Shantanu Chavan
Navaneethakrishnan Raman
Tamil Kadir Rajavel
3/22/2016
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Linear Programming (LP)
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Economic Dispatch (ED)
◦ ED without loss
◦ ED with loss
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Hydro Scheduling
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Literature Review
◦ Commercial Solvers
◦ Research Paper Review
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What is LP?
LP deals with optimizing objective function
satisfying various constraints on decision
variables
Optimization may be in terms of minimization or
maximization
Typical LP formulation:
min or max f=cTx s.t. Ax≤ b; x≥0
c: coefficient matrix (objective function)
A, b: parameters representing constraint functions
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Various Solvers use LP
Optimization Toolbox in MATLAB
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Requires linear objective function
 Uses Simplex Algorithm which moves along
the edges of the polyhedron defined by the
constraints, from one vertex to another, while
decreasing the value of objective function
 What are the limitations?
Lack of ability to incorporate non linear
functions, but this can be overcome by the
technique of piecewise linear approximation
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Linear programming function: LINPROG
Syntax:
◦ x= linprog(c, A, b, Aeq, Beq, LB, UB)
‘x’ solves min cTx such that Ax ≤ b while
additionally satisfying Aeq*x=Beq.
◦ It also defines upper (UB) and lower (LB) bound
for the decision variables
◦
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What is ED?
Method of determining the most efficient,
low-cost and reliable operation of a power
system supplying the load on the system with
the help of generation resources available
What is the Objective?
Minimizing the production cost of thermal
generation satisfying the constraints on
generation levels
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Objective function:
Min C(P)= ∑Ci(Pi)
where, i= 1,2,… Ng (No. of generators)
Ci(Pi)=αi+βiPi+γiPi2 $/hr
PiMin≤Pi≤PiMax
Equality Constraints:
Equality constraints consists of Loadbalance equation,
Pload- ∑ Pi=0
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Non linear nature of cost function can be
addressed in LP
Piecewise linear approximation
e.g. Consider a non linear cost function given
in FIG.1
It can be approximated linearly as shown in
FIG.2, where the slopes s1,s2 of the
respective segments represent the cost
function
More no. of slope segments increase the
accuracy of approximation
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FIG.1
FIG.2
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Formulation of LP in MATLAB:
◦ Objective function:
F= ∑ ∑ sijPij
Where, i=1,2,…Ns (no. of slope segments)
j=1,2,..Ng (no. of generators)
◦ Equality and Inequality constraints:
 Equality Constraints: Beq vector is given by,
Beq= Pload- ∑ Pimin
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Equality and Inequality Constraints:
◦ Inequality Constraints:
LB, UB parameters in MATLAB are used. LB and UB are
the lower and upper bounds of the decision variables
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Final Formulation:
x=linprog(F, A, b, Aeq, Beq, LB, UB)
F=vector containing slope segments for ‘Ng’
generators
A=[], b=[] (since inequality constraints are not used)
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In practical systems, we always have losses
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Generators have to supply the load plus the
losses incurred in the system.
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LP in MATLAB does not have in built ready to
use function to incorporate the losses
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However, loss term can be included in LP
formulation by virtue of including penalty
factor term
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In Lagrange multiplier method, we come
across penalty factor term, pf= 1-(dPLoss/dPi)
as, pf1 (dℒ/dP1)= pf2 (dℒ/dP2)
Include penalty factor term in the objective
function coefficient vector
F= ∑ ∑ pfjsijPij
Where, i=1,2,…Ns (no. of slope segments)
j=1,2,..Ng (no. of generators)
e.g. for Ns=2, Ng=2,
F= pf1(s11P11+s12P12)+pf2(s21P21+s22P22)
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Load- balance equation is given as,
Beq= Pload+Ploss- ∑ Pimin
e.g. for the case mentioned,
Beq= Pload+Ploss- (P1min+ P2min)
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Advantages:
◦ This approach can also be extended to large scale
problems, including hundreds of generation units
◦ Loss function can be introduced as a linear, non
linear function or even as an incremental function
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Deals with supplying load by running both
hydro and steam plant.
Aims at reducing the cost of steam plant.
Two type of Hydro scheduling
Long range hydro scheduling
Short range hydro scheduling
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Timescale ranges from 1week to several
years.
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Due to the timeframe, unknown variables are
treated statistically.
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Typical optimizing variables include load,
inflow etc.
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DP can be used for solving long range
scheduling.
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Timeframe involved is between 1 day and 1
week.
Solved on a hour to hour basis.
Load, inflow, initial conditions are known
beforehand.
Find optimal hourly schedule by optimizing
objective functions based on constraints.
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Utilize the amount of water in reservoir to
minimize operating cost of steam plants.
Usually load is larger than generating
capacity of hydro plant.
Steam plant can have various constraints such
as generation limits.
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Cost function of steam plant is non linear as
seen in ED problem formulation.
Ci(Pi)=αi+βiPi+γiPi2 $/hr.
Cost function is linearized over the steam
plant operating region.
Flow characteristics is assumed to be linear.
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Objective function:
F= ∑∑ si(j)Pi(j)
i=1,2,…,Ns, the number of slope segments
j=1,2,…,N, the number of load periods
Equality constraints:
There are 2 sets of equality constraints.
Load balance equation:
∑Ps,i(j) +Ph(j) =Pload(j)- Ps,min(j)
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Hydraulic continuity constraint:
Vj-1+rj-qj=Vj
Vj: reservoir volume at period j.
rj : net inflow to reservoir during period j
qj : water discharge during period j
Losses can be included in terms of spillage
discharge rate.
In our project, spillage discharge is not
considered.
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Inequality constraints:
These are mainly included as the lower
bounds(LB) and the upper bounds(UB)
Storage limits: VjMin≤Vj≤VjMax
Generation limits:
PhMin ≤ Ph ≤PhMax (Hydro plant)
PsMin ≤ Ps ≤PsMax (Steam plant)
Flow limits:
qj≥ 0; Ph=0 when qj=0
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Constant term in flow equation, prevents the
flow qj from reaching zero value.
This affects the performance of LP.
Difference between the answer found using
LP and DP.
Sometimes DP solution can be better
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Where as absence of constant term, provides
leverage to LP in terms of satisfying the
condition- Ph=0 when qj=0
Answer through LP in this case can be better
as compared to that found using DP
Salient feature of LP is that every point in the
range of can be accessed to find the best
possible answer
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MATLAB
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CPLEX
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Mathematica
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Analytica
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MOSEK
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IBM ILOG CPLEX Optimization Studio
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Originally developed by Robert E. Bixby
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Used by over 50% of the world's largest
companies, 1000’s of Universities, and
1000's of application providers
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Solves
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Integer programming problems
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Very large linear programming problems
◦ Primal or dual variants of the simplex method
◦ Barrier interior point method
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Convex quadratic programming problems
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Convex quadratically constrained problems
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Features
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CPLEX IDE:
◦ CPLEX Optimizer for mathematical programming
◦ CPLEX CP Optimizer for constraint programming
◦ Optimization Programming Language (OPL)
◦ Interactive Optimizer
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Concert – A modeling layer
◦ Interfaces to C++, C#, Java. And Python.
◦ Connectors to Microsoft Excel and MATLAB
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Accessible through modeling systems such as
AMPL, GAMS, etc.
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Advantages
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Industry standard
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Wealth of readily available info
◦ Whitepapers
◦ Tutorials
◦ Webcasts
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IBM Academic initiative
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IBM Academic initiative
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No-Charge Access to IBM ILOG Optimization
Products
Course modules
◦ Linear & integer programming
◦ Constraint programming
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Discussion forums
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Web-based support
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LINEAR PROGRAMMING FOR POWER-SYSTEM
NETWORK SECURITY APPLICATIONS
B . Stott
J.L.Marinho
CEPEL, Ilha do Fundao, Rio de Janeiro, Brazil
IEEE Transactions on Power Apparatus and
Systems, Vol. PAS-98, No. 3 May/June 1979
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LP method for security dispatch and
emergency control calculations on large
power systems.
Handles multi-segment generator cost curves
neatly and efficiently.
Deals with shortcomings of non-linear
programming methods such as low-speed,
unreliability and difficulties in recognizing
infeasibility.
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Requires initial power system operating state
along with overloads
Branch flow limits are known.
For high accuracy, LP iteration with ac load
flow solution can be implemented
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Relief of network overloads by active power
control
Includes Generation shifting, Load shedding,
etc.
All control actions can be summed up as bus
injection changes represented as
where ‘n’ is number of controllable bus generations
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Objective function:
where ‘C’ is vector containing bus-generation incremental cost
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Constraints: The power balance equation is,
where ‘ßi’ is the incremental loss factor
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Limit constraints: ∆Pjmin ≤ ∆Pj ≤ ∆Pjmax
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Various objective functions:
Uses a single variable for each controlled bus,
instead of representing each cost-curve
segment by a separate LP variable.
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Control Priorities
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Soft Limits
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Ineffective Control Actions
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Interchange Restrictions
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This method efficiently handles any convex
objective function, piecewise modeled to any
desired accuracy
Security-constrained economic
dispatch/control using LP.
Applications of the method maybe extended:
◦ network-constrained reactive-power control
◦ including transformers, etc.
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References
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[1] A. J. Wood and B. F. Wollenberg, Power Generation Operation and Control. John Wiley
& Sons, Inc., New York, NY, 2nd Edition, 1996.
[2] Molly E. Ison, Frederick Wurtz, Commercial Linear Programming Solvers and Their
Applications to Power System Optimization. Power and Energy Society General Meeting Conversion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE, Pittsburgh,
PA, pp. 1 - 7, July 2008.
[3] “General Algebraic Modeling System.”, Wikipedia. Available:
http://en.wikipedia.org/wiki/General_Algebraic_Modeling_System
[4] “Economic Dispatch: Concepts, Practices and Issues.”, Presentation to the Joint Board
for the Study of Economic Dispatch. FERC Sta
Palm Springs, California. November 13, 2005
[5] Stephen Boyd, Lieven Vandenberghe, Convex Optimization. Cambridge University Press, New York, pp. 146 - 148, 2009. Available:
http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
[6] “Optimization Toolbox.”, The MathWorks Inc. Available:
http://www.mathworks.com/access/helpdesk/help/toolbox/optim/
[7] B . Stott, J.L.Marinho, CEPEL, Ilha do Fundao Rio de Janeiro, Brazil, Linear
Programming for power-system network security applications. IEEE Transactions on
Power Apparatus and Systems, Vol. PAS-98, No. 3 May/June 1979.
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