# Least Cost System Operation Economic Dispatch 2

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"Least Cost System Operation Economic Dispatch 2"

```					Least Cost System Operation:
Economic Dispatch 2

Smith College, EGR 325
March 10, 2006       1
Overview
• Complex system time scale separation
• Least cost system operation
– Economic dispatch first view
– Generator cost characteristics
• System-level cost characterization
• Constrained optimization
– Linear programming
– Economic dispatch completed
2
Time Scale Separation
1. Decide what to build
2. Given the plants that are built  decide
which plants to have warmed up and
ready to go this month, week...
3. Given the plants that are ready to
generate  decide which plants to use to
meet the expected load today, the next 5
minutes, next hour...
4. Given the plants that are generating 
Decide how to maintain the supply and
demand balance cycle to cycle              3
Economic Dispatch Recap
• Economic dispatch determines the best
way to minimize the current generator
operating costs
• Economic dispatch is not concerned with
determining which units to turn on/off (this
is the unit commitment problem)
• Economic dispatch ignores the
transmission system limitations
4
Constrained Optimization
& Economic Dispatch

5
Mathematical Formulation of Costs
• Generator cost curves are not actually smooth
• Typically curves can be approximated using
– piecewise linear functions

Ci ( PGi )   i   PGi   PGi
2
\$/hr (fuel-cost)
dCi ( PGi )
ICi ( PGi )                 2 PGi \$/MWh
dPGi
6
Mathematical Formulation of Costs
• The marginal cost is one of the most important
quantities in operating a power system
• Marginal cost = incremental cost: the cost of
producing the next increment (the next MWh)
• How do we find the marginal cost?

7
Economic Dispatch
• An economic dispatch results in all the
generator generating at a level where they
have equal marginal costs (for a lossless
system)

IC1(PG,1) = IC2(PG,2) = … = ICm(PG,m)

8
Incremental Cost Example
For a two generator system assume
C1 ( PG1 )  1000 20 PG1  0.01PG1
2
\$ / hr
C2 ( PG 2 )  400 15 PG 2  0.03PG 2
2
\$ / hr
Then
dC1 ( PG1 )
IC1 ( PG1 )               20  0.02 PG1 \$/MWh
dPG1
dC2 ( PG 2 )
IC2 ( PG 2 )                15  0.06 PG 2 \$/MWh
dPG 2
9
Incremental Cost Example

If PG1  250 MW and PG2  150 MW Then
C1 (250)  1000 20  250  0.01  250 2  \$ 6625/hr
C2 (150)  400 15  150  0.03  150 2    \$6025/hr
Then
IC1 (250)  20  0.02  250  \$ 25/MWh
IC2 (150)  15  0.06  150  \$ 24/MWh

10
Economic Dispatch: Formulation
• The goal of economic dispatch is to
– determine the generation dispatch that
minimizes the instantaneous operating cost
– subject to the constraint that total generation
m
Minimize    CT     Ci ( PGi )
i 1
Initially we'll
Such that                          ignore generator
m                     limits and the
 PGi  PD  PLosses   losses          11
i=1
Unconstrained Minimization
• This is a minimization problem with a
single inequality constraint
• For an unconstrained minimization a
necessary (but not sufficient) condition for
a minimum is the gradient of the function
must be zero, f (x)  0
• The gradient generalizes the first
derivative for multi-variable problems:
 f (x) f (x)     f (x) 
f ( x )    x , x ,       ,
 1          2       xn     12
Minimization with Equality Constraint
• When the minimization is constrained with an
equality constraint we can solve the problem using
the method of Lagrange Multipliers
• Key idea is to modify a constrained minimization
problem to be an unconstrained problem
That is, for the general problem
minimize f (x) s.t. g(x)  0
We define the Lagrangian L(x,λ )  f (x)  λ T g(x)
Then a necessary condition for a minimum is the
L x (x,λ )  0   and   L λ (x,λ )  0        13
Economic Dispatch Lagrangian
For the economic dispatch we have a minimization
constrained with a single equality constraint
m                       m
L(PG ,  )       Ci ( PGi )   ( PD   PGi )   (no losses)
i 1                    i 1
The necessary conditions for a minimum are
L(PG ,  )            dCi ( PGi )
               0    (for i  1 to m)
PGi                   dPGi
m
PD   PGi  0
i 1
14
Economic Dispatch Example
What is economic dispatch for a two generator
system PD  PG1  PG 2  500 MW and
C1 ( PG1 )  1000 20 PG1  0.01PG1
2
\$ / hr
C2 ( PG 2 )  400 15 PG 2  0.03PG 2
2
\$ / hr
Using the Largrange multiplier method we know
dC1 ( PG1 )
  20  0.02 PG1    0
dPG1
dC2 ( PG 2 )
    15  0.06 PG 2        0
dPG 2
500  PG1  PG 2  0                              15
Economic Dispatch Example, cont‟d
We therefore need to solve three linear equations
20  0.02 PG1       0
15  0.06 PG 2       0
500  PG1  PG 2  0
0.02    0     1  PG1   20 
 0     0.06 1  PG 2    15 
                             
 1
        1 0      500 
            
 PG1     312.5 MW 
 P    187.5 MW 
 G2                    
  
         26.2 \$/MWh 
                                        16
Constrained Optimization &
Linear Programming

17
Linear Programming Definition
• Optimization is used to find the “best”
value
– “Best” defined by us, the analysts and
designers
• Constrained opt  Linear programming
– Linear constraints
– Complicates the problem
• Some binding, some non-binding
• Visualize via a „feasible region‟
18
Formulating the Problem
•   Objective function
•   Constraints
•   Decision variables
•   Variable bounds
•   Standard form
– min cx
– s.t. Ax = b
xmin <= x <= xmax
19
Formulating the Problem
• For power systems:
min CT = ΣCi(PGi)
s.t. Σ(PGi) = PL
PGi min <= PGi <= PGi max

20
Constrained Optimization
& Economic Dispatch
The Lagrangean

21
Formulating the Lagrangean
• Rewrite the constrained optimization
problem as an unconstrained optimization
problem !
– Then we can use the simple derivative
(unconstrained optimization) to solve
• The task is to interpret the results correctly

22
Formulating the Lagrangean
• We are minimizing gradients of both
multivariate equations
– CT &     ΣPGi = PL
• For both equations to be at a minimum
these gradients must be linearly dependent
vectors
• CT – λw = 0
• with w ≡ ΣPG – PL = 0
• The “Lagrangean multiplier”
– λ is defined to be the scaling variable that
brings CT and w into linear alignment        23
Lagrangean Example
max g(x) = 5x12x2
s.t. h(x) = x1 + x2 = 6 or x1 + x2 – 6 = 0

Formulate L =
L = g(x) – λh(x)
Find ?
dL/dx1, dL/dx2, dL/dλ
x1 = 4, x2 = 2, λ = 80
24
Economic Dispatch & the Lagrangean
min CT = ΣCi(PGi)
s.t. Σ(PGi) = PL
PGi min <= PGi <= PGi max

Then L = ?

L  CT   PGi  PL 
25
Economic Dispatch Example
• What is the economic dispatch for the two
generator problem with
What is economic dispatch for a two generator
system PD =PGD  PG 2  500 MW and
PG1 + PG2 P = 500MW
1

C1 ( PG1 )     1000 20 PG1  0.01PG12
\$ / hr
C2 ( PG 2 )    400 15 PG 2  0.03PG 2
2
\$ / hr
Using the Largrange multiplier method we know
dC1 ( PG1 )
      20  0.02 PG1        0
dPG1                                            26
What is economic dispatch for a two generator
EconomicPDispatch Example
system P  P   500 MW and
D    G1     G2

• PG1 )  1000 20 PG1  0.01P
C1 (Formulate theLagrangean G12
\$ / hr
• ( P ) derivatives  0.03P 2
C2 Take  400 15PG 2                       \$ / hr
G2                     G2
• Solve
Using the Largrange multiplier method we know
dC1 ( PG1 )
  20  0.02 PG1    0
dPG1
dC2 ( PG 2 )
    15  0.06 PG 2        0
dPG 2
500  PG1  PG 2  0
27
Economic Dispatch Example, cont‟d
We therefore need to solve three linear equations
20  0.02 PG1       0
15  0.06 PG 2       0
500  PG1  PG 2  0
0.02    0     1  PG1   20 
 0     0.06 1  PG 2    15 
                             
 1
        1 0      500 
            
 PG1     312.5 MW 
 P    187.5 MW 
 G2                    
  
         26.2 \$/MWh 
                                        28
Economic Dispatch: Formulation
• We find that
– PG1 = 312.5MW;
– PG2 = 187.5MW
•  = \$26.2/MWh

29
Discussion
• Key results for Economic Dispatch?
– Incremental cost of all generating units is
equal
– This incremental cost is the Lagrangean
multiplier, 
– „‟ is called the „System ‟ and is the system-
wide cost of generating electricity
• This is the price charged to customers

30
Power System Control Center

31
Power System Control Center

32
New England Power Grid Operator

33
Regional Prices and Constraints

34
The Hong Kong Trade Development Council

35
36
Summary
• Economic dispatch is used to determine
the least cost means of using existing
generating plants to meet electric demand
• To calculate the economic dispatch for a
power system, the techniques of linear
programming + the Lagrangean are used
• Now to a review of the production cost
homework results...
37

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