Designing Resilient Electrical Distribution GridsDesigning Resilient Electrical Distribution Grids...
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Designing Resilient Electrical Distribution Grids Emre Yamangil, Russell Bent and Scott Backhaus Rutgers University & Los Alamos National Laboratory Grid Science Winter School & Conference, Jan. 12th Presidential Policy Directive - Critical Infrastructure Security and Resilience “The ability to prepare for and adapt to changing conditions and withstand and recover rapidly from disruptions. Resilience includes the ability to withstand and recover from deliberate attacks, accidents, or naturally occurring threats or incidents.” A Simplified Model Given a graph G =(V , E) where V corresponds to node based upgrades (i.e. building facilities and microgrid generation capacity) and E corresponds to line based upgrades (i.e. building new lines, hardening lines and building switches) we want to find: min Budget(G 0 ) s.t. G 0 ⊆ G T s ⊆ G 0 ∀s ∈ S T s ∈ Trees(G) ∀s ∈ S CriticalDemand(T s ) ≥ MinCriticalDemand ∀s ∈ S TotalDemand(T s ) ≥ MinTotalDemand ∀s ∈ S Algorithm 1: Greedy 1 input: A set of disasters S; 1 for s ∈ S do 2 σ s ← Solve(P 0 (s)); 3 σ * (x)= max{σ s (x)|∀s ∈ S}, ∀x ∈X ; 4 Update σ * (x i ) with switches to preserve feasibility; 5 return σ * Algorithm 2: Scenario Based Decomposition 1 input: A set of disasters S and let S 0 = S 0 ; 1 while S \ S 0 6= ∅ do 2 σ * ← Solve P(S 0 ) exactly (SBD) or with VNS (SBVNDS); 3 I ← s 1 , s 2 ... s |S\S 0 | s ∈ S \ S 0 : l (P 0 (s i ,σ * )) ≥ l (P 0 (s i+1 ,σ * )); 4 if l (P 0 (I (0),σ * )) ≤ 0 then 5 return σ * ; 6 else 7 S 0 ← S 0 ∪ I (0); 8 return σ * Algorithm 3: Variable Neighborhood Search 1 input: σ 0 , MAX T IME, MAX R ESTARTS and MAX I TERATIONS; 1 Let σ LP ← Solve(P LP ), σ * ← σ 0 , restart ← false; 2 while t < MAX T IME and i < MAX R ESTARTS do 3 j ← 0; 4 n ←|x ∈X : |σ * (x) - σ LP (x)|6 = 0|; 5 J ← π 1 ,π 2 ...π |J| ∈X : |σ * (π i ) - σ LP (π i )|≤|σ * (π i+1 ) - σ LP (π i+1 )|; 6 if restart then 7 i ← i + 1; 8 step ← 4n d , k = |X | - step; 9 shuffle(J ) 10 else 11 step ← n d , k = |X | - step; 12 while t < MAX T IME and j ≤ MAX I TERATIONS do 13 σ 0 ← Solve(P(σ * , J (1,..., k)); 14 if f (σ 0 ) < f (σ * ) then 15 σ * ← σ 0 ; 16 i ← 0; 17 restart ← false; 18 j ← MAX I TERATIONS; 19 else 20 j ← j + 1; 21 k = k - step 2 ; 22 if j > MAX I TERATIONS then 23 restart ← true; 24 return σ * 1 Yamangil, Bent, Backhaus (2015), Resilient Upgrade of Electrical Distribution Grids, in proceedings of AAAI-15, AAAI press. Withstand Develop new tools, methodologies, and algorithms to enable the design of resilient power distribution systems, using: I asset hardening I system design I building new lines I building switches I building microgrid facilities I building microgrid generation capacity ⇒ binary decisions, mixed-integer programming. Nomenclature Parameters N set of nodes (buses). E set of edges (lines and transformers). S set of disaster scenarios. D s set of edges that are inoperable during s ∈ S. D 0 s set of hardened edges that are inoperable during disaster s ∈ S. c ij cost to build a line between bus i and j. 0 if line already exists. κ ij cost to build a switch on a line between bus i and j. ψ ij cost to harden a line between bus i and j. ζ i,k cost of generation capacity on phase k at bus i. α i cost to build a generation facility at node i. Q ijk line capacity between bus i and bus j on phase k. P ij set of phases for the line between bus i and bus j. P i set of phases allowed to consume or inject at bus i. β ij parameter for controlling maximum flow variation between the phases. d i,k demand for power at bus i for phase k. G i,k existing generation capacity on phase k at node i. Z i,k maximum amount of generation capacity on phase k that can be built at node i. C the set of sets of nodes that includes a cycle. λ fraction of critical load that must be served. γ fraction of all load that must be served. L set of buses whose load is critical. Variables x ij determines if line i, j is built. τ ij determines if line i, j has a switch. t ij determines if line i, j is hardened. z i,k determines the capacity for generation on phase k at node i. u i determines the generation capacity built at node i. x s ij determines if line i, j is used during disaster s. τ s ij determines if switch i, j is used during disaster s. t s ij determines if line i, j is hardened during disaster s. z s i,k determines the capacity for generation on phase k at bus i during disaster s. u s i indicates if the generation capacity is used at node i during disaster s. g s i,k generation produced for bus i on phase k during disaster s. l s i,k load delivered at bus i on phase k during disaster s. y s i j determines if the jth load at bus i is served or not during disaster s. f s ij,k flow between bus i and bus j on phase k during disaster s. ¯ x s ij determines if at least one edge between i and j is used during disaster s. ¯ τ s ij determines if at least one switch between i and j is used during disaster s. x s ij,0 determines if there exists flow on line i, j from j to i, during disaster s. x s ij,1 determines if there exists flow on line i, j from i to j, during disaster s. (a) Urban (b) Rural Figure : Each problem contains three copies of the IEEE 34 system to mimic situations where there are three normally independent distribution circuits that could support each other during extreme events. These problems include 100 scenarios, 109 nodes, 118 possible generators, 204 loads, and 148 edges, resulting in problems with > 90k binary variables. The cost of single and three phase underground lines is between $40k and $1500k per mile and we adopt the cost of $100k per mile and $500k per mile, respectively. The cost of single and three phase switches is estimated to be $10k and $15k, respectively. Finally, the installed cost of natural gas-fired CHP in a microgrid is estimated to be $1500k per MW. Urban, Hardened lines are not damageable (a) CPLEX Greedy SBD SBVNDS CPU OBJ OBJ CPU OBJ CPU OBJ 10% 19984.7 322.9 1044.5 465.8 322.9 289.9 353.7 25% 166352 635.4 1643.5 8028.3 635.4 811.4 635.4 50% TO X 2021.2 2840.7 647.7 791.3 647.7 75% TO X 1874.2 991.1 652.1 692.5 652.1 100% TO X 1934.4 712.7 654.1 662.5 654.1 Urban, Hardened lines are damaged at a 1 100 rate (c) CPLEX Greedy SBD SBVNDS CPU OBJ OBJ CPU OBJ CPU OBJ 10% 159166 445.8 1061.7 2232.9 445.8 2721.3 476.5 25% TO X 1441.9 14299.2 662.9 2994.7 701.5 50% TO X 1571.2 2848.7 646.0 1917.7 760.2 75% TO X 1787.3 16040.6 687.6 1481.4 687.6 100% TO X 2744.8 24270.3 1320.5 2157.5 1330.5 Urban, Hardened lines are damaged at a 1 10 rate (e) CPLEX Greedy SBD SBVNDS CPU OBJ OBJ CPU OBJ CPU OBJ 10% TO X 859.1 5265.1 460.8 2505.7 594.1 25% TO X 1742.2 12530.3 961.2 2843.2 961.2 50% TO X 3133.8 34822.7 1417.2 3363.5 1555.2 75% TO X 3472.0 TO X 7486.5 1894.2 100% TO X 10479.1 TO X 32289.8 7959.4 (a) Urban - 1/100 Rate (b) Urban - 1/10 Rate Scenario Definition We assume each scenario can be associated to a subset of the lines of the power distribution system that are inoperable: (c) Damaged line (d) Fragility Resilient Distribution Grid Design min X ij∈E c ij x ij + X ij∈E κ ij τ ij + X ij∈E ψ ij t ij + X i∈N α i u i + X i∈N ,k∈P i ζ i,k z i,k s.t. x s ij ≤ x ij ∀ij ∈E , s ∈S τ s ij ≤ τ ij ∀ij ∈E , s ∈S t s ij ≤ t ij ∀ij ∈E , s ∈S z s i,k ≤ z i,k ∀i ∈N , k ∈P i , s ∈S u s i ≤ u i ∀i ∈N , s ∈S z i,k ≤ M i,k u i ∀i ∈N , k ∈P i (x s ,τ s , t s , z s , u s ) ∈Q(s) ∀s ∈S x,τ, t, u ∈{0, 1} Minimize budget 1st stage: construction 2nd stage: assets in use flow delivery Set of Feasible Distribution Networks Q(s)= {x s ,τ s , t s , z s , u s : - x s ij,0 Q ijk ≤ f s ij,k ≤ x s ij,1 Q ijk ∀ij ∈E , k ∈P ij x s ij,0 + x s ij,1 ≤ x s ij ∀ij ∈E (τ s ij - 1)Q ijk ≤ f s ij,k ≤ (1 - τ s ij )Q ijk ∀ij ∈E , k ∈P ij X k∈P ij f ij,k |P ij | (1-β ij ) ≤ f s ij,k 0 ≤ X k∈P ij f ij,k |P ij | (1+β ij ) ∀ij ∈E , k 0 ∈P ij x s ij = t s ij ≤ 0 if ij ∈D 0 s 1 else ∀ij ∈D s l s i,k = n i X j=0 y s i j d i j ,k ∀i ∈N , k ∈P i 0 ≤ g s i,k ≤ z s i,k + G i,k ∀i ∈N , k ∈P i g s i,k - l s i,k - X j∈N f s ij,k = 0 ∀i ∈N , k ∈P i 0 ≤ z s i,k ≤ Z i,k u i ∀i ∈N , k ∈P i X ij∈E (C) (x s ij - τ s ij ) ≤|V |- 1 ∀C ∈C τ s ij ≤ x s ij ∀ij ∈E X i∈L,k∈P i l s i,k ≥ λ X i∈L,k∈P i d i,k X i∈N \L,k∈P i l s i,k ≥ γ X i∈N \L,k∈P i d i,k x s , y s ,τ s , u s , t s ∈{0, 1}} 3-phase real flow Modeled as multicommod- ity flow with phase variation Hardened lines can still be damaged Distribution network ⇒ tree Minimum service requirement as resilience criteria Rural, Hardened lines are not damageable (b) CPLEX Greedy SBD SBVNDS CPU OBJ OBJ CPU OBJ CPU OBJ 10% 33083.5 2337.0 3274.8 1837.9 2337.0 503.3 2337.0 25% 32170.8 2390.3 3427.6 571.0 2390.3 457.8 2390.3 50% 20840.3 2397.6 3449.9 471.2 2397.6 421.2 2397.6 75% 15556.1 2400.4 3452.7 337.5 2400.4 299.8 2400.4 100% 17225.9 2400.6 2780.6 385.8 2400.6 346.9 2400.6 Rural, Hardened lines are damaged at a 1 100 rate (d) CPLEX Greedy SBD SBVNDS CPU OBJ OBJ CPU OBJ CPU OBJ 10% 77947.9 2363.0 3375.4 759.0 2363.0 576.9 2363.0 25% TO X 8238.6 TO X 919.4 6744.3 50% TO X 12336.0 TO 9288.9 4361.8 7121.0 75% TO X 23099.5 TO X 23142.6 11500.0 100% TO X 16600.7 TO X 5879.5 9797.3 Rural, Hardened lines are damaged at a 1 10 rate (f) CPLEX Greedy SBD SBVNDS CPU OBJ OBJ CPU OBJ CPU OBJ 10% TO X 7503.3 141718.0 4325.9 7756.8 4424.8 25% TO X 18021.3 TO X 21993.5 7371.9 50% TO X 28865.0 TO 12017.7 74729.0 12031.2 75% TO X 31887.0 TO 13522.2 107165.0 13500.8 100% TO X 32901.9 TO 16794.4 114354.0 16778.2 (e) Rural - 1/100 Rate (f) Rural - 1/10 Rate http://public.lanl.gov/rbent/ [email protected]
Transcript of Designing Resilient Electrical Distribution GridsDesigning Resilient Electrical Distribution Grids...
Designing ResilientElectrical Distribution Grids
Emre Yamangil, Russell Bent and Scott BackhausRutgers University & Los Alamos National Laboratory
Grid Science Winter School & Conference, Jan. 12th
Presidential Policy Directive - CriticalInfrastructure Security and Resilience
“The ability to prepare for and adapt to changingconditions and withstand and recover rapidly fromdisruptions. Resilience includes the ability to withstandand recover from deliberate attacks, accidents, ornaturally occurring threats or incidents.”
A Simplified Model
Given a graph G = (V,E) where V corresponds to nodebased upgrades (i.e. building facilities and microgridgeneration capacity) and E corresponds to line basedupgrades (i.e. building new lines, hardening lines andbuilding switches) we want to find:min Budget(G′)s.t. G′ ⊆ G
Ts ⊆ G′ ∀s ∈ STs ∈ Trees(G) ∀s ∈ SCriticalDemand(Ts) ≥ MinCriticalDemand ∀s ∈ STotalDemand(Ts) ≥ MinTotalDemand ∀s ∈ S
Algorithm 1: Greedy1
input: A set of disasters S;1 for s ∈ S do2 σs ← Solve(P′(s));3 σ∗(x) = max{σs(x)|∀s ∈ S}, ∀x ∈ X ;4 Update σ∗(xi) with switches to preserve feasibility;5 returnσ∗
Algorithm 2: Scenario Based Decomposition1
input: A set of disasters S and let S′ = S0;1 while S \ S′ 6= ∅ do2 σ∗← Solve P(S′) exactly (SBD) or with VNS (SBVNDS);3 I←
⟨s1, s2 . . . s|S\S′|
⟩s ∈ S \ S′ : l(P′(si, σ
∗)) ≥ l(P′(si+1, σ∗));
4 if l(P′(I(0), σ∗)) ≤ 0 then5 returnσ∗;6 else7 S′← S′ ∪ I(0);
8 returnσ∗
Algorithm 3: Variable Neighborhood Search1
input: σ′, MAXTIME, MAXRESTARTS and MAXITERATIONS;1 Let σLP ← Solve(PLP), σ∗← σ′, restart← false;2 while t < MAXTIME and i < MAXRESTARTSdo
3 j← 0;4 n← |x ∈ X : |σ∗(x)− σLP(x)| 6= 0|;5 J←
⟨π1, π2 . . . π|J|
⟩∈ X :
|σ∗(πi)− σLP(πi)| ≤ |σ∗(πi+1)− σLP(πi+1)|;6 if restart then7 i← i + 1;8 step← 4n
d , k = |X | − step;9 shuffle(J)
10 else11 step← n
d, k = |X | − step;
12 while t < MAXTIME andj ≤ MAXITERATIONS do
13 σ′← Solve(P(σ∗, J(1, . . . , k));14 if f(σ′) < f(σ∗) then15 σ∗← σ′;16 i← 0;17 restart← false;18 j← MAXITERATIONS;19 else20 j← j + 1;21 k = k− step
2 ;22 if j > MAXITERATIONS then23 restart← true;
24 returnσ∗
1Yamangil, Bent, Backhaus (2015), Resilient Upgrade of Electrical Distribution
Grids, in proceedings of AAAI-15, AAAI press.
WithstandDevelop new tools, methodologies, and algorithms toenable the design of resilient power distributionsystems, using:
I asset hardeningI system design
I building new linesI building switchesI building microgrid facilitiesI building microgrid generation capacity⇒ binary decisions, mixed-integer programming.
NomenclatureParametersN set of nodes (buses).E set of edges (lines and transformers).S set of disaster scenarios.Ds set of edges that are inoperable during s ∈ S.D′s set of hardened edges that are inoperable during disaster s ∈ S.cij cost to build a line between bus i and j. 0 if line already exists.κij cost to build a switch on a line between bus i and j.ψij cost to harden a line between bus i and j.ζi,k cost of generation capacity on phase k at bus i.αi cost to build a generation facility at node i.Qijk line capacity between bus i and bus j on phase k.Pij set of phases for the line between bus i and bus j.Pi set of phases allowed to consume or inject at bus i.βij parameter for controlling maximum flow variation between the phases.di,k demand for power at bus i for phase k.Gi,k existing generation capacity on phase k at node i.Zi,k maximum amount of generation capacity on phase k that can be built at node i.C the set of sets of nodes that includes a cycle.λ fraction of critical load that must be served.γ fraction of all load that must be served.L set of buses whose load is critical.Variablesxij determines if line i, j is built.τij determines if line i, j has a switch.tij determines if line i, j is hardened.zi,k determines the capacity for generation on phase k at node i.ui determines the generation capacity built at node i.xs
ij determines if line i, j is used during disaster s.τ s
ij determines if switch i, j is used during disaster s.tsij determines if line i, j is hardened during disaster s.zs
i,k determines the capacity for generation on phase k at bus i during disaster s.us
i indicates if the generation capacity is used at node i during disaster s.gs
i,k generation produced for bus i on phase k during disaster s.lsi,k load delivered at bus i on phase k during disaster s.
ysij determines if the jth load at bus i is served or not during disaster s.
f sij,k flow between bus i and bus j on phase k during disaster s.
x̄sij determines if at least one edge between i and j is used during disaster s.τ̄ s
ij determines if at least one switch between i and j is used during disaster s.xs
ij,0 determines if there exists flow on line i, j from j to i, during disaster s.xs
ij,1 determines if there exists flow on line i, j from i to j, during disaster s.
(a) Urban (b) Rural
Figure : Each problem contains three copies of the IEEE 34 system to mimicsituations where there are three normally independent distribution circuits that couldsupport each other during extreme events. These problems include 100 scenarios,109 nodes, 118 possible generators, 204 loads, and 148 edges, resulting inproblems with > 90k binary variables. The cost of single and three phaseunderground lines is between $40k and $1500k per mile and we adopt the cost of$100k per mile and $500k per mile, respectively. The cost of single and three phaseswitches is estimated to be $10k and $15k, respectively. Finally, the installed cost ofnatural gas-fired CHP in a microgrid is estimated to be $1500k per MW.
Urban, Hardened lines are not damageable (a)CPLEX Greedy SBD SBVNDS
CPU OBJ OBJ CPU OBJ CPU OBJ10% 19984.7 322.9 1044.5 465.8 322.9 289.9 353.725% 166352 635.4 1643.5 8028.3 635.4 811.4 635.450% TO X 2021.2 2840.7 647.7 791.3 647.775% TO X 1874.2 991.1 652.1 692.5 652.1100% TO X 1934.4 712.7 654.1 662.5 654.1
Urban, Hardened lines are damaged at a 1100 rate (c)
CPLEX Greedy SBD SBVNDSCPU OBJ OBJ CPU OBJ CPU OBJ
10% 159166 445.8 1061.7 2232.9 445.8 2721.3 476.525% TO X 1441.9 14299.2 662.9 2994.7 701.550% TO X 1571.2 2848.7 646.0 1917.7 760.275% TO X 1787.3 16040.6 687.6 1481.4 687.6100% TO X 2744.8 24270.3 1320.5 2157.5 1330.5
Urban, Hardened lines are damaged at a 110 rate (e)
CPLEX Greedy SBD SBVNDSCPU OBJ OBJ CPU OBJ CPU OBJ
10% TO X 859.1 5265.1 460.8 2505.7 594.125% TO X 1742.2 12530.3 961.2 2843.2 961.250% TO X 3133.8 34822.7 1417.2 3363.5 1555.275% TO X 3472.0 TO X 7486.5 1894.2100% TO X 10479.1 TO X 32289.8 7959.4
(a) Urban - 1/100 Rate (b) Urban - 1/10 Rate
Scenario DefinitionWe assume each scenario can be associated to asubset of the lines of the power distribution system thatare inoperable:
(c) Damaged line (d) Fragility
Resilient Distribution Grid Design
min∑ij∈E
cijxij +∑ij∈E
κijτij +∑ij∈E
ψijtij
+∑i∈N
αiui +∑
i∈N ,k∈Pi
ζi,kzi,k
s.t.xs
ij ≤ xij ∀ij ∈ E, s ∈ Sτ s
ij ≤ τij ∀ij ∈ E, s ∈ Stsij ≤ tij ∀ij ∈ E, s ∈ Szs
i,k ≤ zi,k ∀i ∈ N , k ∈ Pi, s ∈ Sus
i ≤ ui ∀i ∈ N , s ∈ Szi,k ≤ Mi,kui ∀i ∈ N , k ∈ Pi(xs, τ s, ts, zs, us) ∈ Q(s) ∀s ∈ Sx, τ, t, u ∈ {0, 1}
Minimizebudget
1st stage:construction
2nd stage:assets in useflowdelivery
Set of Feasible Distribution Networks
Q(s) = {xs, τ s, ts, zs, us :− xs
ij,0Qijk ≤ f sij,k ≤ xs
ij,1Qijk ∀ij ∈ E, k ∈ Pij
xsij,0 + xs
ij,1 ≤ xsij ∀ij ∈ E
(τ sij − 1)Qijk ≤ f s
ij,k ≤ (1− τ sij)Qijk ∀ij ∈ E, k ∈ Pij∑
k∈Pij
fij,k
|Pij|(1−βij)
≤ f sij,k′ ≤
∑k∈Pij
fij,k
|Pij|(1+βij)
∀ij ∈ E, k′ ∈ Pij
xsij = tsij ≤
{0 if ij ∈ D′s1 else ∀ij ∈ Ds
lsi,k =
ni∑j=0
ysijdij,k ∀i ∈ N , k ∈ Pi
0 ≤ gsi,k ≤ zs
i,k + Gi,k ∀i ∈ N , k ∈ Pi
gsi,k − ls
i,k −∑j∈N
f sij,k = 0 ∀i ∈ N , k ∈ Pi
0 ≤ zsi,k ≤ Zi,kui ∀i ∈ N , k ∈ Pi∑
ij∈E(C)
(xsij − τ
sij) ≤ |V| − 1 ∀C ∈ C
τ sij ≤ xs
ij ∀ij ∈ E∑i∈L,k∈Pi
lsi,k ≥ λ
∑i∈L,k∈Pi
di,k∑i∈N\L,k∈Pi
lsi,k ≥ γ
∑i∈N\L,k∈Pi
di,k
xs, ys, τ s, us, ts ∈ {0, 1}}
3-phase realflow
Modeled asmulticommod-ity flow withphasevariation
Hardenedlines can stillbe damaged
Distributionnetwork⇒tree
Minimumservicerequirementas resiliencecriteria
Rural, Hardened lines are not damageable (b)CPLEX Greedy SBD SBVNDS
CPU OBJ OBJ CPU OBJ CPU OBJ10% 33083.5 2337.0 3274.8 1837.9 2337.0 503.3 2337.025% 32170.8 2390.3 3427.6 571.0 2390.3 457.8 2390.350% 20840.3 2397.6 3449.9 471.2 2397.6 421.2 2397.675% 15556.1 2400.4 3452.7 337.5 2400.4 299.8 2400.4100% 17225.9 2400.6 2780.6 385.8 2400.6 346.9 2400.6
Rural, Hardened lines are damaged at a 1100 rate (d)
CPLEX Greedy SBD SBVNDSCPU OBJ OBJ CPU OBJ CPU OBJ
10% 77947.9 2363.0 3375.4 759.0 2363.0 576.9 2363.025% TO X 8238.6 TO X 919.4 6744.350% TO X 12336.0 TO 9288.9 4361.8 7121.075% TO X 23099.5 TO X 23142.6 11500.0100% TO X 16600.7 TO X 5879.5 9797.3
Rural, Hardened lines are damaged at a 110 rate (f)
CPLEX Greedy SBD SBVNDSCPU OBJ OBJ CPU OBJ CPU OBJ
10% TO X 7503.3 141718.0 4325.9 7756.8 4424.825% TO X 18021.3 TO X 21993.5 7371.950% TO X 28865.0 TO 12017.7 74729.0 12031.275% TO X 31887.0 TO 13522.2 107165.0 13500.8100% TO X 32901.9 TO 16794.4 114354.0 16778.2
(e) Rural - 1/100 Rate (f) Rural - 1/10 Rate
http://public.lanl.gov/rbent/ [email protected]
