Optimal Secondary frequency control
connecting AGC with economic dispatch
Changhong Zhao
Na Li
Lijun Chen
Steven Low
LIDS, MIT Telecommunication
CMS
CU-Boulder
Caltech
CMU
02/04/2014
Frequency control
economic
dispatch
unit
commitment
secondary
freq control
primary
freq control
sec
min
dynamic model
e.g. swing eqtn
5 min
60 min
day
power flow model
e.g. DC/AC power flow
year
Frequency control
economic
dispatch
secondary
freq control
Optimally schedule gens/loads,
and the inter-area power flows
Maintain the nominal freq and
the inter-area power flows
∆ Total Gen = ∆Total Load
sec
min
dynamic model
e.g. swing eqtn
5 min
power flow model
e.g. DC/AC power flow
Area 1
Area 2
Area 3
Inter-area
power flows
Frequency control
To Improve
secondary
freq control
sec
1. Optimally schedule
gens/loads within a area
economic
dispatch
min
dynamic model
e.g. swing eqtn
Maintain the nominal freq and
the inter-area power flows
2. Optimally (re)schedule gens/loads
among
it is permitted)
∆ Total areas
Gen =(if
∆Total
Load
5 min
power flow model
e.g. DC/AC power flow
Area 1
Area 2
Area 3
Inter-area
power flows
Frequency control
To Improve
1. Optimally schedule
gens/loads within a area
economic
dispatch
secondary
freq control
Maintain the nominal freq and
the inter-area power flows
2. Optimally (re)schedule gens/loads
among areas (if it is permitted)
sec
min
dynamic model
e.g. swing eqtn
5 min
power flow model
e.g. DC/AC power flow
Motivation:
Power system efficiency
Frequency control
To Improve
1. Optimally schedule
gens/loads within a area
economic
dispatch
secondary
freq control
Maintain the nominal freq and
the inter-area power flows
2. Optimally (re)schedule gens/loads
among areas (if it is permitted)
sec
min
dynamic model
e.g. swing eqtn
5 min
power flow model
e.g. DC/AC power flow
Challenge:
Physical dynamics + existing
control mechanisms (AGC)
Physical dynamics: Swing Dynamics
-
Swing
+
ωi
: Mechanical power
: Load
ωi : Frequency
Pij : Power Flow
Mi, Di: Constant parameters
Bus i: control area/
balance authority
+
-
Swing
ωj
Variables: the deviations from
reference (steady state) values
Power Flow dynamics
-
Swing
+ -
Pij
Flow
ωi
+
-
-1
+
-
Swing
ωj
: Mechanical power
: Load
ωi : Frequency
Pij : Power Flow
Tij: Constant parameter
Assumptions:
Lossless (resistance=0)
Fixed voltage magnitudes
Small deviation of angles
Turbine-Governor Control
+
1/R1
-
Governor
-
Swing
+ -
Pij
Flow
ωi
+
-
-1
+
1/R2
Governor
+
-
Swing
ωj
: Mechanical power
: Load
ωi : Frequency
Pij : Power Flow
: Power command input
Ti, Ri: Constant parameters
(Area Control Error) ACE-based AGC
B1
+
+
1/R1
+
ACE
Governor
-
Swing
+ -
Pij
-1
+
+
B2
Flow
ωi
+
-
-1
ACE
+
-
1/R2
Governor
+
-
Swing
ωj
: Mechanical power
: Load
ωi : Frequency
Pij : Power Flow
: Power command input
Ki, Bi: Constant parameters
Recap: System Dynamics with AGC
i
Pi m
Swing dynamics
Pij
Pi L
Power Flow Dynamics
Turbine-Governor Control
ACE-based AGC
Suppose the system is in steady state (all variables = 0)
Disturbance, e.g. Pi L
Drive the system to a new steady state
(i)
Economically Dispatch
(ii)
Our objective to minimize generation cost
j
Recap: System Dynamics with AGC
i
Pi m
Swing dynamics
Pij
Pi L
Power Flow Dynamics
Turbine-Governor Control
ACE-based AGC
Suppose the system is in steady state (all variables = 0)
Disturbance, e.g. Pi L
Drive the system to a new steady state
(i)
(ii)
Generation cost
Power flow balance
Economic
AGC
j
Recap: System Dynamics with AGC
i
Pi m
Swing dynamics
Pij
Pi L
Power Flow Dynamics
Turbine-Governor Control
ACE-based AGC
Question: How to modify AGC to be economic AGC?
Drive the system to a new steady state
(i)
(ii)
Economic
AGC
j
Results: Economic AGC
i
Pi m
Swing dynamics
Pij
Pi L
Power Flow Dynamics
Drive the system to a new steady state
(i)
(ii)
Economic
AGC
j
Results: Economic AGC
i
Pi m
Swing dynamics
Power Flow Dynamics
Theorem(Li, Chen, Zhao, Low 2013):
Any trajectory  P M (t ),  (t ), P(t )  converges to  P M * , * , P* 
• P M * is optimal to economic dispatch
*
•  0
• P* is a feasible power flow
Pij
Pi L
j
Results: Economic AGC
i
Pi m
Swing dynamics
Pij
Pi L
Power Flow Dynamics
Local computation
Local communication
— Additional local variable
— Frequency, Power flow, Power Command: local measurable signals
— A decentralized algorithm; Local information and communications
— Not just local…
j
Not Just Local…
i
Pi m
Swing dynamics
Pij
Pi L
Power Flow Dynamics
Only use information/signals that are easy to measure or calculate
j
Extension to load freq control
i
Pi m
Swing dynamics
Power Flow Dynamics
Drive the system to a new steady state
(i)
(ii)
Load disutility
Pij
Pi L
Similar mechanisms
apply to load control
Optimal
gens/loads
control
j
Tool: reverse/forward engineering
System Dynamics & Existing Control
Economic Generation Control
(i)
(ii)
Analogy ?
solve
Optimization Problem
Reverse
Tool: reverse/forward engineering
System Dynamics & Existing Control
Economic Generation Control
(i)
(ii)
Modified
Optimization Problem
Equivalent
Forward
Tool: reverse/forward engineering
System Dynamics & Modified Control Economic Generation Control
(i)
(ii)
solve
Modified
Optimization Problem
Equivalent
Forward
Case Study
1
2
4 control areas: a 39-bus New
England Transmission Network
Source: Ilic, et.al. IEEE TPS, vol.
8, no. 1, 1993
4
•
•
•
•
3
At time t = 10s, load increase 0.2 pu at area 4
(Nonlinear) power flow model; Nonzero resistance
Sample rate =15s (e.g., ACE is reset for every 15s)
Each area has a same cost function (optimal value should be
that each area increase generation by 0.05 pu )
Mechanical Power, PM (pu)
Mechanical Power Generation
0.3
0.2
Area 1
Area 2
Area 3
Area 4
0.1
0
-0.1
0
0.3
Mechanical Power, PM(pu)
ACE-based AGC
100
200
300
400
600
Area 1
Area 2
Area 3
Area 4
Economic AGC
0.2
500
0.1
0
-0.1
0
100
200
300
Time (s)
400
500
600
Total generation cost
1.4
Generation cost
1.2
1
ACE-based AGC
0.8
0.6
Economic AGC
0.4
0.2
Optimal value of economic dispatch
0
0
100
200
300
Time (s)
400
500
600
Frequency
Frequency (Hz)
60.01
60.005
ACE-based AGC
60
Area 1
Area 2
Area 3
Area 4
59.995
59.99
59.985
Frequency (Hz)
0
60.01
100
200
300
60.005
400
500
600
Economic AGC
60
Area 1
Area 2
Area 3
Area 4
59.995
59.99
59.985
0
100
200
300
Time (s)
400
500
600
Conclusion
economic
dispatch
secondary
freq control
primary
freq control
1. Optimally schedule
gens/loads within a area
Maintain the nominal frequency
and the inter-area power flows
2. Optimally (re)schedule gens/loads
among areas (if it is permitted)
Minor Modification
(additional local computation and communication
based on local measurable signals)
Back Up
System dynamics
Economic Energy Control
 c (P )
m
min
i
i
i
over
Pi m , 
s. t.
Pi m = Pi L 
i  0
Partial primal-dual gradient algorithm
min
Reverse
Engineering over
s. t.

1
2
( Pi m )2  D2i i 2

i
Pi M , 
Pi M = Pi L  Dii 
Pi M = Pi L
P P
j:i  j
ij
k :k i
ki
P P
j:i  j
ij
k :k i
ki
Economic AGC
Economic Energy Control
 c (P )
m
min
i
i
i
over
Pi m , 
s. t.
Pi m = Pi L 
P P
j:i  j
ij
k :k i
i  0
Forward
Engineering
Partial primal-dual gradient algorithm
min

1
2
( Pi m )2  D2i i 2
 
Pi M , 
s. t.
Pi M = Pi L  Dii 
Pi M = Pi L

i
i
over
ci ( Pi m )  D2i i 2
P P
j:i  j
ij
k :k i
Pi M  Pi L 
ki

j:i  j
ij


k :k i
ki
ki