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A Two Time-Scale Approach for Voltage
Regulation in Unbalanced Distribution Systems
Brett A. Robbins
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
May 8, 2015
Introduction
Distribution systems are undergoing significant transformations in
structure and functionality
I
advanced communication, sensing, and control
I
variable generation, e.g., photovoltaics
I
storage-capable loads, e.g., electric vehicles, batteries
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Motivating Example
Residential energy usage and common component ratings
I
average residential energy consumption is 30 kWh/day
I
average residential solar installation is 5.7 kW in 2010
I
common electric vehicles specifications
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Motivating Example
Residential energy usage and common component ratings
I
average residential energy consumption is 30 kWh/day
I
average residential solar installation is 5.7 kW in 2010
I
common electric vehicles specifications
Vehicle
Battery [kWh]
Charger [kW]
Tesla Model S
BMW i3
Nissan Leaf
Chevrolet Volt
85
22
24
17.1
9.6, 19.2
7.6
6.6
7.2
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Motivating Example
Residential energy usage and common component ratings
I
average residential energy consumption is 30 kWh/day
I
average residential solar installation is 5.7 kW in 2010
I
common electric vehicles specifications
Configuration
Voltage [V]
Power [kW]
AC Level 1
AC Level 2
DC Level 1
DC Level 2
120
240
200-500 DC
200-500 DC
1.92
19.2
40
100
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Motivating Example: System Base Load
Aggregate Load
Aggregate Load [MW]
4
3
2
1
0
4
8
12
16
20
24
Time [h]
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Motivating Example: System Base Load
Bus Loads
Bus Load [kW]
90
60
30
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: PV Profiles
Power Generation [p.u.]
Ideal Conditions
1
0.5
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: PV Profiles
Power Generation [p.u.]
Shaded Conditions
1
0.5
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: PV Profiles
Aggregate Load [MW]
Base Load
6
4
2
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: PV Profiles
Aggregate Load [MW]
Ideal PV Injections
6
4
2
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: EV Charging Schedules
Aggregate Load [MW]
Worst-Case Uncontrolled
6
4
2
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: EV Charging Schedules
Aggregate Load [MW]
Randomly Delayed Start Times
6
4
2
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: EV Charging Schedules
Aggregate Load [MW]
Optimal Scheduling
6
4
2
0
0
4
8
12
16
20
24
Time [h]
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Motivating Example: Bus Voltages
1.05
Voltage [p.u.]
1
0.95
0.9
0.85
0
4
8
12
16
20
24
Time [h]
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Motivating Example: Bus Voltages
1.05
Voltage [p.u.]
1
0.95
0.9
0.85
0
4
8
12
16
20
24
Time [h]
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Motivating Example: Bus Voltages
1.05
Voltage [p.u.]
1
0.95
0.9
0.85
0
4
8
12
16
20
24
Time [h]
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Motivating Example: Bus Voltages
1.05
Voltage [p.u.]
1
0.95
0.9
0.85
0
4
8
12
16
20
24
Time [h]
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Motivating Example: Bus Voltages
1.05
Voltage [p.u.]
1
0.95
0.9
0.85
0
4
8
12
16
20
24
Time [h]
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Problem Statment
We seek to develop a control strategy that accomplishes the
following:
I
mitigate the variability introduced by the DERs
I
regulate system bus voltages
I
achieve other operational objectives
I
incorporate both conventional hardware and reactive power
support provided by power electronics
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Overview
Introduction
Control Architecture
Slow Time-Scale Control
Fast Time-Scale Control
Case Studies
Conclusion
Control Architecture
Fast Time-Scale Control
time
t0
t1
t2
Slow Time-Scale Control
We envision a hierarchical approach that categorizes devices as:
I
Slow Time-Scale: Conventional voltage regulation devices
are periodically dispatched to reduce mechanical wear
I
Fast Time-Scale: Devices connected to the grid through
power electronic interfaces, e.g., DERs
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Control Architecture
Fast Time-Scale Control
time
t0
t1
t2
Slow Time-Scale Control
Then, we use time-scale separation to control the system:
I
Slow Time-Scale Control: Simultaneously dispatch
mechanical devices and reactive power to define V r
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Control Architecture
Fast Time-Scale Control
time
t0
t1
t2
Slow Time-Scale Control
Then, we use time-scale separation to control the system:
I
Slow Time-Scale Control: Simultaneously dispatch
mechanical devices and reactive power to define V r
I
Fast Time-Scale Control: Regulates the system voltages to
V r and detects when to rerun the slow time-scale control
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Radial Voltage Profile
Voltage [p.u.]
1.05
1
0.95
Distance From Feeder
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Radial Voltage Profile
Voltage [p.u.]
1.05
Tap Settings
1
0.95
Distance From Feeder
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Radial Voltage Profile
Voltage [p.u.]
1.05
Tap Settings
1
Reactive Power
Support
0.95
Distance From Feeder
10/17
Radial Voltage Profile
Voltage [p.u.]
1.05
Tap Settings
1
Reactive Power
Support
0.95
Distance From Feeder
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Overview
Introduction
Control Architecture
Slow Time-Scale Control
Fast Time-Scale Control
Case Studies
Conclusion
Slow Time-Scale Optimization
The rank-relaxed optimization will have the form
Minimize Cost Function C (·)
such that
Slack Bus Voltage
Power Flow on the
Power Flow on the
Power Flow on the
Power Flow on the
Power Flow on the
and
Uncontrollable Buses
Controllable Buses
Virtual Secondary-Side
Uncontrollable Primary-Side Buses
Controllable Primary-Side Buses
Voltage, Reactive Power, and Tap Limits
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Slow Time-Scale Optimization
The rank-relaxed optimization will have the form
min
W 0, q̃, Sps
f0 (W ) +
X
fi (W , q̃)
i∈{1,2,3,4}
such that
2
[W ]00 = (V s ) ,
fui (W ) = 0,
∀i ∈ Nu \Np
fci (W , q̃i ) = 0,
∀i ∈ Nc \Np
fst0 (W , Spt st0 ) = 0,
∀t ∈ T
fpt (W , Spt st0 ) = 0,
∀t ∈ {T | pt ∈ Nu }
fpt0 (W , Spt st0 , q̃t ) = 0,
∀t ∈ {T | pt ∈ Nc }
and
V2
q ri
≤
≤
[W ]ii
q̃i
2
≤
V ,
∀i ∈ N
≤
q ri ,
∀i ∈ Nc
a2 [W ]pt pt ≤ [W ]st0 st0 ≤ a2 [W ]pt pt ,
∀t ∈ T
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Overview
Introduction
Control Architecture
Slow Time-Scale Control
Fast Time-Scale Control
Case Studies
Conclusion
Optimal Voltage Regulation Problem
The fast time-scale optimization will have the form
Minimize Voltage Deviation from V r
such that
Slack Bus Voltage
Voltage Magnitude Drop
Active Power Flows
Uncontrollable Reactive Power Flows
Controllable Reactive Power Flows
and
Reactive Power Limits
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Optimal Voltage Regulation Problem
Convex Relaxation: Quadratic Program
min
U
n
X
2
wk (Uk − Ukr )
k=1
such that, for all (i, k) ∈ E ,
U0 = |V s |2
Uk = Ui − 2 (rik Pik + xik Qik ) + ciku (U r , P, Q),
X
Pik =
Pkj + pkd + cikp (U r , P, Q),
k∈N
k∈N
j∈Hk
Qik =
X
Qkj + qkd + cikq (U r , P, Q),
k ∈ Nu
Qkj + qkd + cikq (U r , P, Q) − qk ,
k ∈ Nc
j∈Hk
Qik =
X
j∈Hk
and
q k ≤ qk ≤ q k ,
k ∈ Nc
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123-Bus Unbalanced Three-Phase Distribution System
33
34
49
28
47
32
35
31 30
29
45
43
24
38
21
27
23
22
Feeder
51 52
46
48
26
25
119
50
15
12
2
16
120
14
20
10
4
8
9
3
5
6
13
7
64
17
18
Three-Phase
61
63
11
112
110
66
106
65 104
44 67
42
68
40 41
39 37
58
62 60
1
T1
36
117 118
53 113 115 114 116
109
105
72
70
69
121
92 90
98 96 94
107
100
T2
72
54 55 56 57 59
97
77
19
91
93
95
99
Two-Phase
111
108
101
73
103
102
76
75
79
78
76
83 89
80 82
81
84
85
87
86
88
Single-Phase
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123-Bus Unbalanced Three-Phase Distribution System
Load [MW]
System Response
4
2
0
0
4
8
12
Time [h]
16
20
24
16/17
123-Bus Unbalanced Three-Phase Distribution System
Load [MW]
System Response
4
2
0
0
4
8
12
Time [h]
16
20
24
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123-Bus Unbalanced Three-Phase Distribution System
Load [MW]
System Response
4
2
0
0
4
8
12
Time [h]
16
20
24
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123-Bus Unbalanced Three-Phase Distribution System
Voltage [p.u.]
Load [MW]
System Response
4
2
0
0
4
8
12
Time [h]
16
20
24
0
4
8
12
Time [h]
16
20
24
1
0.95
0.9
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123-Bus Unbalanced Three-Phase Distribution System
Voltage [p.u.]
Load [MW]
System Response
4
2
0
0
4
8
12
Time [h]
16
20
24
0
4
8
12
Time [h]
16
20
24
1.05
1
0.95
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Concluding Remarks
Slow Time-Scale
I
developed transformer models for the semidefinite
programming approach
I
cost functions
I
three-phase extension
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Concluding Remarks
Slow Time-Scale
I
developed transformer models for the semidefinite
programming approach
I
cost functions
I
three-phase extension
Fast Time-Scale
I
feedback-based approach
I
optimization-based approach
I
three-phase extension
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Concluding Remarks
Slow Time-Scale
I
developed transformer models for the semidefinite
programming approach
I
cost functions
I
three-phase extension
Fast Time-Scale
I
feedback-based approach
I
optimization-based approach
I
three-phase extension
Two Time-Scale Architecture
I
unbalanced three-phase simulation
I
implementation and coordination of the two time-scales
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