This document provides an overview of economic dispatch and unit commitment in power systems. It discusses: 1. Economic dispatch is the process of determining generator outputs to meet demand at minimum cost, taking into account generator costs and constraints. It can be solved graphically or using the KKT conditions. 2. Unit commitment determines which generators will operate over different time periods to meet forecasted load at minimum cost, while considering generator operating constraints like minimum up/down times. It is solved using techniques like mixed integer programming and Lagrangian relaxation. 3. Mixed integer programming and Lagrangian relaxation are commonly used optimization methods for unit commitment. Mixed integer programming formulates it as an optimization problem with discrete and continuous variables.

1. TUGAS OPTIMASI DAN KEANDALAN SISITEM
RANGKUMAN ECONOMIC DISPATCH DAN UNIT
COMMITMENT
OLEH :
Doni Wahyudi Indra C. 13050874033
S1 TEKNIK ELEKTRO 2013
SISTEM TENAGA LISTRIK
JURUSAN TEKNIK ELEKTRO
FAKULTAS TEKNIK
UNIVERSITAS NEGERI SURABAYA
2016

2. ECONOMIC DISPATCH
1. Introduction
Economic Dispatch forms the important analysis functions dealing with
Operation in an EMS. Economic Dispatch (ED) is defined as the process of allocating
generation levels to the generating units in the mix, so that the system load is supplied
entirely and most economically. In static economic dispatch, the objective is to
calculate, for a single period of time, the output power of every generating unit so that
all the demands are satisfied at the minimum cost, while satisfying different technical
constraints of the network and the generators.
Economic Dispatch is the process of determination of the output power
generated by the unit or units to supply the specified load in a manner that will
minimize the total cost of fuel. Each generating unit has a unique production cost
defined by its fuel cost coefficients (a, b, c of a+bp+cp2). Economic dispatch is also
defined as the coordination of the production costs of all the participating units in
supplying the total load. The purpose of economic dispatch is to determine the
optimal power generation of the units participating in supplying the load. The sum of
the total power generation should equal to the load demand at the station. In a
simplified case, the transmission losses are neglected. This makes the task of solution
procedure easier. In actual practice, the transmission losses are to be considered. The
inclusion of transmission losses makes the task of economic dispatch more
complicated. A different solution procedure has to be employed to arrive at the
solution.
Economic Dispatch models the electric power system (with one or more
control areas) and most economic manner in real-time operation. The objective is to
minimize the total generation cost (including fuel cost, plus emission cost, plus

3. operation/maintenance cost, plus network loss cost) by meeting the following
operational constraints:
• System load demand
• Downward-and-upward regulating margin requirements of the system
• Lower and upper economic limits of each generating unit
• Maximum ramping rate of each generating unit
• Unit's restricted operating zones (up to three restricted zones per unit)
• Emission allowance of the system (SO2, CO2, NOx)
• Network security constraints (maximum mW power flows of transmission lines)
• Supporting multiple I/O curves (incremental Heat Rate) and emission cost
curves for different fuels.
The program operates in real-time, according to a schedule you determine Automatic
Generation Control (AGC), e.g. every two (2) minutes. It can also be scheduled more
frequently by AGC under various circumstances, such as changing the control mode of a
unit in AGC, or the system base load deviation exceeding a pre-specified threshold, or a
request to execute on demand by a operator.
2. Definition
Economic dispatch is the short-term determination of the optimal output of a
number of electricity generation facilities, to meet the system load, at the lowest possible
cost, subject to transmission and operational constraints. The Economic Dispatch Problem
is solved by specialized computer software which should honour the operational and
system constraints of the available resources and corresponding transmission capabilities.
In the US Energy Policy Act of 2005 the term is defined as "the operation of generation
facilities to produce energy at the lowest cost to reliably serve consumers, recognising any
operational limits of generation and transmission facilities".[1]

4. The main idea is that in order to serve load at minimum total cost, the set of
generators with the lowest marginal costs must be used first, with the marginal cost of the
final generator needed to meet load setting the system marginal cost. This is the cost of
delivering one additional MWh of energy onto the system. The historic methodology for
economic dispatch was developed to manage fossil fuel burning power plants, relying on
calculations involving the input/output characteristics of power stations.
3. Generation Model
The electric power system representation for Economic Dispatch consists of
models for the generating units and can also include models for the transmission
system. The generation model represents the cost of producing electricity as a
function of power generated and the generation capability of each unit. We can
specify it as:
a. Unit cost function:
Fi = Fi (Pi )
where Fi = production cost
Fi(.) = energy to cost conversion curve
Pi = production power
b. Unit capacity limits:
Pi ≤ Pimax
Pi ≥ Pimin
c. System constraints (demand – supply balance)
N

5. ∑Pi =D
i=1
4. Formulation of the LaGrangian
We are now in a position to formulate our optimization problem. Stated in
words, we desire to minimize the total cost of generation subject to the inequality
constraints on individual units (2) and the power balance constraint (3). Stated
analytically, we have:
N
F (P )
Minim
ize:
Subject to: (1)~(3)
The equality constraint h(x) = b for the general case was allowed to contain multiple
constraints. Here, in the EDC problem, we see that there is only one equality
constraint, i.e., h and b are both scalars. This implies that lambda is a scalar also. The
Lagrangian function, then, is:
N N
=∑Fi (Pi ) +λ(D−∑Pi )
i=1 i=1
5. KKT Conditions
Application of the KKT conditions to the LaGrangian function of (4) results in:
∂Fi (Pi )/∂Pi =λ i=1,2…N
N
D−∑Pi = 0

6. i=1
The KKT conditions provide us with a set of equations that can be solved. The
unknowns in these equations include the generation levels P1, P2, …, Pn and the
LaGrange multipliers
a total of (n+1) unknowns. We note that (5) provides n equations, (6) provides one
equation. Thus, we have a total of (n+1) equations.
6. KKT Conditions for a 2-unit system
To illustrate more concretely, let’s consider a simple system having only two
generating units. The LaGrangian function, from (4), is:
F1(P1) +F2(P2) +λ(D−P1 −P2)
The KKT conditions, from (5) and (6) become:
∂F1(P1)/∂P1 =λ
∂F2 (P2 )/∂P2 =λ
D=P
1
+P
2
7. Graphical Solution
Recall the first KKT condition when applied to the generation system, if we
assume that all binding inequality constraints have been converted to equality, then
the equations becomes
∂F1(P1)/∂P1 =∂F2(P2)/∂P2 =λ
This equation implies that for all regulating generators (i.e. units not at their limits.)
each generators incremental costs are the same and are equal to lambda.

7. This very important principle provides the basis on which to apply the
graphical solution method. The graphical solution is illustrated in Figure 1 (note that
"ICC" means incrementalcost-curve). The unit's data are simply plotted adjacent to
each other. Then, a value for lambda is chosen (judiciously) and the generations are
added. If the total generation is equal to the total demand "PT" then the optimal
solution has been found. Otherwise, a new value for lambda is chosen and the process
repeated. The limitations of each unit are included as vertical lines since the rulers
must not include generation beyond unit capabilities. The unit is simply fixed at the
value crossed.
Figure Graphical Solution of EDC

8. UNIT COMMITMENT
1. Unit Commitment
Unit commitment (UC) aims to schedule the most cost-effective combination of generating
units to meet forecasted load and reserve requirements, while adhering to generator and
transmission constraints. Generally, UC is completed for a time horizon of one day to one week
and determines which generators will be operating during which hours. This commitment
schedule takes into account the inter-temporal parameters of each generator (minimum run
time, minimum down time, notification time, etc.) but does not specify production levels,
which are determined five minutes before delivery. The determination of these levels is known
as economic dispatch and it is “the least-cost usage of the committed assets during a single
period to meet the demand”.
Problem Definition
The objective is to minimize the total system cost, F, of generating power from N units
over a specific time horizon, T. The total cost from each generator at a given period of time is
the fuel cost, Ci, plus the any start-up costs, Si, that may be incurred during the period. (All
equations are based on )
The fuel costs, Ci, are dependent on the level of power generation Pi(t). The start-up costs, Si,
are dependent on the state of the unit, xi, which indicate the number of hours the unit has been
on (positive) or off (negative), and the discrete decision variable, ui, which denotes if power
generation of the unit at time t is up (1) or down (-1) from the unit at time t+1.
System-wide Constraints
The objective function is to be minimized subject to a series of system and generator
constraints. First, the system demand, Pd(t), must be met.

9. There also must be sufficient spinning reserve, ri, to ensure reliability. The required
system spinning reserve is designated as Pr. The actual spinning reserve for a unit i is zero if
the unit is off or
.
Generator Constraints
Thus far, the generators have been referred to as a general collection of power
producing units, but in actuality the system is comprised of several types of power generators
(CCGT, coal-fired, nuclear, etc.) that each operate differently. Even within these types, each
individual unit will have its own physical characteristics. Several of the parameters to be
considered as generator constraints are minimum run time, minimum down time, ramp rates,
and notification time. The notification time is the hours or minutes needed to start the unit
from a hot, cold or intermediate state [1].
The state of the unit changes if a start-up or shut-down occurs. This is described in this model
by the variables mentioned above, xi and ui.
if if
If no change of state occurs, the hours accumulate. If a shut-down or start-up does take place,
the count starts over again.
Generators have a certain range of operation between a minimum and maximum load. Below
the minimum load level the unit cannot stably produce electricity.
if if

10. Certain types of generators have the ability to ramp up or down to higher or lower outputs.
The ramp rate, which is generally expressed in MW/hr, is here used as the maximum change
in load levels between two consecutive hours, .
if and
Because of physical characteristics generators cannot be turned off and on immediately. They
have a minimum run time and a minimum down time .
if
if
The above equations outline the very basic UC problem. Actual systems must
consider additional factors, such as emission, fuel and transmission constraints. Security-
constrained unit commitment (SCUC) determines an optimal schedule, and also ensures that
delivery of that schedule is physically feasible based on the constraints of the transmission
network. Methods of solving the UC problem will be discussed in section 3.
2. Optimization
Unit commitment is a large-scale problem, often dealing with hundreds of
generating units in a region, making it difficult to find the optimal solution in an acceptable
amount of time. Several optimization methods are currently used and many more are being
researched. A recent literature review identified nine of these methodologies: priority list
method, dynamic programming, Lagrangian relaxation, genetic algorithms, simulated
annealing, particle swarm optimization, tabu-search method, fuzzy logic algorithm, and
evolutionary programming [9]. This list of approaches is not exhaustive and does not highlight
that much of the present research includes a combination of methods. In this section, the
primary focus will be on the two approaches most used in the US markets: mixed integer
programming (MIP) and Lagrangian relaxation (LR).
In 1999, unit commitment was solved primarily by linear programming (LP) and
Lagrangian relaxation, but due to approximations these yielded suboptimal solutions. At the
time, the use of mixed integer programming had been dismissed because of the unacceptable

11. solving time. Improvements to the method eventually led to PJM testing MIP against its
present software in its day-ahead market. The annual bid cost savings were estimated at $60
million. PJM adopted the method into its day-ahead market in 2004 and its real-time market
in 2006 [4]. CAISO implemented the MIP technique during its market redesign in 2009 [5].
Many markets have since implemented MIP in their markets.
Mixed Integer Programming
To better understand the capabilities of mixed integer programming it is important
to recognize that MIP is a special class of linear programming. Momoh [10] defines a linear
programming problem as “the problem of allocating a number m of resources among 1,2,…,n
activities in such a way as to maximize the worth from all the activities.” In such a problem,
all constraints and all relationships between decision variables and activities are linear. There
are four implicit assumptions to the linear programming approach: proportionality, additivity,
divisibility and certainty [10]. The assumption of divisibility states that decision variables do
not have to be discrete values. Integer programming (IP) and mixed integer programming are
the special cases in which all or some of the decision variables are discrete values. If the
integer values can only be zero or one, as is the case in on/off scheduling, they are called
binary decision variables.
The three main approaches to solving LP problems are the graphical approach, variations of
the simplex method and the tableau approach.
Formulation of MIP and Methods of solving
The basic MIP formulation in the case of UC, aims to minimize an objective function that
consists of continuous, y, and discrete, z, decision variables.
min
subject to: where and
There are several methods of solving MIP problems, but the two considered here are the most
used in power systems.
The cutting-plane technique views the MIP problem as a LP relaxation and
iteratively narrows down its feasible region until the constraints are satisfied and an optimal

12. solution is found. The first step is to consider the MIP as a non-integer LP problem, which
has a feasible region that includes all the feasible solutions of the MIP problem [11]. This LP
relaxation can be solved by the simplex method. The resulting optimal solution of the
relaxation can only be the optimal solution of the MIP if the constraints are met. If they are
not met, an additional linear inequality constraint (a cut) is considered, refining the feasible
region of the LP relaxation [11]. The process is then repeated until a solution is reached.
As its name implies, the branch-and-bound technique is made up of two major
steps. Branching is the task of breaking down the master problem to smaller sub-problems.
This is done by considering the vector z, in the MIP problem presented above, as having
binary variables, 0 or 1. The resulting subproblems can be formulated in a tree structure and
solved using straight-forward LP methods to find the optimal solution [11]. Solving all sub-
problems would require a considerable amount of time so a second step known as bounding
is applied. The bounds of a sub-problem are defined by its optimal solution and objective
(minimization or maximization). If a feasible solution exists for the first sub-problem
encountered it is considered the best solution and is referred to as the first incumbent. The
bounds of subsequent solutions are compared with the incumbent. The incumbent is replaced
if a better bound is found, otherwise the branch is fathomed [10].
The efficiency of the cutting-plane technique is dependent on cuts made, while the
efficiency of the branch-and-bound algorithm depends on how the bounds are found and how
the tree is searched [11]. The current leading approach to solving MIP problems is a
combination of these two techniques known as the branch-and-cut [6].
Lagrangian Relaxation
Momoh [10] describes Lagrangian relaxation as a way of incorporating the complicated
constraints of an optimization problem into the objective function by using a penalty term.
Formulation of the Lagrangian and Dual Functions
Based on the problem definition presented in section 2.1, the corresponding Lagrangian,
L, is given as:

13. In the above equation, only the system-wide constraints, not the generator constraints, are
relaxed by using Lagrangian multipliers, λ(t) and μ(t), which correspond to the demand and
reserve requirements, respectively. By using the duality theorem, the above can be broken
down into a “two-level maximumminimum optimization problem” [2].
The low level problem focuses on the minimization of the Lagrangian. If the Lagrangian
multipliers are fixed, the minimization of L can be decomposed into a set of N simpler
minimization problems; one problem for each generating unit, i=1,2,...N [12]. The low-level
sub-problem can be written for unit i as:
min L, with
The minimization sub-problems are subject to their corresponding generator constraints and
can be solved using dynamic programming [2], [12].
The high level problem focuses on the maximization of the dual function, θ. The dual
function includes the optimal Lagrangian, Li*(λ,μ).
max , with
Because discrete decision variables are used in the low level optimization, the dual function
may not be differentiable at all points. For this reason, a subgradient algorithm is used for
updating the multipliers [2].
The sub-problems are solved first and the multipliers are updated in the high level
optimization. The values of the Lagrangian form a lower bound for feasible solutions of the

14. original problem. The duality gap is the difference between the optimal solution to the
original (primal) problem and the optimal solution of the dual problem [13].
3. Methods and Constraints Optimez Research
Researchers continue to refine the UC problem by investigating new combinations
of algorithms. The primary focus is still to achieve better and faster optimization while
incorporating a large and often complicated set of constraints. For example, Duo et al. [12]
examined the use of evolutionary programming techniques to improve on the initial LR
solution. Other researchers focus on including more realistic constraints, such as considering
the restricted operating zones of a generator’s inputoutput curve [17]. For a complete
understanding of methodological developments see the recent literature review [9].
Security-constrained unit commitment is an extension of the basic UC problem that
includes transmission constraints, such as those pertaining to branch-flow4 and bus
voltages.5 Fu et al. [20] propose using Benders decomposition to yield a UC problem with
a transmission security check (subproblem). In the iterative process, security violations in
the network are identified, additional constraints are added to the UC problem, and then the
UC problem is solved again. This continues until the constraints are met.
4. Uncertainty
As the generation mix shifts to include more renewable energy and as demand
becomes more price sensitive6, there is increased uncertainty in real-time resource
commitment and dispatch. For example, the UC solution reached the day before would not
have taken into account significant changes in wind power generation on the day. Bertsimas
et al. [21] described the current research of uncertainty challenges as being divided into two
approaches. The first approach, which is widely used in the power industry, is called the
reserve adjustment method. It involves obtaining a commitment schedule based on the

15. forecasted demand but requiring a much larger reserve capacity – economically inefficient
solution. The second approach uses stochastic programming [21]. This technique is best
illustrated by an example from wind power generation. The wind forecast and corresponding
error distribution are used to produce a single “predicted scenario and large numbers of error
scenarios” for wind power generation [22]. A large number of scenarios is necessary to have
confidence in the approach, but this significantly increased the complexity of an already
large-scale optimization problem.

16. BIBLIOGRAPHY
www.srmuniv.ac.in/sites/.../Chapter4EconomicDispatch.pdf
shodhganga.inflibnet.ac.in/bitstream/10603/38951/7/07_chapter2.pdf
www.ijareeie.com/upload/2015/june/103_Analysis.pdf
https://en.wikipedia.org/wiki/Economic_dispatch
https://en.wikipedia.org/wiki/Unit_commitment_problem_in_electrical_power_productio
n