Employing Distributed Resources in Smart Grids Smart...
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Employing Distributed Resources in Smart Grids
Smart & Cool project
Morten Juelsgaard,Rafael Wisniewski, Jan Bendtsen,
Palle Andersen, Christoffer Sloth, Jayakrishnan Pillai
Dept. of Automation and Control, Aalborg University
ACL presentation, KTH, 16/09/2013
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Who am I
◮ Ph.d. student at Aalborg University,Dept. of Automation and Control
◮ Employed at the project called ’Smart and Cool’
◮ Project period: 15/8/11 - 14/8/14
◮ Visiting researcher at ACL during 1/9-20/12 under strictsupervision of prof. Mikael Johansson
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Where is Aalborg
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Some background: Tendencies
◮ Decreasing use of fossil fuels, both with respect to energyproduction and -consumption
◮ Production: Wind, solar, hydro etc. are promoted over oil,gas and coal
◮ Consumption: electric vehicles (EVs), electric heatpumps (EHPs), etc., are preferred over traditionalcombustion engines and oil-fired boilers
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Some background: Consequences
◮ Production: Decreasing controllability, increasing volatilityand uncertainty,
◮ Consumption: Increased load, increased risk of congestion,increasing losses
Emerging problems:
How to maintain balance, security of supply, satisfactory grid
operation, etc. during and after this shifting paradigm
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Some background: Suggested solution
◮ It has been suggested that part of the solution may bederived from smart grids
◮ By smart grid we mean a future electric grid, with advancedinfrastructure for metering and control:
◮ Near real-time measurements of consumption◮ Control rights for various classes of consumers◮ Advanced communication structure to support the above
◮ This allows demand management
◮ The underlying existence of a smart grid, is the vantagepoint of our work
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This talk
◮ Some of the problems we have been working on:
1. Distributed energy balancing2. Consumption coordination for loss minimization3. Voltage control through consumption coordination
◮ Some initial thoughts on what I will be doing here
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1. Distributed balancing
◮ We consider a portfolio operator, managing a number ofpower production and consumption units
◮ The portfolio should collectively obey a predeterminedoperating schedule
◮ Imbalances can be eliminated by adjusting either productionor (flexible) consumption
◮ Each unit presents individual operating costs, and-constraints
Type 1Type 2
Consumers
External actors
Grid
Σ
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1. Distributed balancing
The overall problem is:
minimize∑
i
fi (xi )
subject to xi ∈ Xi∑
i
xi = d
The solution must be ob-tained distributed:
x1
x2
x3
x4
x5
x6
xn
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1. Distributed balancing
Dual decomposition:
Obtain:x⋆i = arg inf
xi∈Xi
(fi (xi ) +λT xi ), ∀i
Update Lagrange variables:λ := λ+ α(
∑
i x⋆
i − d)
Lagrange update may beformed from average.
The solution must be ob-tained distributed:
x1
x2
x3
x4
x5
x6
xn
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2. Coordination for loss minimization
We consider a community of households in the low-voltagedistribution grid:
r1
r2
rm
PCC
Grid
tie-connection
All households are characterized by both flexible (controllable)and inflexible consumption, and some further represent localpower production
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2. Coordination for loss minimization
The task is to schedule flexible consumption such as to
◮ Minimize local objectives on cost and comfort
◮ Obey local constraints
◮ Minimize joint objective on cost of losses
◮ Obey grid capacity constraints
The goal is to efficiently utilize the grid, while increasing thepossible installation of e.g. EVs and solar panels
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2. Coordination for loss minimization
The task is to schedule flexible consumption such as to
minimize∑
i
fi(xi ) + l(x1, . . . , xn)
subject to xi ∈ Xi , ∀i
|∑
i∈rj
xi | ≤ ν j , ∀j
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2. Coordination for loss minimization
The cost of losses is a joint objective for the community, and weuse this for coordination:
◮ Coordination among consumers
◮ Coordination between consumptionand local production
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2. Coordination for loss minimization
A centralized coordination scheme requires
◮ Knowledge about each individual control objectives
◮ Access to all dynamic models
◮ Knowledge about local constraints
unlikely that one entity will be able to collect all this
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2. Coordination for loss minimization
We enforce a distributed approach, using ADMM∗ iterations
◮ each consumer optimizes own objective, subject to localconstraints
◮ coordination is conducted by iteratively updating anddistributing tariff functions
◮ this task may be performed by the distribution systemoperator
Distribution System
Consumers
OperatorUpdate tariff, basedon local results
Local optimizationincluding tariff
∗Alternating Direction Method of Multipliers16
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3. Voltage control through coordination
We consider again the low voltage distribution grid:
us
z1
i1
ppv,1q̃pv,1
p1,
q1
p̃ev,1
un
zm−1
in
ppv,nq̃pv,n
pn,
qn
p̃ev,n
zm
u1
z2
q̃ev,1 q̃ev,n
(pi , qi): inflexible consumption , (p̃i , q̃i ): inflexible consumption(ppv,i , q̃pv,i): local photo voltaic production, zi : grid impedances
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3. Voltage control through coordination
Massive installation of EVs and photo-voltaics (PVs), may carryundesired scenarios:
|u(h)|
[pu]
Distance to transformer (h) [-]
0 10 20 30 40 50
0.9
1
1.1
1.2
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3. Voltage control through coordination
The task is again utilize flexibility such as to
◮ Minimize local objectives on cost and comfort
◮ Obey local constraints
◮ Minimize joint objective on cost of losses
◮ Obey constraints on voltage variation
The goal is to efficiently utilize the grid, while increasing thepossible installation of e.g. EVs and solar panels
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3. Voltage control through coordination
Consumer modeling:
◮ Consider the set of consumers H = {1, . . . , n}, over thetime period T = {1, . . . ,T}
◮ Consumer flexibility is given by EV charging and PVreactive capabilities
◮ We let Hev ⊆ H and Hpv ⊆ H denote households with EVsand and solar panels installed, respectively
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3. Voltage control through coordination
Consumer modeling:
◮ Let Eev,h(t) denote EV state of charge (SOC):
Eev,h(t) = Eev,h(tev,h) +
t∑
τ=tev,h
Ts p̃ev,i (τ), ∀h ∈ Hev,
where tev,h denotes time of plug-in, and Eev,h(tev,h) denotesplug-in SOC.
◮ EV charging is flexible within the constraints:
Eev,h(T ) = Edem,h, Emin,h ≤ Eev,h(t) ≤ Emax,h,
pmin,h ≤ p̃ev,h(t) ≤ pmax,h, q̃ev,h(t) = p̃ev,h(t) tan(acos(ψh)),
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3. Voltage control through coordination
Consumer modeling:
◮ Solar panel active power output is defined by weatherconditions
◮ Reactive power is controllable with the constraint:
|q̃pv,h(t)| ≤√
s2max,h − p2pv,h(t), ∀t ∈ T
where smax,h is the inverter apparent power limit.
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3. Voltage control through coordination
Consumer modeling:
◮ For each h ∈ H and t ∈ T , the active and reactive power is
ph(t) = ph(t) + p̃ev,h(t)− ppv,h(t),
qh(t) = qh(t) + q̃ev,h(t)− q̃pv,h(t).
◮ Current drawn by each consumer is
ih(t) = f (ph(t), qh(t), uh(t)) =
(
ph(t) + jqh(t)
uh(t)
)†
, ∀t ∈ T ,
◮ We let
i(t) = (i1(t), . . . , in(t)) ∈ Cn, u(t) = (u1(t), . . . , un(t)) ∈ Cn,
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3. Voltage control through coordination
Grid modeling:
◮ We consider grids,represented as trees:
z1
z2
us
z4
z5
z3
z6
z7
i1(t) i2(t) i3(t) i4(t)
◮ Here zi = ri + jxi denotes gridimpedance
◮ 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
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