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Employing Distributed Resources in Smart Grids
Smart & Cool project
Morten Juelsgaard,Rafael Wisniewski, Jan Bendtsen,
◮ For h ∈ H, let Zh denote theimpedance indices on theunique simple path between thetransformer and consumer h
◮ Define matrix J ∈ Cn×n
[Jx,y ] =
∑
h∈Zx
zh, x = y
∑
h∈Zx∩Zy
zh, x 6= y ,
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3. Voltage control through coordination
Grid modeling:
◮ Let Jr = Re(J), then total active losses are
∑
t∈T
i(t)†Jri(t) > 0.
◮ Voltage throughout is
u(t) = us − Ji(t), ∀t ∈ T .
◮ Voltage is constrained by
umin ≤ |u(t)| ≤ umax, ∀t ∈ T
with | · | denoting element wise complex magnitude
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3. Voltage control through coordination
Centralized problem formulation:Given:sets: H,Hev,Hpv, signals: ph(t), ppv,h(t), qh(t) ∀ h ∈ H, t ∈ T ,matrices: J, Jr , values: ψh, tev,h for each h ∈ Hev,
solve
minimizep̃ev,h(t), q̃pv,h(t)
T∑
t=1
i(t)†Jri(t)
subject to umin ≤ |u(t)| ≤ umax,
Eev,h(T ) = Edem,h,
Emin,h ≤ Eev,h(t) ≤ Emax,h,
pmin,h ≤ p̃ev,h(t) ≤ pmax,h
q̃pv,k(t) ≤√
s2max,k − p2pv,k(t),
ij(t) = f (pj(t), qj(t), uj(t)),
for all t ∈ T , j ∈ H, h ∈ Hev and k ∈ Hpv.
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3. Voltage control through coordination
Simplified problem formulation:
◮ We have considered only the centralized case
◮ Our approach so far:◮ Convexify main problem, by e.g. linear approximations◮ Employ sequential convex programming with iterative
update of convexifications
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3. Voltage control through coordination
Numerical Example: A feeder from Aasted in Northern Jutland
z1
z6
z7 z11z15 z21
z20
z19
z32
z33
z41
us
z2
z5
z14
z12
z10
z8
z18
z16 z22
z23
z31
z29
z28
z26
z25
z24
z38
z34
z40
z39
z45
z42
14
57
810
11
13
14
15
16
18
19
21
22
23
24
28
29
30
31
34
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3. Voltage control through coordination
Numerical Example: Provided datappv(t)[pu]
t
13:56 18:58 24:00 04:52 09:540
2
4
6
8
ph(t)[pu]
t
13:56 18:58 24:00 04:52 09:540.2
0.3
0.4
0.5
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3. Voltage control through coordination
Numerical Example: Setup
◮ 34 households, 21 hrs, 10 min. sample intervals
◮ 3 cases to be examined
A. Flexibility only from EVsB. Flexibility only from PVsC. Flexibility from mixture of EVs and PVs
◮ Comparison to the current situation, where flexibility is notutilized
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3. Voltage control through coordination
Numerical Example: ResultsBenchmark; HA
ev = {30 − 34}, HBev = {1− 7, 29 − 34}
uh(t)[pu]
t
uh(t)[pu]
13:56 18:58 24:00 04:52 09:54
0.9
1
1.1
0.9
1
1.1
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3. Voltage control through coordination
Numerical Example: ResultsCoordinated; HA
ev = {1− 34}, HBev = {1− 7, 29 − 34}
uh(t)[pu]
t
uh(t)[pu]
13:56 18:58 24:00 04:52 09:54
0.9
1
1.1
0.9
1
1.1
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3. Voltage control through coordination
Numerical Example: ResultsCoordinated; HA
ev = {1− 34}, HBev = {1− 7, 29 − 34}
uh(t)[pu]
t
uh(t)[pu]
13:56 18:58 24:00 04:52 09:54
0.9
1
1.1
0.9
1
1.1
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3. Voltage control through coordination
Numerical Example: Results, mixed example
|uh(t)|
[pu]
t
pev,h(t)[kW
]
Bench.Opt.
ph(t)[kW
]
13:56 18:58 24:00 04:52 09:54
0.9
1
1.1
0
5
10
−10
0
10
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4. Pending work
◮ Distributing the voltage coordination algorithm
◮ Communication structure should be neighbor-to-neighbor
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4. Pending work
Simplified problem sketch:
us
r1 rmr2
xc1
x i1 xo1
l1(x i1, xo1 )
f1(xc1 )
u1
xc2
x i2 xo2
l2(x i2, xo2 )
f2(xc2 )
u2
xcm
x im xom
lm(x im, xom)
fm(xcm)
um
x i , xo : Power entering and leaving each cable section,xc consumed power, l : power losses in each cable section,f : cost function of each consumer, r : cable resistance,u: connection point voltage
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4. Pending work
The simplified problem may be formulated as
minimizem∑
j=1
(fj(xcj ) + lj(x
ij , x
oj ))
subject to xcj ∈ Xj ,
x ij + xoj + lj(xij , x
oj ) = 0
x ij + xcj + xoj+1 = 0
u = us − Ax i
umin ≤ u ≤ umax
for j = 1, . . . ,m, where u = (u1, . . . , um), xi = (x i1, . . . , x
im) and
A ∼
r1r1 r2...
.... . .
r1 r2 · · · rm
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4. Pending work
◮ Optimization of grid sections and consumers, are decoupled
◮ Couplings introduced through nodal and voltage constraints
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4. Pending work
A first approach could be ADDM:
minimizem∑
j=1
(f +j (xcj ) + l+j (x ij , xoj )) + g+(z i , zo , zc)
subject to x i = z i , xo = zo , xc = zc
with superscript + denoting extended value function, and:
g+(z i , zo , zc) =
0, if z ij + zcj + zoj+1 = 0 ∀j ∧
umin ≤ us − Az i ≤ umax
∞, otherwise
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4. Pending work
◮ The x-updates can now be run in parallel
◮ However, the z-update does not decompose well on accountvoltage coupling across entire line
◮ Current suggestion involves an axillary variable y , andreformulating constraints:
umin ≤ y ≤ umax, y = us − Axi = e1 − Cy − Dxi
with e1 = (1, 0, . . . , 0)T and
C =
[
0T 0I 0
]
D =
r1. . .
rm
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4. Pending work
This allows a reformulation of the ADDM problem
minimize
m∑
j=1
(f +j (xcj ) + l+j (x ij , xoj )) + g+(z i , zo , zc , y)
subject to x i = z i , xo = zo , xc = zc
now with:
g+(z i , zo , zc , y) =
0, if: z ij + zcj + zoj+1 = 0 ∀j ∧
y = ei − Cy − Dz i ∧
umin ≤ y ≤ umax
∞, otherwise
here, there is still coupling across the line, but the coupling isnow reduced so that it is between neighbors only.
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Thank you
In summary:
◮ We have discussed issues related to future smart grids
◮ We have seen how many of these naturally form distributedcontrol and optimization problems
◮ I have outlined some of the approaches we have employedso far, and presented a few of our results