This paper is dedicated to the application of the multigrammatical framework to the assessment of vulnerability of energy infrastructures affected by impacts destroying (reducing capabilities of) their facilities (power plants, fuel producing plants, power transmission lines, fuel transporting pipes, as well as networking devices of both electricity and fuel subsystems of an energy infrastructures). A basic graph representation of energy infrastructures is considered, and technique of their multigrammatical representation is introduced. Criterial base for recognition of the energy infrastructures vulnerability, being a generalization of the similar criterial base developed regarding industrial infrastructures is proposed. Techniques of multigrammatical modelling reservation of energy infrastructures and their recovery after impacts is proposed. Directions of future research in this area are announced.
1. Introduction
The multigrammatical framework (MGF), introduced and described in [
resilience/vulnerability to various destructive impacts (malfunctions, technogenic catastrophes, natural hazards,
acts of terror, mutual sanctions etc.). A unified approach to the solution of this class of problems regarding
largescale industrial systems (ISs) was described in [
This paper is dedicated namely to the application of the MGF to some considered from the substantial and mathematical points of view in [22-27] actual tasks from the area of resilience of energy infrastructures. Amounts of electric power (EP) to be delivered by an EI on demand of external customers at some predefined period of time in a general case are restricted by amounts of primary resources – crude oil, natural gas, and other possible energy carriers (ECs) – available for EP generation, as well as by limited bandwidths of links forming electric grids and fuel pipelines. A problem in question is, given a demand of costumers, i.e. amounts of power and fuel to be consumed by them during a considered time period (this demand will be named also an order), an EI segment, including fuel producing and power generating facilities, links providing power transmission and fuel transfer through distributed areas, as well as terminal units delivering fuel and power to their consumers, primary resources available for power generation, a destructive impact, eliminating some part of a considered EI segment and the aforementioned resources, to assess whether a part of a considered segment and resources, remained after an impact, would be capable to produce and deliver amounts of power and fuel necessary to consumers (in other words, to complete an order). If so, then an EI will be named resilient to this impact. Otherwise an EI will be named vulnerable to it. The objective of this paper is to develop a criterial base providing the assessment of EIs vulnerability to destructive impacts. Everywhere below in this paper we shall consider an EI as a closed system, which operate without direct application of any external resources or their application for replenishment of EI own (internal) resources spent whilst order completion.
Section 2. Filtering unitary multiset grammars being a basic tool for consideration and solution of the problem in question are described in the Section 3. A multigrammatical representation of energy infrastructures is proposed in the Section 4 whilst criteria of vulnerability of energy infrastructures to destructive impacts – in the Section 5. Modelling reservation of EIs and their recovery after impacts is considered in the Section 6. A Conclusion is dedicated to the future directions of the MGF development and it’s application to various issues concerning resilience of critical infrastructures and key resources. 2. Basic graph representation of energy infrastructures An energy infrastructure is usually considered consisting of two strongly interconnected and mutually supplying segments producing fuel and electricity [20-22].
PPs), power transforming-distributing substations (PTDSs), and power terminal units (PTUs), delivering electric power to it’s consumers. All these elements are connected by links, named power transmission lines (PTLs), each such line having it’s own technical parameters (voltage, length, power losses during transmission etc.), and are joined to electric grids, which , in fact, in aggregate form ElcI [21-23].
A fuel infrastructure (FI) [24, 25, 28, 29], similarly to an ElcI, includes fuel producing plants (FPPs), working out fuel from some primary energy carriers (PECs), and fuel distribution stations (FDSs), as well as fuel terminal units (FTUs). All these elements are connected by pipes, which, in a general case, as PTLs, have individual technical parameters (diameter, length, pressure, amounts of EP consumed, fuel losses during transfer etc.). Fuel produced by FPPs is used by power plants and other consumers. To limit a complexity of consideration here, we shall not expand a FI down to production crude oil and natural gas from oil and gas fields and their transportation via oil and gas pipelines to FPPs; we shall assume that certain amounts of primary energy carriers (PECs), used for fuel production, are accumulated at fuel storages (FSs) collocated with FPPs, and these amounts are a part of a resource base (RB) of an EI.
ElcI and FI are joined with one another by terminal units: any element of an FI consumes an electric power delivered to it by some PTU, whilst any PP is operating due to a FTUs delivering fuels needed for power generation (in a general case there may be several energy carriers utilized by a single power plant). Also there are PTUs and FTUs delivering power and fuels to external consumers. Regarding a considered time period (hour, day etc.), any FPP may produce certain amounts of various fuels, as well as any PP may produce certain amounts of EP with various technical parameters. Any output of any element of EI is assumed consistent with a link transferring resource from it to another element, which input, in turn, is assumed consistent with the aforementioned link which is an incoming for this another element and thus delivering to it the aforementioned resource. This overlapping of EI elements and boundary points of EI links is a background for modelling a circulation of an EP and fuel via EI. Any link has a limited bandwidth (or throughput capacity) as an integral technical parameter, determining maximal amount of power (if it is a PTL) or fuel (if it is a pipe) which may be transmitted (transferred) via this link during a considered time period. Also, as it was mentioned above, there are some power losses occurring during it’s transmission via a PTL; similar losses of fuel are inherent to fuel transferring pipes.
EI may be represented by an weighted oriented graph with nodes corresponding to EI elements, and marked edges corresponding to EI links. This graph, in turn, in the algebraic representation is a ternary relation
× × , where is a set of EI elements (PPs, PTDSs, PTUs, FPPs, FDSs, FTUs, FSs), and is a set of positive rational numbers representing bandwidths of EI links (PTLs and pipes). So < , ′, >∈ means that an element is capable to transmit (transfer) to an element ′ amount of resource (EP or fuel) by link (PTL or pipe) < , ′ > no more than
units (kilowatt∙hours in the case of EP, and barrels, cubic meters, kilograms, tons etc. in the case of various fuels) during a considered time period. There may be the only triple < , ′, > ∈
for any link < , ′ > , i.e. a link has the only bandwidth (throughput capacity).
EI as well as some amounts of resources stored at an EI resource base; naturally, an impact may be represented by some subset of nodes and edges eliminated from an initial graph .
transformation-distribution stations, seven power terminal units, a fuel producing plant, a fuel storage, two fuel distribution stations, and three fuel terminal units (figure 1(a)). (Sequential numbers of FDSs and FTUs, as well as names of fuel storage and fuel producing plant are denoted by bold symbols).
There are also three external power customers. Generated power from a PP is delivered to both PTDSs, the first of which (enumerated “1”) delivers received power to four PTUs ( “1”, “2”, “3” and “7”), and the second (“2”) delivers accepted power to five PTUs (“4”, “5”, “6”, “8” and “9”). PTUs deliver power to the following considered graph, including bandwidths (throughput capacities) is contained in the table 1.
the impact reduces bandwidth of the link between the PTDS “2” and the PTU “6” from 300 kWh to 100 kWh.
Having this basic graph representation of EIs we may move to the MGF application to the assessment of resilience/vulnerability of EIs. To introduce proposed a criterial base for this assessment let us remind some necessary notions and denotations concerning syntax and semantics of filtering unitary multiset grammars (FUMGs) being a simplest MGF tool for formalizing and solution of many actual tasks from the applied systems analysis and operations research areas.
PTU5 PTU6 PTU7 FS FPP FDS1 FDS1 FDS1 FDS2 FTU1 FTU2 FTU3 PTU8 PTU9 EPC1 EPC2 EPC3 FPP FDS1 FDS2 FTU2 FTU3 FTU1 PP FC1 FC2 Channel upper threshold values of bandwidths (throughput capacities) 200 kWh 100 kWh 300 kWh 300 kWh 100 kWh 200 tons of crude oil 200 tons of the fuel 100 tons of the fuel 50 tons of the fuel 50 tons of the fuel 100 tons of the fuel 100 tons of the fuel 50 tons of the fuel 50 tons of the fuel
Following [
=< 0, >, 0 = { 1 ∙ 1, … , ∙ }, (1) (2) (3) (4) (5) (6) is called a kernel, and , called a scheme, is a finite set of rules which are applied for generation new multisets from already generated. (Everywhere below objects are denoted , , , whilst their multiplicities being positive rational numbers – as , , called a multiobject). A rule has a form
etc.; a construction ∙ representing collection of objects is where and ′, called respectively the left part and the right part of a rule, are multisets, and ≠ {∅}. By we shall designate below a set of all objects having place in rules entering a scheme of an MG .
and operations of addition and subtraction of multisets, denoted respectively + and - . Let ̅be a multiset. A rule (3) is applicable to ̅,if and a result of an application is a multiset MS ̅′from an MS ̅by application a rule ∈ , that is denoted as → ′, ̅ , ̅′ = ̅- + ′,
̅ ̅′, or, if the only MG is considered, then, as in the classic string-operating grammars [28, 29], ̅ ̅′, ̅∗ ̅′.
If a generation chain is non-empty, a denotation +
instead of ∗ is used.
A set of multisets (SMS), generated by an MG =< 0, >, is denoted and is defined as follows: = { | 0 }.(
terminal sets (STMS) will be denoted ̅. Obviously, ̅ .
Unitary multiset grammars (UMGs) are a simplified version of a partial case of MGs, called context-free multigrammars. A scheme of an UMG is a set of unitary rules (URs), where an UR is recorded as whilst a fact, that an MS ̅′is generated from an MS ̅by any (including empty) sequence of generation steps, called a generation chain, is recorded as 0 → 1 ∙ 1, … , ∙ ,
{1 ∙ 0 } → { 1 ∙ 1, … , ∙ }.
= ∪ , ∩ = {∅}, +, (7) (8) (9) (10) (11) (12) (13) (14) that is equivalent to The left part of an UR being an object 0 is called it’s header, whilst the right one – it’s body. A set of nonterminal objects, each being a header of at least one UR, is denoted ; and a set of all other objects, presenting only in bodies of URs and called terminal , is denoted : ∗
+ where + is a set of non-empty strings in some primary alphabet used for construction of objects’ names. Everywhere below bold letters in objects’ names will be assumed entering an alphabet , and bold letters “(“, ”)”, ”[“, ”]”, ”:” will be delimiters entering and used for construction of object names entering a set .
UMGs may be classified by number of URs having the same header. If an UMG =< 0, > is such that in a scheme there exists at least one non-terminal object being of header of > 1 URs, then this UMG is named alternating; otherwise, i.e. if any non-terminal object is a header of the only one UR, then this UMG is named non-alternating. Evidently, if an UMG is non-alternating, then it defines a one-element STMS, i.e. |̅ |=1. If an UMG is alternating, then in a general case it defines a set containing no less than one TMS, i.e. |̅ |≥1.
Also UMGs may be cyclic or non-cyclic. An UMG = 〈 0, 〉 will be called cyclic, if there exists a generation chain 0 ⇒ ⇒ ′ such that ⊆ ′, or, just the same, ′ = + , where ⊇ {∅}. As may be seen, a cyclic UMG in a general case, when ⊇ {∅} but ≠ {∅}, defines an infinite STMS ̅. All UMGs which are not cyclic, are named acyclic. Any acyclic UMG = 〈 0, 〉 defines a finite STMS ̅.
By finite or infinite number of elements of an STMS defined by an UMG = 〈 0, 〉 it may finitary (in this
case |̅ |< ∞) or infinitary (in this case |̅ |= ∞). As it is known from [
Alternating UMGs are a standard tool for representation of alternative structures of complex (composite)
objects or ways of solution of some task. This class of UMGs is for a long time used for modelling industrial
systems and infrastructures [
Example 2. Consider the UMG =< 0, >, where 0 = {2 ∙ ( )}, and the scheme contains three unitary rules 1, 2 and 3: 1: ( ) → 1 ∙ (
400 ∙ ( 2: (
100 ∙ ( 3: ( 80 ∙ ( ), 1 ∙ (
), 50 ∙ ( ) → 1 ∙ (
), 60 ∙ ( ) → 1 ∙ ( ), 70 ∙ ( ), 4 ∙ ( : ), 1 ∙ ( : ), 1 ∙ ( ).
), ), ), 4 ∙ ( ),
The kernel of this UMG represents the order, which objective is to obtain two autos, whilst the scheme represents the so called manufacturing technological base of some industrial facility capable to complete such orders. The UR 1 represents the structure of auto, which consists of frame, engine, 4 wheels and 4 doors, as well as resources necessary for assembling this auto: 400 kilowatt∙hours of electric power and 50 minutes of operation of autos assembling line (AL). The URs 2 and 3 represent structure of engine (motor and fuel tank), and two alternative ways of it’s manufacturing by two engines assembling lines, the first consuming 100 kilowatt∙hours and 60 minutes, and the second – 80 kilowatt∙hours and 70 minutes for one engine. According to the semantics of UMGs, ̅ = { 2,0, 0,2, 1,1}, where 2,0 represents total of resources necessary for manufacturing both autos by the first way (involving the first engines AL), 0,2 – similar value when both engines are assembled by the second such AL, and 1,1 – when engines are assembled in parallel by separate ALs. Evidently, where ∎ 2,0 = + {1000∙ ( 0,2 = + {960 ∙ ( 1,1 = + {980 ∙ ( ), 60 ∙ ( ), 120 ∙ ( ), 140 ∙ ( : : )}, )}, = { 2 ∙ ( ), 2 ∙ ( 2 ∙ ( ), 8 ∙ ( ), 100 ∙ ( ), 8 ∙ ( : }.
(15) (16) (17) (18) (19) (20) (21) (22) =< 0, , >,
We shall use below filtering unitary multiset grammars (FUMGs) as a basic mathematical tool for representation and
solution of tasks in question. According to [
, ̅ = ( ̅ ′ ↓ ≤) ↓ , where symbol ↓ denotes an operation of filtration: an STMS, generated by an UMG ′, is filtered by a set of BCs, and then a resulting subset, including TMSs, satisfying all BCs entering ≤, is filtered by a set of OCs, so, finally, ̅ includes TMSs satisfying not only all BCs but also all OCs.
Example 3. Let us consider now the FUMG =< 0, , >, where 0 and are the same as above, and = { ( : ) > 0}. condition (BCs) is recorded as recorded as = , where ∈ { or ′ , where ∈ {≥, >, <, ≤, =, ≠}, whilst an optimizing condition (OC) is
, }. So in a general case
where ≤ is a set of BCs, and is a set of OCs. Semantics of filters, in fact, is very similar to semantics of
relational query languages if to consider a set ̅ ′ as a specific database (however, infinite in a general case);
also, due to application of OCs, filters provide natural representation of various tasks from the area of
mathematical programming and, in general, operations research [
From the substantial point of view this filter provides selection of such ways of order completion where no one engines AL is out of operation (both such assembling lines are involved). So, obviously, ̅ = { 1,1}. If (23) (24) (25) (26) (27) (28) = { ( ) = } i.e. such ways of order completion are preferable which consume minimal amount of electric power, then ̅ = { 0,2}. In the case = { ( ̅ = { 1,1} ↓ { ( : ) =
}, } = { 1,1}. ∎
4. Basic multigrammatical representation of energy infrastructures Let us begin from an electricity infrastructure.
a measurement unit of EP transmitted via PTLs (kilowatt∙hour), and is a string in an alphabet representing a geographical point, where an element of an ElcI is located (it may be designated by a unique symbolic name associated with specific geographic coordinates in a special database, or directly by these coordinates). So a multiobject ∙ ( : ) represents kilowatts generated or consumed at a point (position, place) .
Let us begin our consideration from power terminal units. Any PTU in order to deliver one unit of power to a consumer, switched to this PTU, must receive it from a closest PTDS, connected with it by a PTL. So a unitary rule, representing this fragment of an ElcI, would be as follows: ( ), ∙ [ ,
. (Let us note that the sense of (29) is fully similar to the sense of (10) regarding
industrial systems and called a technological interpretation of unitary rules [
If a PTDS, located at a point , is connected to power terminal units, located at points 1, … , , then this fragment of an ElcI is represented by following unitary rules: 1 ) → 1 ∙ ( ) → ∙ ( ), 1 ∙ [ ), ∙ [ ,
1], , ]. where
is a location of a power plant.
: ) → 1 ∙ ( 1: 1), … , ∙ ( : ), Similarly may be represented fragments of an ElcI, consisting of connected PTDSs. In this case a string a representation of a location of a delivering power transforming-distributing substation, whilst – locations of such PTDSs, which are consuming power transformed and transmitted by it: 1, … , ), 1 ∙ [ ), ∙ [ , , 1], ]. ), 1 ∙ [ , ), ∙ [ , 1], ], ), 1 ∙ [ , 1], ), ∙ [ , ],
power. Any tree-like fragment of an ElcI, containing some PP and connected with it PTDSs, may be represented by following URs: and, if there are some power terminal units connected to a power plant directly, i.e. without any intermediate PTDSs, then also (
( (( ( ( ( ( ( ( where multiobject ∙ ( : ) represents a PTU of an electricity infrastructure, located at a point and providing operation of an FTU located at a point during delivery of one unit of a resource from a point to a point . This amount of power is consumed during a resource transfer via a pipe, which start point is and final point is . In a general case, due to losses of fuel during it’s transfer via a pipe, ≥ 1 units of fuel are needed to be delivered to a pump at a start point of this pipe.
Distributing facilities (namely, FDSs) of fuel infrastructure may be represented similarly to PTDSs: 1 ∙ ( ∙ (
( : ∙ ( : 1: 1), 1 ∙ (
… , : ), ∙ ( ) → ), 1], ]. 1], ], that means that delivered resource, incoming to any FDS, is distributed to + pipes by application of corresponding needed amounts of electric power. The first pipes provide fuel transfer to another FDSs whilst the last – to FTUs. As above, [ , ], = 1, … , , are pipes, which start point is and final points are . Similarly, [ , ], = 1, … , , are pipes, which start point is and final points are . Presence of objects ( : ) in all unitary rules (36) and objects ( : ′ ) in all unitary rules (37) means that power terminal units, belonging to an electricity infrastructure, would be installed and operate at some predefined points and ′ respectively to make possible physical contact with FDSs and FTUs and their power supply during transfer of resource .
amounts of electric power consumed, i.e. and ′ , as well as losses of a fuel during it’s transfer via this pipe, i.e. and ′ .
As it is clear, the described techniques may be applied until places of origination of energy carriers, i.e. fuel production plants, working out pipeline gas and various oil derivatives, used as a fuel by power plants. As it was assumed above, PECs, used for fuel production, are accumulated at fuel storages collocated with FPPs. So operation of any such FPP may be represented as follows: where 1, … , are points, where fuel storages with PECs 1, … , are located, so namely regarding these places power terminal units would be installed, thus providing relocation of amounts of these PECs necessary to an FPP for production of one unit of fuel at a location . The aforementioned relocation would be possible if needed amounts of electric power, i.e. 1,…, kilowatt∙hours, would be available at points 1, … , where respective PTUs are operating. In turn, to produce one unit of a fuel an FPP itself would consume kilowatt∙hours from a power terminal unit located at a point .
One more nuance connected with a multigrammatical representation of an energy infrastructures and
assessment of their resilience is representation of active states of EI elements. To represent the fact that any
producing or transmitting (transferring) facility (PP, FTP, PTDS, FTDS, PTU, FTU) to carry out it’s functions
would be in an active state we shall apply techniques proposed and described in [
[
. Let us illustrate techniques of construction such set given a graph representation of an EI.
Example 4. Consider the EI segment represented by the graph at Fig. 1a and Table 1. It may be also represented as a following set of unitary rules: ( ( ( ( ( ( ( ( : :
) → 1 ∙ ( : ) → 3 ∙ ( : : ) → 1.05 ∙ ( : ) → 1.01 ∙ ( : ) → 1.02 ∙ ( : ) → 1.01 ∙ ( : ) → 1.01 ∙ ( : ) → 2.9 ∙ (
As seen, the URs (43) − (44) represent knowledge about the PTDSs “1” and “2”, which are located respectively at the points and , and are connected by the PTLs, represented by the multiobjects 1 ∙ [ , ] and 1 ∙ [ , ], with the power plant located at the place ; there are no valuable losses of the EP during it’s transmission from the PP to both PTDSs, so the same amount of the EP which is given into any PTL by the PP is received by a PTDS; hence, the multiplicities of the object ( : ) in both URs (43) and (44) are equal to 1. The multiobjects 1 ∙ (+ ) and 1 ∙ (+ ) represent a fact that both PTDSs would be in active states to receive the EP from the producing it power plant and to deliver the EP to the connected with them power terminal units or PTDSs. The knowledge about PTUs “1” − “9”, connected with the respective PTDSs in full accordance with the graph representation of the considered segment of the EI, is represented by the URs (45) − (53). The UR (54) represents, that the power plant may produce one kilowatt∙hour consuming for this objective 3 tons of the fuel (represented by the multiobject 3 ∙ ( : )), receiving it via the pipe (represented by the MO 1 ∙ [ , ]) from the fuel terminal unit “1” located at the point , and being in the active state, that is represented by the MO 1 ∙ (+ ). The URs (55)−(57) represent knowledge about the fuel terminal units “1” − “3”. The UR (55) represents the knowledge about the resources necessary to the FTU “1” for receiving one ton of fuel from the fuel distributing station “2” located at the place via the pipe represented by the MO 1 ∙ [ , ]. Due to the fuel losses during transfer, the FDS “2”, delivering the fuel to the FTU “1”, gives into the pipe, represented by the MO 1 ∙ [ , ], 1.05 ton of the fuel, that is represented by the MO 1.05 ∙ ( : ). The FTU “1” to receive one ton of the fuel consumes 20 kilowatt∙hours of the EP from the power terminal unit located at the point , that is represented by the MO 20 ∙ ( : ). And, as usual, the FTU “1” must be in the active state, that is represented by the MO 1 ∙ (+ ). The URs (56)−(57) in the same manner represent the knowledge about the fuel terminal units “2” and “3” which are provided by the EP from the PTU “2”, and this PTU consumes the same 20 kilowatt∙hours for one ton of the received fuel. The URs (58)−(59) represent the similar knowledge about the fuel distributing stations “1” and “2” provided by the EP from the PTUs “3” and “2” respectively; the FDS “1” consumes 40 kilowatt∙hours of EP from the PTU “7” located at the point , that is represented by the MO 40 ∙ ( : ), and the FDS“2” consumes 30 kilowatt∙hours of EP from the PTU “2”, located at the point , that is represented by the MO 30 ∙ ( : ). The FDS “2” receives the fuel from the FDS “1” via the pipe represented by the MO 1 ∙ [ , ]. The FDS “1”, in turn, receives the fuel from the fuel producing plant via the pipe represented by the MO 1 ∙ [ , ] consuming 40 kilowatt∙hours of EP from the PTU “3”, located at the point , and this is represented by the MO 40 ∙ ( : ). At last, the UR (60) represents the knowledge about the FPP which is capable to produce one ton of the fuel receiving 2.9 tons of crude oil from the fuel storage via a pipe represented by the MO 1 ∙ [ , ] and consuming 50 kilowatt∙hours of EP from the PTU “4”, located at the point , and this is represented by the MO 50 ∙ ( : ). Finally, as seen, the considered segment of the EI, consuming crude oil from the fuel storage, provides external consumers by the electric power and the fuel, respectively, via the PTUs “5”, “6” and “7”, and via the FTUs “2” and “3”. ∎
A resource base of any EI may be represented as a multiset including multiobjects of the following three types: 1) ∙ ( : ) for all fuel storages entering a considered EI, that means units of materiel resource (PEC or produced fuel) are available at some FS located at a place ; 2) ∙ [ , ′] for all links having place in a considered EI, that means a value is a bandwidth (throughput capacity) of a link [ , ′], i.e. a maximal amount of EP or materiel resource, which may be transmitted (transferred) via this link during a considered time period (in the case [ , ′] is a PTL this amount is measured in kilowatt∙hours whilst in the case [ , ′] is a pipe this amount may be measured in barrels, cubic meters, kilograms, tons etc.); 3) ∙ (+ ) for all elements of a considered EI, thus establishing for any such element a maximal number of operation cycles which might be executed by it at a considered time period (in other words, is fixing a maximal productivity of an element ; a multiobject ∙ (+ ) will be referred below as an operation resource of an element ).
= {100 ∙ ( 300 ∙ [ 300 ∙ [ 200 ∙ [ ,
100 ∙ [ 100 ∙ (+ 100 ∙ (+
: ), 1000∙ [ , ], 400 ∙ [ , ] , 300 ∙ [ ], 200 ∙ [ , , ], 100 ∙ [ ), 100 ∙ (+ ), 10 ∙ (+
], ], As seen, the fuel storage entering the considered segment of an EI contains 100 tons of crude oil; the PTL connecting the power plant and the PTDS “1” during a considered time period provides transmission no more than 1000 kilowatt∙hours of EP that is represented by the MO 1000∙ [ , ]; the PTL connecting the PP and the PTDS “2” provides transmission no more than 1100 kilowatt∙hours of EP that is represented by the MO 1100∙ [ , ]; similarly are represented the upper threshold values of bandwidths of all other PTLs of the considered segment of the EI. The pipe, connecting the fuel storage and the fuel producing plant, provides delivery of no more than 200 tons of crude oil that is represented by the MO 200 ∙ [ , ]; the pipe, connecting the fuel producing plant and the fuel distributing station “1”, provides delivery of no more than 200 tons of the fuel that is represented by the MO 200 ∙ [ , ]; similarly are represented the upper threshold values of throughput capacities of all other pipes of the considered segment of the EI. Any element of the ElcI, entering this segment, during a considered period of time may execute 100 operation cycles, that is represented by the MOs 100 ∙ (+ ), … , 100 ∙ (+ ); any element of the FI, entering this segment, during a considered period of time may execute 10 operation cycles, that is represented by the multiobjects 10 ∙ (+ ), … , 10 ∙ (+ ). ∎
After specifying a resource base, an EI may be considered as a free industrial system =< {∅}, , >
in the sense [
= { ≤ | ∙ ∈ } ∪ { = 0 | ∈ ̅& ⋶ }, all). having a form in such a way that this FUMG generates a set of terminal multisets each representing some collection of resources sufficient for an order completion by some definite cooperation of manufacturing devices (the second operand of a join is obligatory to eliminate ways of an order completion which satisfy restrictions implied by an available resource base of an EI, but need some additional resources which are absent at an RB at As now may be seen, a unitary multiset grammar =< , > defines a set ̅ of terminal multisets each { 1 ∙ ( 1: 1), … , ∙ ( :
), 1 ∙ [ 1, ′ 1],…, ∙ [ , ′ ], 1 ∙ (+ 1), … , ∙ (+ )}, where 1,…, are amounts of, respectively, resources 1,…, (PECs stored at FSs, fuels, produced by FPPs, as well as EP, produced by PPs) to be available at places 1, … , (via PTUs, FTUs, or directly from fuel storages); 1 … , are amounts of energy carriers and electric power to be transferred (transmitted) via, respectively, links [ 1, ′ 1],…,[ , ′ ] (PTLs and pipes) during a considered time period; 1 … , are numbers of operation cycles of, respectively, facilities 1, … , involved in a completion of an order . So every TMS ∈ ̅ corresponds to some specific way of an order completion (in a general case there may be several ways identical by resource consumption and facilities involvement).
We shall represent an EI current resource base as a sum of three multisets the first =
+ + , = { 1 ∙ ( : ), … , ∙ ( : ) } representing amounts of resources having place at an EI fuel storages, the second representing current bandwidths and throughput capabilities of an EI links, and the third = { 1 ∙ [ , ′ ],…, ∙ [ , ′ ]}
=
{ ∙ (+ ), … , ∙ (+ )}
representing current operation resource of an EI facilities. (Bold indices , . . , , , … , , , … , , used in
(64) −(67), differ from ordinary indices 1, . . , , 1, … , , 1, … , , used in (63)).
5. Cyclicity of FUMGs, representing energy infrastructures, and their finitarization
Let us note, that industrial systems are represented through a technological interpretation of unitary rules [
Here we propose a simple solution of this problem based on the so called terminalization of non-terminal
objects introduced in [
In the first case power storages (PSs) similar to fuel storages are presumed, and, like FSs, they may be represented by multiobjects ′ = {∅} | ′ | = ∞. ( components until some atomary (non-splitted) elements (spare parts, microchips, etc.); thus FUMGs representing
consumes EP generated by it’s electricity segment (namely, ElcI), whilst the last one consumes fuel necessary for EP production. Thus FUMGs representing EIs are essentially cyclic, and sets of multisets generated by their application are in a general case infinite: for a core UMG ′ =< , > of a FUMG =< , , > it would be valid that means a PS located at a place may provide on demand up to kilowatt∙hours.
processes of power supply, we have proposed the simplest way of finitarization of FUMGs representing EIs. Now, evidently, despite a set ′ remains infinite, a set ′ in a general case would be non-empty, thus representing at least one way of an order completion by application of a priori accumulated power; from the (68) (69) (70) (71) (72) where =< , >.∎
Speaking informally, an EI is not capable to complete an assigned order , if there exists no one way of
generation (production) and delivery of necessary amounts of EP and fuels, consuming for this objective such
amounts of primary energy carriers which are not greater than available at fuel storages of this EI, and also
capabilities of EI facilities and links are sufficient for these generation (production) and delivery. Following [
= { ≤ | ∙ ∈ } ∪ { = 0 | ∈ ̅& ⋶ } (the second operand of a join is obligatory to eliminate ways of order completion which satisfy restrictions implied by an available resource base of an EI, but need some additional resources which are absent at an RB at all).
Statement 2. Energy infrastructure =< {∅}, , > is not capable to complete an assigned order , if mathematical point of view, this means that a core UMG ′ =< , > of a FUMG =< , , > generates at least one terminal TMS.
6. Criteria of vulnerability of an energy infrastructure to a destructive impact Let us begin from the initial task, which verbal formulation is as follows: given amounts of primary energy carriers at fuel storages of an EI and demand on an electric power and fuels from it’s external consumers (an order to be completed by an EI), to assess whether an EI is or is not capable to complete an order (i.e. to provide these consumers by required amounts of EP and fuels).
criterion [
Statement 1. An energy infrastructure =< {∅}, , > is not capable to complete an assigned order , if (∀ ∈ ) ⊂ , (73) (74) (75) (76) (77) where =< , , >.∎
EIs to destructive impacts. Following [
Namely, means, that an impact eliminates units of a resource from an FS located at a place . Similarly, means, that an impact reduces for units a maximal amount of EP or fuel which may be transmitted (transferred) at a considered time period via a PTL (pipe) with start point and final point ′. Finally, to represent destruction of any producing or transmitting (transferring) facility (PP, FTP, PTDS, FTDS, PTU, FTU) we may apply the same techniques including to an MS multiobjects like ∙ (+ ) representing that an element of EI would be affected, and a result of this action would be reduction of an operation resource of this element by units. Obviously, a case of an entire destruction of any component of an EI may be easily represented by inclusion to a multiset an object ∙ ( ), where is a number maximal for the used implementation of
is valid.
Example 5 segment of an EI as well as reduces amount of the crude oil at the fuel storage by 20 tons, and also reduces bandwidths (throughput capacities) of: PTL [ , ] by 100 kilowatt∙hours, PTL [ , ] by 200 kilowatt∙hours, and pipe [ , ] by 10 tons of fuel. So the result of this impact will be { ∙ ( )} - { ∙ ( )} = {∅} (78) where =< , >.∎
regarding available after an impact), then an energy infrastructure is vulnerable to an applied impact. Similarly to the Statement 3, this criterion may be represented by the application of FUMGs.
Statement 4. An energy infrastructure =< {∅}, , > is vulnerable to an impact , applied before a beginning of an assigned order completion, if (79) (80) (81) (82) where =< , , ′ >, and ′ = { ≤ | ∙ ∈ - } ∪ { = 0 | ∈ ̅& ⋶ - }. ∎
Let us consider now a more general case, when an impact is applied to an EI at some time moment inside time period of an order completion.
Just to this moment some part of an order may be completed, as well as a respective part of an EI resource base would be already consumed. If so, then there is not difficult to formulate statements being a corollaries of the Statements 3 and 4 and representing criteria of vulnerability of an EI affected by a destructive impact inside a time period of an order completion.
Statement 5. An energy infrastructure =< {∅}, , > is vulnerable to an impact , applied inside a time period of a completion an assigned order , when a part of this order is already completed and a part of an EI resource base is already consumed, if
Statement 6. An energy infrastructure =< {∅}, , > is vulnerable to an impact , applied inside a time period of a completion an assigned order , when a part of this order is already completed and a part of an EI resource base is already consumed, if (∀ ∈ - ) - -
⊂ . ∎ -
= {∅}, where -
=< , , ′′ >, and ′′ = { ≤ | ∙ ∈ - - } ∪ { = 0 | ∈ ̅& ⋶ - - }. ∎
7. Modelling reservation and recovery of energy infrastructures Taking into account a possibility of application of destructive impacts of various nature to components of energy infrastructure, an EIs’ management is usually preparing in advance some additional reserved facilities and primary or produced resources, which are made available promptly after an impact is detected, and such measure in many cases provides as effective as possible mitigation of consequences of the aforementioned impact (up to making this order feasible by the adjusted itself EI). A background of such adaptability is some redundancy implanted to an EI before or during it’s operation [30-34]. The most usual measure implemented by EIs’ designers and management are the so called backup power systems, providing EP generation for a time periods when the affected segments of EIs are recovered, and also bypasses, providing electricity or fuel flows by some workarounds if a preordered routes are broken by an impact.
Namely, for any UR (18) representing a PTU, located at a point , it is sufficient to join to a set one more UR ( representing a fact that this PTU may receive EP not only from a PTDS located at a point , but also from a PTDS located at a point ′. (It is assumed that there is a technological solution providing such opportunity). In a general case there may be ≥ 1 such alternative PTDSs capable to deliver EP to this PTU, and this is possible regarding any PTU entering a considered EI. (83) (84) (85)
PTDS, located at a point
, it is sufficient to join to a set one more UR ( representing a fact that this PTDS may receive EP not only from a PTDS located at a point , but also from a
′. As in the case of PTUs, there may be ≥ 1 such alternative PTDSs capable to deliver EP to this PTDS, and this is possible regarding any PTDS entering a considered EI.
The described technique without any changes may be applied also to PTUs and PTDSs, connected directly with additional (reserve) power plants: ( ( : ) → ′1 ∙ ( ′1: ′1), … , ′ ′ ∙ ( ′ ′: ′ ′) ,
, may receive EP not only from a PP located at a point , as defined by a respective UR from a set (21), but also from a PP located at a place ′, as well as PTU, located at a point
and entering a set of URs (22), may receive EP from some PP located at a place ′′, which may differ from ′ or be the same. As in all considered above cases, there may be ≥ 1 such alternative power plants capable to deliver EP to these PTUs and PTDSs, and this is possible regarding any PTU and PTDS connected with several power plants.
Any reserve power plant, entering a considered ElcI and located at a point ′, may be represented by an UR : ′) → ′1 ∙ ( ′1: ′1), … , ′ ′ ∙ ( ′ ′: ′ ′) , where, as in (23), ′1, … , ′ ′ are amounts of resources ′1, … , ′ ′, which must be delivered to locations ′1, … , ′ ′
respectively in order to generate one kilowatt∙hour of electrical power at a location ′ , from which, in turn, it may be delivered by PTLs to PTDSs (PTUs), closest to this PP. Let us note, that a reservation may be implemented not only by inclusion to an ElcI some additional power plants but also by implementation of alternative ways of EP generation and associated with them resources, by which a PP must be supplied. In this case to an UR (23) a unitary rule ( (
) and alternative body, representing a respective supply set, necessary for this way implementation, is joined to a set .
complexity, not only of tree-like structure, as it was considered above in the Section 4.
Namely, for any UR (24) representing a FTU, located at a point , it is sufficient to join to a set one more UR ( ) → ′∙ ( : ′), ′∙ ( representing a fact that this FTU may receive a resource not only from a FDS located at a point also from a FDS located at a point
′. (As in the case of ElcI, it is assumed that there is a technological solution providing such opportunity). In a general case there may be ≥ 1 such alternative FDSs capable to deliver a resource
to this FTU, and this is possible regarding any FTU entering a considered FI.
Reservation of distributing facilities (namely, FDSs) of fuel infrastructure may be represented similarly to reservation of TDSs: ( Any such UR represents the fact, that an FDS, located at a point , may receive required amounts of resource not only from an FDS located at a point , as defined by a respective UR from a set (25), but also from an FDS located at a place ′. As in all considered above cases, there may be ≥ 1 such alternative FDSs capable to deliver resource to this FDS, and such opportunity is possible regarding any FDS connected with several supplying it FDSs.
At last, any reserve FPP, producing fuel used by power plants for EP generation, located at a point ′, may be represented by an UR ′1 ∙ ( ′ ∙ ( ′ :
′ ′), ′ ′ ∙ ( ( : ′) → ′∙ ( : ′1: ′1), ′1 ∙ ( … ,
′), ′1 ∙ ( ′ ∙ ( ′ :
′ ′), ′ ′ ∙ ( ( : ′∙ ( : ′1: ′1), ′1 ∙ ( … , ) → ′), where ′1, … , ′ ′ are points, where fuel storages with PECs ′1, … , ′ ′ are located, so namely regarding these places power terminal units would be installed, thus providing relocation of amounts of these PECs necessary to an FPP for production of one unit of fuel at a location ′. The aforementioned relocation would be possible if needed amounts of electric power, i.e. ′1,…, ′ ′ kilowatt∙hours, would be available at points ′1, … , ′ ′ where respective PTUs are operating. In turn, to produce one unit of a fuel a reserve
additional FPPs but also by implementation of alternative ways of fuel producing and associated with them
with the same header ( : ) and alternative body, representing a respective alternative supply set, necessary for this way implementation, is joined to a set .
EIs without any corrections. The main difference is that UMGs representing such EIs are alternating, and thus is a multi-element set of terminal multisets, i.e. | | ≥ 1.
However, in practice an EI operates by some subset of it’s components, and this subset as a whole has an ordinary concentric tree-like structure whilst the rest components stay in a reserve until an impact, after which some or even all of reserve components may be joined (switched) to an affected EI. To implement this approach it is sufficient to represent a reserved EI as a ternary tuple (for short “quadraple”) =< {∅}, , , >, where is a reserve resource base, any part (submultiset) of which may be added to an RB reduced by an impact, transforming it from - to - + . Following (64)−(67), a multiset may be represented as a sum of three non-intersecting multisets similar to MSs , , : (94) (95) =
+ + , = { 1 ∙ ( : ), … , ∙ (
: ) } representing amounts of resources (PECs and fuels) having place at EI reserve fuel storages, as well as amounts EP accumulated by backup power systems (in this case nothing but (96) (97) (98) (99) the first summand
); the second of the third representing reserve throughput capabilities of an EI (PTLs and pipes kept out of operation until an impact), and representing reserve operation resource of an EI (facilities kept ready to operate if necessary). Thus addition to the reduced RB an MS
provides join to an affected EI some amounts of fuel located at reserve FSs, an MS − switching to an EI transporting network some reserve links (PTLs and pipes), and, at last, an MS − join to an EI some reserve producing, generating and transmitting/transferring facilities. After this addition criteria (80) − (82) may be applied to an EI =< {∅}, , - + > and an order . If this EI is not vulnerable then there exists an opportunity to recover it after impact, and thus there arises naturally a task of computation “the best” of all possible multisets providing recovery of the affected EI to a state sufficient for an order completion. Otherwise it is clear that available reserve is insufficient for EI recovery and order completion. 8. Modelling rechargeable power storages and their application Until now, introducing the concept of a power storage, we have not determined nor verbally, nor strictly, how power is accumulated in any specific PS, as well as how the last is recharged after or during order completion (for example, in a case of a car accumulator it is recharged during or, more correct, by a car motion). Let us consider a case of rechargeable power storages and their multigrammatical modelling..
Namely, we shall associate with any order , which objective is meeting a demand on predefined by an external consumer collections of resources located at predefined places, a so called internal order ′, which objective is addition of a collection ′ to a resource base, i.e. full or partial (or even redundant) replenishment of resources spent whilst order completion. The simplest way of an internal order interpretation is a replacement of an RB , remained after external order completion, by + ′. However, such approach is not satisfactory, because it, in fact, applies a concept of an energy infrastructure as an open system – a collection ′ is not a part of EI resource base and is applied from systems which are external regarding EI. The second reason for rejection of this approach is that a collection ′ is not at all correlated with an order ; it would be assigned to by an EI control system in some arbitrary way (the most natural one is ′ ∈ , that means full replenishment of spent resources).
So it would be necessary to develop such techniques of representation of logic of internal order ′ construction and completion which, from one side, would provide it’s compliance with an order , and, from another one, would provide replenishment of namely power storages, applied for an order completion, by consumption of any other resources having place in an EI resource base. Such an approach fully fits the reality, where PSs are recharged on a regular basis by consumption of other resources entering an EI RB. Thus, firstly, a presumption that an EI is a closed system would be satisfied, and, secondly, all power storages, applied whilst order completion, would be recharged by means of only internal capabilities of an energy infrastructure.
The proposed multigrammatical representation of this important feature of EIs and their fragments is as follows. Let ∈ is a collection of resources consumed for an order completion, so ⊆ . We shall define a submultiset ′ of a multiset in such a way that multiobjects entering ′ represent amounts of power delivered from PSs during an order completion (all such multiobjects, obviously, have a form ∙ ( : ) ), so ′ = { ∙ ( : ) | ∙ ( : ) ∈ }
Namely these amounts of electric power would be replenished in power storages before the next order would income to an EI, so this multiset is nothing but a needed internal order, i.e., for the first glance, ′ = ′. however, the substantial difficulty, breaking (101), is that to be an order, completed by some chain of energy transfers and transmissions, an MS ′ in a general case would contain non-terminal objects, i.e. objects, being headers of unitary rules, representing an EI. At the same time an MS ′, being a submultiset of an MS , contains only terminal objects. To avoid this deadlock, we propose the following solution. To define logic of PSs replenishment (recharge) a set would contain URs of a form (100) (101) (∗ (102) where bold symbol " ∗ " means that to replenish one kilowatt∙hour at PS, located at a place , it is sufficient to complete an order being a set, containing all multiobjects, entering a body . So all URs like (102), in fact, define logic of PSs replenishment. If so, then, evidently, ′ = { ∙ (∗ : ) | ∙ ( : ) ∈ ′}, (103) so after an initial order completion, which results in delivery of determined by amounts of electric power, an internal order ′ is completed, resulting in replenishment of power storages, applied during completion. As may be seen, power storages, applied during an internal order ′ completion, are not replenished; otherwise a process of replenishment may become recursive and too complicated in an implementation. From the practical point of view this is quite natural. Let us note, as a conclusion of this Section, that not all power storages entering a considered energy infrastructure are rechargeable (i.e. in a multigrammatical representation of an EI not all URs with headers ( : ) are supplemented by URs (102) with headers (∗ : )); all the rest PSs are presumed of a single use, so they may be replaced after consumption of all initially accumulated power.
As it was mentioned above, a criterial base, introduced in the Section 6, provides an assessment of vulnerability
of energy infrastructures to destructive impacts: if an EI and an impact satisfy formulated conditions, then an EI
is vulnerable; but if the aforementioned conditions are not satisfied, then it does not mean that an affected EI is
substantially resilient to an impact. Let us underline, that we do not confirm, that in the case ≠ {∅} an EI
is capable to complete an order , because in a general case there would be also assessed time delays associated
with production and delivery of materiel resources (though regarding ElcI and electric power, circulating via it’s
networks and grids, in a general case such delays may be ignored). So, if an order includes a deadline for
delivery of all necessary resources to external consumers being a source of this order, then, despite amounts of
resources available for an order completion may be sufficient for this objective, even an optimal schedule of an
order completion may not provide timely delivery of all necessary resources to consumers. This is an implication
of non-additivity of time; time is an additive resource regarding separate device (facility), whilst regarding an EI
as a whole it is non-additive because different devices may operate in parallel. So an EI, not satisfying the
introduced above criteria and thus being not vulnerable in the above sense, in a general case may be not resilient
to an impact. By this reason, if in a general case an order includes restrictions on duration of it’s completion,
then to assess EI resilience it would be necessary to apply more general mathematical tools than unitary multiset
grammars. Such tools named temporal multiset grammars were for the first time announced in [
resources, to assess, what maximal subset of a full set of consumers may be provided by power and fuel in accordance with their demand. Another variation of this task is to assess whether some predefined part of consumers may be provided by power and fuel according to their demand while all the rest consumers may be provided by some part of their demand not less than some threshold values.
EI will be considered, and also a task of an assessment of “the best” part of an order which may be completed given remained after an impact part of an EI (both with and without reserve).
may be applied also to heating systems, heating and cooling systems, combined heat and power systems [35, 36]
and water supply systems [37-39]. An application of the MGF to these systems as well as to a sewer-mining
[40] is described in short in [
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