4a) and (4c), we can obtain Σ j Î S α j , i k , t =0 and ζ k , i t =0. ii) For any Carr j , if it just departed from some other node i ( i ≠ i ), i.e. , x j , i , t − =1 and x j , i , t =0, we know that ζ k , i , t − =0, and then, based on (3b), γ k , j , t =0; if Carr j is traveling during time spans t − t , it is clear from (3c) that γ k , j , t − =0 and γ k , j , t =0; in addition, if Carr j is parked at some node, Fig. 2. Illustration of SMESS behavior. (a) A Carr arrives at a node; (b) a Carr is being parked at a node; (c) a Carr departs from a node; and (d) a Carr is traveling without contacting to any node. Fig. 1. Evolution from MESS to SMESS. γ k , j , t =0 due to (3a). From i) and ii) , Mod k at node i cannot go anywhere but instead stay only at node i . Thus, no additional constraints are needed for Scenario 3 for nodes . Operation of Mods , , , , , , S , , , k i t k i t k i t c d t k i ζ+ ≤ ∀ ∈ ∈ ∈ (5a) c.S c.S, , , , ,maxd.S d.S, , , , ,max SS,Mod , , , , ,Mod , ,
00 , , , k i t k i t kk i t k i t kk k i t k i t k k i t
P c PP d P t k iS Q S ζ ζ ≤ ≤ ≤ ≤ ∀ ∈ ∈ ∈− ≤ ≤ (5b) ( ) ( )
S S , , k i t k i t k i t ki i
P P Q S t k ∈ ∈ − + ≤ ∀ ∈ ∈ ∑ ∑ (5c) ( )
S S c c.S d.S d, , 1 , , , ,,min , ,max ,, , k t k t k k i t k i t k ki ik k t k
SOC SOC e P P e t ESOC SOC SOC t k − ∈ ∈ = + − ∆≤ ≤ ∀ ∈ ∈ ∑ ∑ (5d) Constraint (5a) restricts the charging/discharging mode of Mods. Constraints (5b)-(5c) bound the power output, and the reactive power support of Mods is considered, as in [10]. Constraint (5d) restricts the state of charge (SOC) of Mods. D. Operation of MEGs
G G, , , , ,max G GG G, , , , ,max m i t m i t mm i t m i t m
P x P t m iQ x Q ≤ ≤ ∀ ∈ ∈ ∈ ≤ ≤ (6a) ( ) ( )
G G , , m i t m i t mi i
P Q S t m ∈ ∈ + ≤ ∀ ∈ ∈ ∑ ∑ (6b) Constraints (6a)-(6b) bound the power output of MEGs. E. Scheduling of FTs
The gross fuel demand of an MEG at a node is restricted as , , , , ,max G G m i t m i t m
B x B t m i ≤ ≤ ∀ ∈ ∈ ∈ (7a) , , G DP m i t
B t m i = ∀ ∈ ∈ ∈ (7b) Manufacturers commonly provide the fuel consumption rate of MEGs at several load levels, e.g. , 1/4, 1/2, 3/4, and full load, as in [24]. Hence, we can formulate the gross fuel demand of an MEG as the following piecewise linearization:
G GG
G, , , , , ,G, , , 1 , , , m i t m l m i t m li im i t m l m li
B y P zif P p p ∈ ∈−∈ = + ∈ ∑ ∑∑ (7c) Equation (7c) can be further formulated as follows [22], [25]: ( ) ( ) ( ) ( )( )( ) { }
G GG GGG
G, , , , , , , ,G, , , , , , , ,G, 1 , , , ,G, , , , ,G m t l m i t m l m i t m li im i t m l m i t m l m t li im l m i t m t lim i t m l m t li
M B y P zB y P z Mp P MP p Mt m l L τ τττ ∈ ∈∈ ∈− ∈∈ − − ≤ − + − + ≤ − − ≤ − − ≤ −∀ ∈ ∈ ∈ ∑ ∑∑ ∑∑∑ (7d) { } , , G1,3, , m t ll L t m τ ∈ = ∀ ∈ ∈ ∑ (7e) Similar to the SOC for Mods, we introduce the “state of fuel (SOF)” for FTs, MEGs, nodes, and fuel depots. The fuel carried by an MEG may not always be enough to feed itself and the extra fuel demanded can be derived as “if gross fuel demand Σ i Î G B m , i , t is less than the fuel on MEG m itself SOF m , t − · F m , then MEG m is self-sufficient during time span t and no extra fuel is needed; otherwise, the extra fuel Σ i Î G B m , i , t − SOF m , t − · F m is needed.” We can introduce a binary variable b m , t as the indicator of Σ i Î G B m , i , t < SOF m , t − · F m , and the above logic is formulated as ( )( ) ( ) GG G G , , , , G DP, , , , 1 ,max ,, , ,max ,, , , , , 1 ,G m i t m i tm m t m i t m t m m m tim i t m m ti m i t m i t m t m m m ti i
B B iF b B SOF F B bB B bB B SOF F F bt m + −∈+∈ + −∈ ∈ ≤ ≤ ∀ ∈− ≤ − ≤ − ≤ − ≤ − − ≤∀ ∈ ∈ ∑∑ ∑ ∑ (7f) The SOF of relevant facilities are restricted by (7g)-(7n). ( ) G DP , , 1 , , , , , ,G , m t m t m i t m i t m i t mi SOF SOF B B G Ft m +− ∈ = − − −∀ ∈ ∈ ∑ , (7g) G DP , , 1 , , F , , h t h t h i t hi
SOF SOF D F t h − ∈ = − ∀ ∈ ∈ ∑ (7h) ( ) F G , , 1 , , , , , ,G DP , , i t i t h i t m i t m i t ih m
SOF SOF D B G Ft i +− ∈ ∈ = + − + ∀ ∈ ∈ ∑ ∑ (7i) , , , , G G DP , , , m i i m i t x t m i ι ≤ ∈ ∈ ∈ (7j) , , , , , , G G DP , , , m m i t m i t m m i t F G F t m i ι ι− ≤ ≤ ∀ ∈ ∈ ∈ (7k) , , , , F G DP , , , h i i h i t x t h i ι ≤ ∈ ∈ ∈ (7l) ( ) ( ) ,in , , , , ,out , ,F G DP , , , h h i t h i t h h i t t D tt h i υ ι υ ι− ∆ ≤ ≤ ∆∀ ∈ ∈ ∈ (7m) , G F G DP i t SOF t i ≤ ≤ ∀ ∈ ∈ (7n) III. D YNAMIC DS R ECONFIGURATION AND O PERATION A. Dynamic DS Reconfiguration
We use the flow-based model presented by [26] to formulate the constraints for dynamic DS reconfiguration as follows: ( ) ( ) , ,, , x xr rr r i ii i t i i t x ri i i i f f t i i ′ ′′ ′∈ ∈ − = − ∀ ∈ ∈ ∑ ∑ (8a) ( ) ( ) , ,, , x xx xx x i ii i t i i t x ri i i i f f t i i ′ ′′ ′∈ ∈ − = ∀ ∈ ∈ ∑ ∑ (8b) ( ) ( ) { } , ,, ,
0, , \ , \ , x x i ii i t ii t x r x ri i i i f f t i i i i i ′ ′′ ′∈ ∈ − = ∀ ∈ ∈ ∈ ∑ ∑ (8c) ( ) , , , , , , x x i ii i t i i t ii t ii t x r t f f i i i i λ λ ′ ′ ′ ′ ∀ ′≤ ≤ ≤ ≤ ∈ ∈ ∈ (8d) ( ) ( ) , ,, i i t ii ti i t λ λ ′ ′′ ∈ + = − ∀ ∈ ∑ (8e) ( ) , , , , , , i i t ii t i i t t i i λ λ µ ′ ′ ′ ∀ ∈ ′+ = ∈ (8f) ( ) , , , , , i i t i i t t i i κ µ ′ ′ ∈ ′≤ ∀ ∈ (8g) Constraints (8a)-(8g) always ensure the radiality of the DS, and specific explanation can be found in [26]. B. Operation of the DS
The constraints for DS operation are given as (9a)-(9m), based upon the linearized DistFlow model [8], [26], [27]. ( ) G IN d.S c.S G, 1, , , , , 2, , , , , i t i k i t k i t i m i tk m
P a P P a P t i ∈ ∈ = − + ∀ ∈ ∈ ∑ ∑ (9a) G IN S G, 1, , , 2, , , , , i t i k i t i m i tk m
Q a Q a Q t i ∈ ∈ = + ∀ ∈ ∈ ∑ ∑ (9b) ( ) ( )
IN L, , , , ,, , , , i i t i t i t i t ii ti i i i
P P P P t i δ ′ ′′ ′∈ ∈ + − = ∀ ∈ ∈ ∑ ∑ (9c) ( ) ( ) IN L, , , , ,, , , , i i t i t i t i t ii ti i i i
Q Q Q Q t i δ ′ ′′ ′∈ ∈ + − = ∀ ∈ ∈ ∑ ∑ (9d) , , 1 , , i t i t t i δ δ − ≥ ∀ ∈ ∈ (9e) ( ) ( )( ) ( ) ( ) i t i t i i t i i i i t i i i i ti t i t i i t i i i i t i i i i t V V P r Q x M t i iV V P r Q x M κκ ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ≥ − + − − ′∀ ∈ ∈ ≤ − + + − (9f) , , i i t i V V V t i ≤ ≤ ∀ ∈ ∈ (9g) 5 ( ) , , , i i t i i t i i t i i
P Q S t i i κ ′ ′ ′ ′ ′+ ≤ ∀ ∈ ∈ (9h) ( ) ( ) , 1 2 FO , 1 2 FC, FO , , FC
0, , ; 1, , ;,0 , ; , i i t i i ti t i t i t i i i i ti i κ κδ δ η = ∀ ∈ = ∀ ∈ ∀ ∈ = ∀ ∈ ≥ ∀ ∈ (9i) , r i t t η = ∀ ∈ (9j) ( ) ( ) { } GG , 1, , , 2, , , 1, 2, G, 1, , , 2, , ,
1, , \ i t i k i t i m i t i ik mi t i k i t i m i tk mr a a x a aa a xt i i i ρ ζρ ζ ∈ ∈∈ ∈ ≥ + + + ≤ +∀ ∈ ∈ ∈ ∑ ∑∑ ∑ (9k) ( ) ( ) ( ) ( ) ( ) { } , , , , ,, ,, , , , ,, , , , \ i t i t ii t i t i i ti i i ii t i t ii t i t i i ti i i ir
Mt i i σ η κ η κσ η κ η κ ′ ′ ′ ′′ ′∈ ∈′ ′ ′ ′′ ′∈ ∈ ≥ + ≤ +∀ ∈ ∈ ∑ ∑∑ ∑ (9l) { } , , , , , , , , , , , \ i t i t i t i t i t i t i t r t i i η ρ η σ η ρ σ≥ ≥ ≤ + ∀ ∈ ∈ (9m) The power input at each node is expressed in a general form by (9a)-(9b). The values of a i , t and a i , t are preset as follow: for i S ∩ G , a i =1 and a i =1; for i S \ G , a i =1 and a i =0; for i G \ S , a i =0 and a i =1; and for i \( S G ), a i =0 and a i =0. Constraints (9c)-(9d) ensure the power flow balance. Constraint (9e) indicates that a load that was already picked up cannot be de-energized. Constraint (9f) denotes the relationship between the voltage magnitudes of two adjacent nodes [8], [26]. Constraint (9g) bounds the voltage magnitude. We regard V i , t as a single variable; thus, (9f)-(9g) are linear. Constraint (9h) limits the power flow on each branch. Components without normal switches are considered by (9i), which restricts the connected/picked-up status of faulted branches/nodes [26]. Specifically, a load with a faulted closed load switch is picked up as long as the node where the load is located is energized. Repair resource dispatch is meaningful; nonetheless, it is beyond the scope of this paper, and relevant research can be found in [8]. Here, we assume that the repair process has already been known or anticipated in advance of scheduling and can be reflected as the shrinking sets FO , FC , FO , FC . For example, if a damaged branch ( i , j ) is to be repaired during time span t , then ( i , j ) is removed from FO afterward. Constraint (9j) assumes that the substation node is always energized. In addition, if the substation loses power from the bulk grid, then (9j) is removed. Constraints (9k)-(9m) denote that any other node is energized if a Mod or an MEG arrives or it is connected to at least one energized node around it. C. Linearization of Nonlinear Constraints
Note that (5c), (6b), (9h), and (9l) are nonlinear. The first three can be simply linearized based on the method in [28] (see (13) in [28]). For the product term in (9l), e.g. , η i' , t · κ ii' , t , according to [22], we introduce a new binary variable χ i' - ii' , t to replace η i' , t · κ ii' , t by adding the following constraints: , , , , , , , i ii t i t ii t i ii t i t i ii t ii t χ η κ χ η χ κ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′− − − ≥ + − ≤ ≤ (10) Thus, (9l) is equivalently linearized. The above method is actually identical to the McCormick envelopes [29] used in [26]. IV. S CHEDULING M ODEL TO B OOST R ESILIENCE
We set the objective function as follows, which was commonly used as a good metric to assess the DS resilience under restoration [5], [8], [26], [30]. ( ) ( ) S SG G dp F G dpG S F S
L, , , ,, , , ,, , , , max i i t i t travel j i tt i j im i t h i tm i h ifuel m i t h i tm i h i w P t vv v δ ϕϕ ι ι ∈ ∈ ∈ ∈∈ ∈ ∈ ∈∈ ∈ ∈ ∈ ∆ −+ + − + ∑ ∑ ∑ ∑∑ ∑ ∑ ∑∑ ∑ ∑ ∑ (11) The first term of (11) is the weighted sum of the picked-up energy of loads during scheduling. Two coefficients φ travel and φ fuel are introduced as penalties with regard to the traveling of MERs and the fuel exchange between MEGs/FTs and nodes, as in [8]. Specifically, the second term of (11), weighted by φ travel , is intended to prevent futile travel of the MERs, and the third term, weighted by φ fuel , is used to prevent redundant fuel exchange between FTs or MEGs and nodes. In general, as stated in [8], the values of φ travel and φ fuel can be set based on the decision-makers’ preference or the analytic hierarchic process [31]. We can also adjust the values of w i , φ travel and φ fuel based on an assessment of interruption cost of loads, travel cost of MERs, and labor cost for handling fuel exchange. Thus, the scheduling model involving SMESSs, MEGs, FTs, and reconfiguration to boost the DS resilience is expressed as: Objective : (11);
Constraints : (1)-(9), along with (10). Particularly, it is simple to prove that the traditional MESS solutions always lie in the feasible region of the above proposed model, and thus, the optimal SMESS solution has at least no worse objective value and performance than those of the MESS. Essentially, a traditional MESS is a special case of an SMESS by letting its Carr and some fixed Mods be always together. Suppose that an MESS comprises Carr 1 and Mods 1 & 2, and then, any scheduling solution of it can be represented as “when Carr 1 is traveling, γ t =1 and γ t =1; and when Carr 1 is parked at node i , ζ t =1 and ζ t =1” in our model. In other words, for an MESS solution, we can always find an equivalent SMESS solution that does not cause any violation of constraints (2)-(4) as well as the others. Thus, the MESS solutions are always included in the feasible region of the proposed model. V. N UMERICAL R ESULTS
We conduct test and case studies based on the modified IEEE 33-node system to verify the effectiveness of the proposed model. The model is coded on the MATLAB R2020b platform with the YALMIP toolbox [32] and solved by Gurobi v9.1.0 on a computer with an Intel Core i5-8400 CPU processor and 32 GB RAM. The parameter
MIPGap of the solver is set as 0.1%. A. Test System
The test system is shown in Fig. 3. Specific data, including impedance of branches and loads at nodes, can be found in [33]. Due to the lack of relevant data, we simply assume that the priority weights of loads are randomly assigned from 1 to 5. The first term of the objective function in (11) is calculated in kW·h. In practice, reaching a high level of electric service restoration rapidly is clearly a primary task and much more urgent than the others for the DS operator in postdisaster conditions; hence, we assign a small value of 0.1 to φ travel and φ fuel . We do this also to focus fully on and display the performance of the SMESS in enhancing DS resilience by alleviating the restriction of the other costs in the objective function. 6 Four Mods, each with a 500 kW/1 MW·h capacity [16], and two tractors as Carrs are scheduled as SMESSs in the test. Each Carr can carry at most two Mods. The charging/discharging efficiency of each Mod is set as 0.95/0.95, and the allowable SOC range is 0.1-0.9. In addition, an 800 kW/1000 kVA MEG is used, with a fuel consumption rate of 64.4 L, 109.8 L, 155.2 L, and 200.6 L per hour at 1/4, 1/2, 3/4, and full load, respectively [24]. We assume that the MEG itself is equipped with a small 500 L fuel tank. A 2000 L FT also participates in the scheduling. The available fuel storage capacities of each node and the depot are set as 5000 L and 10000 L, respectively. As shown in Fig. 3, four nodes {8, 11, 24, 29} support the connection of SMESSs to the DS, while three nodes {3, 22, 27} and the depot support the connection of the MEG/FT. For simplicity, we assume that all the Carrs, MEG and FT have the same speed, i.e. , spend the same time traveling between two locations, and that the time spent traveling from one to another and back are the same. Further, we arbitrarily assume the distances among nodes as follows: for {8, 11, 24, 29}, the time spans spent traveling between nodes 8 and 11, between 11 and 24, and between 24 and 29 are set as 2, while that between the other pairs is 1; for {3, 22, 27, depot}, the time spans spent traveling between nodes 3 and 27, between 3 and the depot, and between 27 and the depot are set as 2, while the others are 1. B. Scenario
In the test, scheduling is performed over 6 h with a time step of Δ t =0.5 h, i.e. , there is a total of 12 time spans. We consider a postdisaster scenario with a total of 23 faults, as depicted in Fig. 3, by which the DS is separated into 5 de-energized islands. The repair of faults is simply considered in a two-step way. As indicated by Table I, some of the faults are cleared after the 6th time span, and the others are cleared after the 11th time span. All the Mods and Carrs of the SMESSs are initially located at node 8, and the initial SOC of each Mod is set as 0.5. The MEG and FT are initially located at node 3 and the depot, respectively, and both have the same initial SOF of 0.1. We set 0.8 as the initial SOF of the depot and 0 for all the nodes of the DS to represent the extreme condition where the DS operator has not prepared fuel within the DS for the MEG operation before disasters, which might often occur due to the poor predictability of many disasters. C. Simulation Results
By solving the proposed model, the main results are given by Fig. 4 to Fig. 7, which can verify the effectiveness of the proposed model. For clarity, we describe the main scheduling process from the perspectives of the five islands in Fig. 3. We look at
Island 4 first, which includes the initial positions of SMESSs. As soon as scheduling begins, Mod 1 is transported by Carr 1 to node 11 (
Island 5 ), Mods 2 and 3 are transported by Carr 2 to node 29 (
Island 2 ), and Mod 4 stays at node 8 for a while. In addition, the MEG immediately moves from node 3 to node 22. In the 1st time span ( i.e. , in t =1), Mod 4 discharges slightly to energize a small part of the loads in Island 4 , and as the MEG arrives, which is of high power and energy capacity (the FT comes soon to supplement the fuel), Mod 4 begins to charge until it is transported by Carr 2 to node 29 in t =5. The MEG acts as the main source of Island 4 until it leaves in t =6; before that, i.e. in t =5, Mod 2 charges intensively to store enough energy to then sustain the power supply of Island 4 after the MEG leaves away. In t =7, the connection of Island 4 to the substation is recovered by the repair of faulted lines. Then, Mods 2 and 3 charge at the maximum power for two time spans and are sent by Carr 1 to node 24 in t =9. For Island 1 , as shown in Fig. 4, no power source is available until Mods 2 and 3 are transported by Carr 1 to node 24 in t =10. For Island 3 , it obtains a power supply when the MEG arrives at node 27 from node 22 in t =7, and all the loads in it are energized. For Island 5 , when scheduling begins, Carr 1 transports Mod 1 from node 8 to node 11. After it arrives, Mod 1 keeps discharging to supply part of the loads of
Island 5 until t =7 when Island 5 is reconnected to the substation; then, Mod 1 begins to charge intensively for two time spans. In t =9, Mod 1 is carried by Carr 2 and transported to node 29. For Island 2 , it obtains a power supply soon after scheduling begins due to the timely arrival of Mods 2 and 3 carried by Carr 2. It is interesting to note that after arriving, Mod 3 charges in t =2 rather than discharging to provide some power supply. This is similar to that described above, where Mod 2 in Island 4 charges intensively before the MEG leaves. Since Mod 2 is going to be carried away from
Island 2 in t =3, before that, Mod 3 “absorbs” enough energy from Mod 2 to sustain the power supply to Island 2 after Mod 2 leaves. Then, in t =6, Mod 4 is brought by Carr 2, and Mod 3 is transported to node 8 by Carr 1. Subsequently, the power supply is held by Mod 4 during the following several time spans, until Mod 1 is brought by Carr 2 and joins in. Regarding the fuel delivery, as shown in Fig. 4 and Fig. 7, the FT refuels itself adequately at the depot when scheduling begins. After that, the FT moves to node 22 and releases all the fuel there in t =3, resulting in an increase in the SOF of node 22. Thus, the MEG obtains extra fuel for power generation to supply Island 4 , even though it has uses up the fuel stored in itself after t =2. Then, in t =5, the MEG fully refuels itself from the fuel at node 22 (see Fig. 7 (a), which shows a large increase Fig. 3. State of the test system before scheduling. TABLE
I T HE S ETS OF F AULTS FO , FC , FO , AND FC DURING S CHEDULING
Faults FO FC FO FC time spans Nodes 7, 25, 30 Nodes 5, 26, 12, 23, 15 Branches (8,9), (11,12), (13,14), (20,21), (4,5), (7,8), (25,29), (3,23), (18,33), (28,29) Branches (5,6), (23,24), (21,22), (29,30), (10,11) time spans Φ Nodes 12, 23, 15 Branches (4,5), (7,8), (25,29), (3,23), (18,33), (28,29) Branches (29,30), (10,11) time span Φ Φ Φ Φ t =6, it moves to node 27 to energize Island 3 for a relatively long duration. D. Case studies and Comparison
Based on the above, the following five cases are studied to verify the advantage of the proposed model in boosting DS resilience over other measures.
Case 1 : No energy storage or generator is available.
Case 2 : Only stationary energy storage systems and generator are available, and scheduling of the FT is considered. Specifically, we fix the four Mods and the MEG at node 8 and node 3, respectively, during the whole scheduling horizon.
Case 3 : Two general MESSs, each with a 1 MW/2 MW·h capacity (to ensure an equivalent capacity of energy resources to that of
Case 5 ), and the MEG is available. Scheduling of the FT is also considered.
Case 4 : The SMESSs and MEG are available, but scheduling of the FT is forbidden. However, in this case, we allow the MEG to refuel itself by going to the depot.
Case 5 : The SMESSs, MEG, and FT are jointly scheduled, as we highlight in this paper and did in the previous test. The revisions to the proposed model required to realize
Cases to are given in the electronic appendix for this paper [34]. By solving the models, the comparison among the five cases regarding the performance in boosting DS resilience is shown in Fig. 8. Clearly, energy resources, whether stationary or with mobility, can give the DS better resilience. As shown in Fig. 8, the line of Case 1 always lies below that of the others. Stationary energy resources can maintain the power supply to some loads after a blackout; however, they cannot serve the de-energized loads electrically isolated from them. As shown in Fig. 8, in the last several time spans, the performance of
Case 2 is reduced to that of
Case 1 since the stationary Mods and MEG cannot supply power to
Islands 1 to , which are isolated from any power source by faulted lines. With mobility, energy resources can further boost the DS resilience to a higher level, as shown when comparing Case 2 with
Case 3 / / , the latter of which have much higher lines in Fig. 8 than the former. In particular, Case 5 results in a higher overall resilience level than that of
Case 3 . This is mainly because, in brief, as extensive faults cause multiple islands, the number of which could exceed the number of power sources ( e.g. , there are 5 islands but 2 MESSs and 1 MEG in
Case 3 ), the energy resources might become overwhelmed when attempting to serve all the de-energized islands. The proposed SMESS method could release and decouple the carrier and energy resources of the MESS ( e.g. , we have more individual energy resources in
Case 5 , even though some of them have a smaller capacity than in
Case 3 ) Fig. 4. Results of time-spatial behaviors of SMESSs, MEG, and FT. Fig. 5. Structures of the DS involving reconfiguration and the picked-up states of loads at different times. Fig. 6. Results of the active power output and SOC (at the end of each time span) of (a) Mod 1, (b) Mod 2, (c) Mod 3, and (d) Mod 4. Fig. 7. Results of (a) active power output, SOF, (b) fuel demand/supplement of MEG; (c) fuel output and SOF of FT; and (d) SOF of node 22 and the depot. Fig. 8. Comparison among
Cases 1 to . Case 4 and
Case 5 , since the fuel shortage MEGs might encounter could be mitigated in time and MEGs could go straight to the de-energized areas without too much consideration regarding whether to refuel themselves. VI. C ONCLUSION
In this paper, we propose the idea of SMESS to further evolve the scheduling of traditional MESS. In the concept of SMESS, a Carr is allowed to carry multiple Mods and the Carr and each of the Mods are scheduled as independent components, thus endowing the scheduling of SMESSs with more feasibility and flexibility than that of MESSs. The constraints for scheduling SMESSs are derived. The fuel delivery is modeled to schedule FTs to guarantee adequate fuel for MEGs operation. Then, aiming at the DS restoration issue, the joint scheduling of SMESSs, MEGs, FTs, and the DS reconfiguration is formulated as an MILP model. Numerical results demonstrate the effectiveness of the proposed method and its good performance in boosting DS resilience. The SMESS concept and the proposed joint scheduling model could result in a more effective and higher-level usage of mobile energy resources. R
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