Towards Continuous-time Optimization Models forPower Systems Operation
Masood Parvania
Assistant Professor, Electrical and Computer EngineeringUniversity of Utah
FERC Technical Conference on Increasing Real-Time and Day-Ahead Market Efficiencythrough Improved Software, Washington, DC
June 27-29, 2016
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 1 / 16
Motivation and Background
Power system operation optimization problem: stochastic,continuous-time, mixed-integer
Current practice: break down the problem into different time scales,from several days ahead to real-time operation, solving discrete-timeoptimization problems for each time scale.
( )N t
Power System Operation: Current Practice
• The current practice to solve power system operation optimization problem is to break it down into different time scales, from several days ahead to real-time.
Piece-wise constant day-
Real-time load deviation
©2016 M. Parvania, and the University of Utah
t
4
Piece-wise constant day-ahead generation trajectory
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 2 / 16
Motivation and Background
Power system operation optimization problem: stochastic,continuous-time, mixed-integer
Current practice: break down the problem into different time scales,from several days ahead to real-time operation, solving discrete-timeoptimization problems for each time scale.
( )N t
Power System Operation: Current Practice
• The current practice to solve power system operation optimization problem is to break it down into different time scales, from several days ahead to real-time.
Piece-wise constant day-
Real-time load deviation
©2016 M. Parvania, and the University of Utah
t
4
Piece-wise constant day-ahead generation trajectory
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 2 / 16
Implications of Discrete-time Operation Models
Implication of discrete-time generation schedule:
Generation trajectory is modeled by zero-order piecewise constant curve
Ramping of units is defined as the finite difference between theconsecutive discrete-time generation schedules (there is no explicitramping trajectory).
The discrete-time generation trajectories do not appropriately capturethe flexibility of generating units to balance the continuous-timevariations (ramping) of load, and leaves out residual that needs to besupplied in real-time operation
If the real-time ramping requirement is beyond the available rampingcapacity → ramping scarcity event
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 3 / 16
Implications of Discrete-time Operation Models
Implication of discrete-time generation schedule:
Generation trajectory is modeled by zero-order piecewise constant curveRamping of units is defined as the finite difference between theconsecutive discrete-time generation schedules (there is no explicitramping trajectory).
The discrete-time generation trajectories do not appropriately capturethe flexibility of generating units to balance the continuous-timevariations (ramping) of load, and leaves out residual that needs to besupplied in real-time operation
If the real-time ramping requirement is beyond the available rampingcapacity → ramping scarcity event
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 3 / 16
Implications of Discrete-time Operation Models
Implication of discrete-time generation schedule:
Generation trajectory is modeled by zero-order piecewise constant curveRamping of units is defined as the finite difference between theconsecutive discrete-time generation schedules (there is no explicitramping trajectory).
The discrete-time generation trajectories do not appropriately capturethe flexibility of generating units to balance the continuous-timevariations (ramping) of load, and leaves out residual that needs to besupplied in real-time operation
If the real-time ramping requirement is beyond the available rampingcapacity → ramping scarcity event
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 3 / 16
Implications of Discrete-time Operation Models
Implication of discrete-time generation schedule:
Generation trajectory is modeled by zero-order piecewise constant curveRamping of units is defined as the finite difference between theconsecutive discrete-time generation schedules (there is no explicitramping trajectory).
The discrete-time generation trajectories do not appropriately capturethe flexibility of generating units to balance the continuous-timevariations (ramping) of load, and leaves out residual that needs to besupplied in real-time operation
If the real-time ramping requirement is beyond the available rampingcapacity → ramping scarcity event
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 3 / 16
Continuous-time Generation and Ramping Trajectories
Instead of discrete-time schedules, assume that a set of K generatingunits are modeled by:– Continuous-time generation trajectories: G(t)=(G1(t), . . . ,GK (t))T
– Continuous-time commitment variables: I(t)=(I1(t), . . . , IK (t))T
Ik(t) =∑Hk
h=1
(u(t − t(SU)k,h )− u(t − t(SD)k,h )
)
We define the continuous-time ramping trajectory of unit k as thetime derivative of its continuous-time generation trajectory:
Gk(t) ,dGk(t)
dt
Vector of continuous-time ramping trajectories of units: G(t)
Explicit definition of ramping trajectories allows us to define costfunctions that are also functions of ramping trajectories:
Ck(Gk(t), Gk(t), Ik(t))
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 4 / 16
Continuous-time Generation and Ramping Trajectories
Instead of discrete-time schedules, assume that a set of K generatingunits are modeled by:– Continuous-time generation trajectories: G(t)=(G1(t), . . . ,GK (t))T
– Continuous-time commitment variables: I(t)=(I1(t), . . . , IK (t))T
Ik(t) =∑Hk
h=1
(u(t − t(SU)k,h )− u(t − t(SD)k,h )
)We define the continuous-time ramping trajectory of unit k as thetime derivative of its continuous-time generation trajectory:
Gk(t) ,dGk(t)
dt
Vector of continuous-time ramping trajectories of units: G(t)
Explicit definition of ramping trajectories allows us to define costfunctions that are also functions of ramping trajectories:
Ck(Gk(t), Gk(t), Ik(t))
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 4 / 16
Continuous-time Generation and Ramping Trajectories
Instead of discrete-time schedules, assume that a set of K generatingunits are modeled by:– Continuous-time generation trajectories: G(t)=(G1(t), . . . ,GK (t))T
– Continuous-time commitment variables: I(t)=(I1(t), . . . , IK (t))T
Ik(t) =∑Hk
h=1
(u(t − t(SU)k,h )− u(t − t(SD)k,h )
)We define the continuous-time ramping trajectory of unit k as thetime derivative of its continuous-time generation trajectory:
Gk(t) ,dGk(t)
dt
Vector of continuous-time ramping trajectories of units: G(t)
Explicit definition of ramping trajectories allows us to define costfunctions that are also functions of ramping trajectories:
Ck(Gk(t), Gk(t), Ik(t))
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 4 / 16
Continuous-time Unit Commitment Model
Continuous-time Unit Commitment:
minK∑
k=1
∫TCk(Gk(t), Gk(t), Ik(t))dt
s.t.K∑
k=1
Gk(t) = N(t) ∀t ∈ T
G k Ik(t) ≤ Gk(t) ≤ G k Ik(t) ∀k , t ∈ T
G k Ik(t) ≤ Gk(t) ≤ G k Ik(t) ∀k , t ∈ Tt(SD)k,h − t(SU)k,h ≥ T (on)
k , t(SU)k,h+1 − t(SD)k,h ≥ T (off)k ∀k , h, t ∈ T
The continuous-time UC model is a constrained variational problemwith infinite dimensional decision space⇒ We need to reduce dimensionality of the problem. Idea:
1 Subdivide T into M intervals: Tm =[tm, tm+1), T =∪M−1m=0 Tm.
2 Map the parameters and decision variables in each interval into afinite-dimensional function space.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 5 / 16
Continuous-time Unit Commitment Model
Continuous-time Unit Commitment:
minK∑
k=1
∫TCk(Gk(t), Gk(t), Ik(t))dt
s.t.K∑
k=1
Gk(t) = N(t) ∀t ∈ T
G k Ik(t) ≤ Gk(t) ≤ G k Ik(t) ∀k , t ∈ T
G k Ik(t) ≤ Gk(t) ≤ G k Ik(t) ∀k , t ∈ Tt(SD)k,h − t(SU)k,h ≥ T (on)
k , t(SU)k,h+1 − t(SD)k,h ≥ T (off)k ∀k , h, t ∈ T
The continuous-time UC model is a constrained variational problemwith infinite dimensional decision space
⇒ We need to reduce dimensionality of the problem. Idea:1 Subdivide T into M intervals: Tm =[tm, tm+1), T =∪M−1
m=0 Tm.2 Map the parameters and decision variables in each interval into a
finite-dimensional function space.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 5 / 16
Continuous-time Unit Commitment Model
Continuous-time Unit Commitment:
minK∑
k=1
∫TCk(Gk(t), Gk(t), Ik(t))dt
s.t.K∑
k=1
Gk(t) = N(t) ∀t ∈ T
G k Ik(t) ≤ Gk(t) ≤ G k Ik(t) ∀k , t ∈ T
G k Ik(t) ≤ Gk(t) ≤ G k Ik(t) ∀k , t ∈ Tt(SD)k,h − t(SU)k,h ≥ T (on)
k , t(SU)k,h+1 − t(SD)k,h ≥ T (off)k ∀k , h, t ∈ T
The continuous-time UC model is a constrained variational problemwith infinite dimensional decision space⇒ We need to reduce dimensionality of the problem. Idea:
1 Subdivide T into M intervals: Tm =[tm, tm+1), T =∪M−1m=0 Tm.
2 Map the parameters and decision variables in each interval into afinite-dimensional function space.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 5 / 16
Continuous-time Trajectories in a Function Space
Assume that in T , except for a small residual error, thecontinuous-time load trajectory N(t) lies in a countable and finitefunction space of dimensionality P, spanned by a set of basisfunctions e(t) = (e1(t), . . . , eP(t))T , that is:
N(t) =P∑
p=1
Npep(t) + εN(t) = NTe(t) + εN(t)
N = (N1, . . . ,NP)T : coordinates of the approximation onto thesubspace spanned by e(t).
To ensure the power balance in continuous-time, any generationtrajectory should have a component that lies in the same subspacespanned by e(t) and one that is orthogonal to it, i.e.,:
Gk(t) =P∑
p=1
Gk,pep(t) + εGk(t) = GT
k e(t) + εGk(t).
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 6 / 16
Continuous-time Trajectories in a Function Space
Assume that in T , except for a small residual error, thecontinuous-time load trajectory N(t) lies in a countable and finitefunction space of dimensionality P, spanned by a set of basisfunctions e(t) = (e1(t), . . . , eP(t))T , that is:
N(t) =P∑
p=1
Npep(t) + εN(t) = NTe(t) + εN(t)
N = (N1, . . . ,NP)T : coordinates of the approximation onto thesubspace spanned by e(t).
To ensure the power balance in continuous-time, any generationtrajectory should have a component that lies in the same subspacespanned by e(t) and one that is orthogonal to it, i.e.,:
Gk(t) =P∑
p=1
Gk,pep(t) + εGk(t) = GT
k e(t) + εGk(t).
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 6 / 16
Spline Representation using Cubic Hermite Polynomials
Cubic Hermite Polynomials: four polynomials in t ∈ [0, 1), formingthe vector: H(t) = (H00(t),H01(t),H10(t),H11(t))T
CubicHermiteSplineInterpola3on
14
0 00 1 10
0 01 1 11
( ) ( ) ( )( ) ( )
p t p H t p H tv H t v H t
= +
+ +
Modeling the continuous-time load and generation trajectories inspline function space of cubic Hermite:
N(t) =M−1∑m=0
HT (τm)NHm , Gk(t) =
M−1∑m=0
HT (τm)GHk,m
NHm and GH
k,m are the vectors of Hermite coefficients.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 7 / 16
Spline Representation using Cubic Hermite Polynomials
Cubic Hermite Polynomials: four polynomials in t ∈ [0, 1), formingthe vector: H(t) = (H00(t),H01(t),H10(t),H11(t))T
CubicHermiteSplineInterpola3on
14
0 00 1 10
0 01 1 11
( ) ( ) ( )( ) ( )
p t p H t p H tv H t v H t
= +
+ +
Modeling the continuous-time load and generation trajectories inspline function space of cubic Hermite:
N(t) =M−1∑m=0
HT (τm)NHm , Gk(t) =
M−1∑m=0
HT (τm)GHk,m
NHm and GH
k,m are the vectors of Hermite coefficients.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 7 / 16
Spline Representation using Bernstein Polynomials
Bernstein Polynomials of degree Q: Q + 1 polynomials in t ∈ [0, 1),forming the vector BQ(t) = (B0,Q , ...,Bq,Q , ...,BQ,Q)T , whereBq,Q(t) =
)tq(1− t)Q−q
Modeling the continuous-time load and generation trajectories inspline function space of Bernstein polynomials of degree 3:
N(t) =M−1∑m=0
BT3 (τm)NB
m , Gk(t) =M−1∑m=0
BT3 (τm)GB
k,m
CubicHermiteSplineInterpola3on
15
The Bernstein and Hermite coefficients are linearly related:
GBk,m = WTGH
k,m , NBk,m = WTNH
k,m
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 8 / 16
Spline Representation using Bernstein Polynomials
Bernstein Polynomials of degree Q: Q + 1 polynomials in t ∈ [0, 1),forming the vector BQ(t) = (B0,Q , ...,Bq,Q , ...,BQ,Q)T , whereBq,Q(t) =
)tq(1− t)Q−q
Modeling the continuous-time load and generation trajectories inspline function space of Bernstein polynomials of degree 3:
N(t) =M−1∑m=0
BT3 (τm)NB
m , Gk(t) =M−1∑m=0
BT3 (τm)GB
k,m
CubicHermiteSplineInterpola3on
15
The Bernstein and Hermite coefficients are linearly related:
GBk,m = WTGH
k,m , NBk,m = WTNH
k,m
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 8 / 16
Why Bernstein Polynomials?
Bernstein coefficients of the derivative of generation trajectory arelinearly related with the coefficients of the generation trajectory:
Gk(t) =M−1∑m=0
BT2 (τm)GB
k,m , GBk,m = KTGB
k,m = KTWTGHk,m
Convex hull property of the Bernstein polynomials: trajectories arebounded by the convex hull formed by the four Bernstein points:
mintm≤t≤tm+1
{BT3 (τm)GB
k,m} ≥ min{GBk,m}
maxtm≤t≤tm+1
{BT3 (τm)GB
k,m} ≤ max{GBk,m}
mintm≤t≤tm+1
{BT2 (τm)GB
k,m} ≥ min{GBk,m}
maxtm≤t≤tm+1
{BT2 (τm)GB
k,m} ≤ max{GBk,m}
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 9 / 16
Why Bernstein Polynomials?
Bernstein coefficients of the derivative of generation trajectory arelinearly related with the coefficients of the generation trajectory:
Gk(t) =M−1∑m=0
BT2 (τm)GB
k,m , GBk,m = KTGB
k,m = KTWTGHk,m
Convex hull property of the Bernstein polynomials: trajectories arebounded by the convex hull formed by the four Bernstein points:
mintm≤t≤tm+1
{BT3 (τm)GB
k,m} ≥ min{GBk,m}
maxtm≤t≤tm+1
{BT3 (τm)GB
k,m} ≤ max{GBk,m}
mintm≤t≤tm+1
{BT2 (τm)GB
k,m} ≥ min{GBk,m}
maxtm≤t≤tm+1
{BT2 (τm)GB
k,m} ≤ max{GBk,m}
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 9 / 16
Representation of Cost Function and Balance Constraint
Piecewise linear continuous-time cost function can be written in termsof the spline coefficients of generation and ramping trajectories:∫
TCk(Gk(t), Gk(t), Ik(t))dt = Ck(Gk , Gk , Ik).
Continuous-time power balance is ensured by balancing the four cubicHermite coefficients of the continuous-time load and generationtrajectory in each interval:
K∑k=1
Gk(t) = N(t) ∀t ∈ T →K∑
k=1
GHk,m = NH
m ∀m
DC power flow constraints can be modeled similarly.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 10 / 16
Representation of Cost Function and Balance Constraint
Piecewise linear continuous-time cost function can be written in termsof the spline coefficients of generation and ramping trajectories:∫
TCk(Gk(t), Gk(t), Ik(t))dt = Ck(Gk , Gk , Ik).
Continuous-time power balance is ensured by balancing the four cubicHermite coefficients of the continuous-time load and generationtrajectory in each interval:
K∑k=1
Gk(t) = N(t) ∀t ∈ T →K∑
k=1
GHk,m = NH
m ∀m
DC power flow constraints can be modeled similarly.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 10 / 16
Representation of Cost Function and Balance Constraint
Piecewise linear continuous-time cost function can be written in termsof the spline coefficients of generation and ramping trajectories:∫
TCk(Gk(t), Gk(t), Ik(t))dt = Ck(Gk , Gk , Ik).
Continuous-time power balance is ensured by balancing the four cubicHermite coefficients of the continuous-time load and generationtrajectory in each interval:
K∑k=1
Gk(t) = N(t) ∀t ∈ T →K∑
k=1
GHk,m = NH
m ∀m
DC power flow constraints can be modeled similarly.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 10 / 16
Continuous-time UC Solution
Function Space-based Unit Commitment
Projection in Cubic Hermite Function Space
Reconstructing Gk(t), , and Ik(t)
Generation Constraints and Bids
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 11 / 16
Simulation Results: IEEE-RTS + CAISO Load
The data regarding 32 units ofthe IEEE-RTS and load data fromthe CAISO are used here.
Both the day-ahead (DA) andreal-time (RT) operations aresimulated.
The five-minute net-load forecastdata of CAISO for Feb. 2, 2015 isscaled down to the originalIEEE-RTS peak load of 2850MW,and the hourly day-ahead loadforecast is generated where theforecast standard deviation isconsidered to be %1 of the loadat the time.
1600
1800
2000
2200
2400
2600
2800
3000
0 2 4 6 8 10 12 14 16 18 20 22 24
Load
(MW
)
Real-Time LoadDA Cubic Hermite loadDA Piecewise constant load
-250-200-150-100
-500
50100150200250
0 2 4 6 8 10 12 14 16 18 20 22 24
RT L
oad
Dev
iaio
n (M
W)
Hour
DA Cubic Hermite loadDA Piecewise constant load
(a)
(b)
0 2 4 6 8 10 12 14 16 18 20 22 24Hour
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 12 / 16
Reduced Operation Cost and Ramping Scarcity Events
Case 1: Hourly UC Model
Case 2: Continuous-time UC Model
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16 18 20 22 24
Gen
erat
ion
Sch
edul
e (M
W)
HourGroup 1: Hydro Group 2: Nuclear Group 3: Coal 350 Group 4: Coal 155 Group 5: Coal 76
Group 6: Oil 100 Group 7: Oil 197 Group 8: Oil 12 Group 9: Oil 20
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16 18 20 22 24
Gen
erat
ion
Sch
edul
e (M
W)
(a)
(b)
Total operation cost and ramping scarcityevents are reduced in Case 2
Case DA Operation Cost ($)
RT Operation Cost ($)
Total DA and RT Operation Cost ($)
RT Ramping Scarcity Events
Case 1 471,130.7 16,882.9 488,013.6 27 Case 2 476,226.4 6,231.3 482,457.7 0
Continuous-time ramping trajectories
-30
-20
-10
0
10
20
30
0 2 4 6 8 10 12 14 16 18 20 22 24
Ram
ping
Tra
ject
ory
(MW
/min
)
Group 1: Hydro Group 2: Nuclear Group 3: Coal 350 Group 4: Coal 155 Group 5: Coal 76 Group 6: Oil 100 Group 7: Oil 197 Group 8: Oil 12 Group 9: Oil 20
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 13 / 16
Reduced Operation Cost and Ramping Scarcity Events
Case 1: Hourly UC Model
Case 2: Continuous-time UC Model
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16 18 20 22 24
Gen
erat
ion
Sch
edul
e (M
W)
HourGroup 1: Hydro Group 2: Nuclear Group 3: Coal 350 Group 4: Coal 155 Group 5: Coal 76
Group 6: Oil 100 Group 7: Oil 197 Group 8: Oil 12 Group 9: Oil 20
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16 18 20 22 24
Gen
erat
ion
Sch
edul
e (M
W)
(a)
(b)
Total operation cost and ramping scarcityevents are reduced in Case 2
Case DA Operation Cost ($)
RT Operation Cost ($)
Total DA and RT Operation Cost ($)
RT Ramping Scarcity Events
Case 1 471,130.7 16,882.9 488,013.6 27 Case 2 476,226.4 6,231.3 482,457.7 0
Continuous-time ramping trajectories
-30
-20
-10
0
10
20
30
0 2 4 6 8 10 12 14 16 18 20 22 24
Ram
ping
Tra
ject
ory
(MW
/min
)
Group 1: Hydro Group 2: Nuclear Group 3: Coal 350 Group 4: Coal 155 Group 5: Coal 76 Group 6: Oil 100 Group 7: Oil 197 Group 8: Oil 12 Group 9: Oil 20
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 13 / 16
Reduced Operation Cost and Ramping Scarcity Events
Case 1: Hourly UC Model
Case 2: Continuous-time UC Model
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16 18 20 22 24
Gen
erat
ion
Sch
edul
e (M
W)
HourGroup 1: Hydro Group 2: Nuclear Group 3: Coal 350 Group 4: Coal 155 Group 5: Coal 76
Group 6: Oil 100 Group 7: Oil 197 Group 8: Oil 12 Group 9: Oil 20
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16 18 20 22 24
Gen
erat
ion
Sch
edul
e (M
W)
(a)
(b)
Total operation cost and ramping scarcityevents are reduced in Case 2
Case DA Operation Cost ($)
RT Operation Cost ($)
Total DA and RT Operation Cost ($)
RT Ramping Scarcity Events
Case 1 471,130.7 16,882.9 488,013.6 27 Case 2 476,226.4 6,231.3 482,457.7 0
Continuous-time ramping trajectories
-30
-20
-10
0
10
20
30
0 2 4 6 8 10 12 14 16 18 20 22 24
Ram
ping
Tra
ject
ory
(MW
/min
)
Group 1: Hydro Group 2: Nuclear Group 3: Coal 350 Group 4: Coal 155 Group 5: Coal 76 Group 6: Oil 100 Group 7: Oil 197 Group 8: Oil 12 Group 9: Oil 20
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 13 / 16
Continuous-time Model Outperforms Discrete-time Models
Simulations are repeated forCAISO’s load data of the entiremonth of Feb. 2015.Half-hourly UC model is alsosimulated.
The proposed modeloutperforms the other two casesin terms of real-time and totaloperation cost reduction, evencompared to the half-hourly UCsolution with twice the binaryvariables.
Computation time for Feb. 2,2015 load data:– Hourly UC: 0.257s– Half-hourly UC: 0.572s– Proposed UC: 1.369s
2
6
10
14
18
22
300 350 400 450 500
Rea
l-tim
e O
pera
tion
Cos
t (Th
ousa
nds $
)
Day-Ahead Operation Cost (Thousands $)
Proposed UC Hourly UCHalf-hourly UC
(a)
0
15
30
45
60
75
Ram
ping
Sca
rcity
Eve
nts
Days
Proposed UCHourly UCHalf-hourly UC
(b)
0Days
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 14 / 16
Continuous-time Model Outperforms Discrete-time Models
Simulations are repeated forCAISO’s load data of the entiremonth of Feb. 2015.Half-hourly UC model is alsosimulated.
The proposed modeloutperforms the other two casesin terms of real-time and totaloperation cost reduction, evencompared to the half-hourly UCsolution with twice the binaryvariables.
Computation time for Feb. 2,2015 load data:– Hourly UC: 0.257s– Half-hourly UC: 0.572s– Proposed UC: 1.369s
2
6
10
14
18
22
300 350 400 450 500R
eal-t
ime
Ope
ratio
n C
ost (
Thou
sand
s $)
Day-Ahead Operation Cost (Thousands $)
Proposed UC Hourly UCHalf-hourly UC
(a)
0
15
30
45
60
75
Ram
ping
Sca
rcity
Eve
nts
Days
Proposed UCHourly UCHalf-hourly UC
(b)
0Days
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 14 / 16
Continuous-time Model Outperforms Discrete-time Models
Simulations are repeated forCAISO’s load data of the entiremonth of Feb. 2015.Half-hourly UC model is alsosimulated.
The proposed modeloutperforms the other two casesin terms of real-time and totaloperation cost reduction, evencompared to the half-hourly UCsolution with twice the binaryvariables.
Computation time for Feb. 2,2015 load data:– Hourly UC: 0.257s– Half-hourly UC: 0.572s– Proposed UC: 1.369s
2
6
10
14
18
22
300 350 400 450 500R
eal-t
ime
Ope
ratio
n C
ost (
Thou
sand
s $)
Day-Ahead Operation Cost (Thousands $)
Proposed UC Hourly UCHalf-hourly UC
(a)
0
15
30
45
60
75
Ram
ping
Sca
rcity
Eve
nts
Days
Proposed UCHourly UCHalf-hourly UC
(b)
0Days
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 14 / 16
Conclusions
Continuous-time models capture the continuous-time variations ofload and renewable resources, and tap the flexibility of generatingunits and other flexible resources to ramp beyond the current linearramping paradigm
Continuous-time models define ramping trajectory as an explicitdecision variable and enable accurate ramping valuation in markets
Enabling the definition of continuous-time marginal electricity price:
16
20
24
28
32
Mar
gina
l Pric
e ($
per
MW
in u
nit o
f tim
e) Continuous-time Price Hourly Price Half-hourly Price
8
12
0 2 4 6 8 10 12 14 16 18 20 22 24
Mar
gina
l Pric
e ($
per
MW
in u
nit o
f tim
e)
Hour
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 15 / 16
Conclusions
Continuous-time models capture the continuous-time variations ofload and renewable resources, and tap the flexibility of generatingunits and other flexible resources to ramp beyond the current linearramping paradigm
Continuous-time models define ramping trajectory as an explicitdecision variable and enable accurate ramping valuation in markets
Enabling the definition of continuous-time marginal electricity price:
16
20
24
28
32
Mar
gina
l Pric
e ($
per
MW
in u
nit o
f tim
e) Continuous-time Price Hourly Price Half-hourly Price
8
12
0 2 4 6 8 10 12 14 16 18 20 22 24
Mar
gina
l Pric
e ($
per
MW
in u
nit o
f tim
e)
Hour
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 15 / 16
Conclusions
Continuous-time models capture the continuous-time variations ofload and renewable resources, and tap the flexibility of generatingunits and other flexible resources to ramp beyond the current linearramping paradigm
Continuous-time models define ramping trajectory as an explicitdecision variable and enable accurate ramping valuation in markets
Enabling the definition of continuous-time marginal electricity price:
16
20
24
28
32
Mar
gina
l Pric
e ($
per
MW
in u
nit o
f tim
e) Continuous-time Price Hourly Price Half-hourly Price
8
12
0 2 4 6 8 10 12 14 16 18 20 22 24
Mar
gina
l Pric
e ($
per
MW
in u
nit o
f tim
e)
Hour
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 15 / 16
Further Reading
M. Parvania, A. Scaglione, “Unit Commitment with Continuous-timeGeneration and Ramping Trajectory Models,” IEEE Transactions onPower Systems, vol. 31, no. 4, pp. 3169-3178, July 2016.
M. Parvania, A. Scaglione, “Generation Ramping Valuation inDay-Ahead Electricity Markets,” in Proc. 49th Hawaii InternationalConference on System Sciences (HICSS), Kauai, HI, Jan. 5-8, 2016.
c©2016 Masood Parvania, and the University of Utah Continuous-time Operation Optimization of Power Systems 16 / 16