ENM323 Energy Systems
Planning
Week 12 – Economic
dispatch & energy supply
chain & unit commitment
problem
Assist. Prof. Zeynep İdil ERZURUM ÇİÇEK
Outline
• Economic dispatch problem
• Energy supply chain
• Unit commitment problem
Economic load dispatch problem
• a key issue in power systems and its goal is to achieve minimum
economic costs by allocating the output of generator units when
satisfying the load demands and the operating constraints.
• minimizes the generation cost with meeting demand
Source:
Fu, C.; Zhang, S.; Chao, K.-H. Energy Management of a Power System
for Economic Load Dispatch Using the Artificial Intelligent
Algorithm.
Electronics
2020,
9,
108.
https://doi.org/10.3390/electronics9010108
Akkaş, Özge Pınar, Yağmur Arıkan, and Ertuğrul Çam. "Load dispatch
for a power system in terms of economy and environment by using
VIKOR method." (2017)
Economic load dispatch problem
• The problem of allocating the customers’ load demands among the
available thermal power generating units in an economic, secure and
reliable way has received considerable attention since 1920 or even
earlier.
• The problem has been formulated as a minimization problem of the
fuel cost under load demand constraint and various other constraints
at a certain time of interest. It has been frequently known as the
static economic dispatch (SED) problem.
Source: Xia, X., and A. M. Elaiw. "Optimal dynamic economic dispatch of generation: A
review." Electric power systems research 80.8 (2010): 975-986.
Economic load dispatch problem – operating
costs
• Factors influencing the minimum cost of power generation
• Operating efficiency of prime mover and generator
• Fuel costs
• Transmission losses
• The most efficient generator in the system dose not guarantee
minimum costs
• May be located in an area with high fuel costs
• May be located far from the load centers and transmission losses are high
• The problem is to determine generation at different plants to
minimize the total operating costs
Economic load dispatch problem – operating
costs
• Generator heat rate curves lead to the fuel cost curves
Economic load dispatch problem – operating
costs
• The fuel cost is commonly express as a quadratic function
• The derivative is known as the incremental fuel cost
Pi: The power output of plant I
𝛼! , 𝛽! , 𝛾! : Fuel cost coefficients for plant i
Economic load dispatch problem – definition
The main cost incurred in power generation is the operating fuel cost,
and the economic load dispatch problem is
•
to minimize the total fuel cost of all the generating units in
operation,
•
satisfying a set of real-time operating constraints.
Classical economic dispatch problem
The fuel cost characteristics of each generating unit i, is represented by
a quadratic equation. The objective of the problem is minimizing cost
function over all plants.
Decision variable
Pi: The power output of plant i
Classical economic dispatch problem
1. equality constraint/real power balance constraint
The total demand is equal to the sum of the generators’ output.
Economic dispatch problem
+ Generation limits
+ Ramp rate limit
+ Multiple fuel cost functions
+ Prohibited operating zone constraints
+ Objective functions with valve-point effects
+…
Economic dispatch problem + POWER
GENERATION LIMIT
2. inequality constraint/real power generation limit
The power output of any generator should not exceed its rating nor be
below the value for stable boiler operation.
Classical economic dispatch problem + RAMP
RATE LIMIT
3. ramp rate limit (the speed at which a generator can increase (ramp
up) or decrease (ramp down) generation)
Classical economic dispatch problem +
LOSSES
• For large interconnected system where power is transmitted over
long distances with low load density areas
• Transmission line losses are a major factor
• Losses affect the optimum dispatch of generation
• One common practice for including the effect of transmission losses is
to express the total transmission loss as a quadratic function of the
generator power outputs:
• Simplest form
Classical economic dispatch problem +
LOSSES
• Bij are called the loss coefficients
• They are assumed to be constant
• Reasonably accuracy is expected when actual operating conditions are close
to the base conditions used to compute the coefficients.
• The economic dispatch problem is to minimize the overall generation
cost, C, which is the function of plant output.
• Constraints:
• The generation equals the total load demand + transmission losses
• Each plant output is within the upper and lower generation limits – inequality
constraints.
Classical economic dispatch problem +
LOSSES
• The mathematical model then become as:
Solution
• Mathematical programming based approaches
• Lagrange relaxation
• Metaheuristics
• Artificial Bee Colony Algorithm
• Differential evolution
• A modified crow search algorithm
• Teaching learning based optimization
• Particle swarm optimization
• Cuckoo search optimization
• Genetic algorithm
•…
Solution for a typical dispatch problem using
the Lagrange multipliers
The simplest problem no losses, no generator limits
The objective function is a quadratic cost function.
(!"#
𝐿 = 𝐶!"!#$ + 𝜆 𝑃%&'#() − ( 𝑃*
*+,
𝜕𝐿 𝜕𝐶!"!#$
𝜕𝐶!"!#$
=
+𝜆 0−1 =0 →
=𝜆
𝜕𝑃*
𝜕𝑃*
𝜕𝑃*
Solution for a typical dispatch problem using
the Lagrange multipliers
(!"#
𝜕𝐶!"!#$ 𝑑𝐶*
𝐶!"!#$ = ( 𝐶* →
=
= 𝜆 ∀𝑖 = 1, … , 𝑛-&(
𝜕𝑃*
𝑑𝑃*
*+,
𝑑𝐶*
𝜆=
= 𝛽* + 2𝛾* 𝑃*
𝑑𝑃*
Solution for a typical dispatch problem using
the Lagrange multipliers
(!"#
(!"#
*+,
*+,
𝑑𝐿
= 𝑃%&'#() − ( 𝑃* = 0 → ( 𝑃* = 𝑃%&'#()
𝑑𝜆
Reaarranging and combining the equations to solve for 𝜆
𝜆 − 𝛽*
𝑃* =
2𝛾*
(!"# . /0$
∑*+,
=
𝑃
%&'#()
12$
𝜆=
#!"# +$
,-$
#!"# *
∑$)*
,-$
3%"&'#( 4∑$)*
Example
• Neglecting system losses and generator limits, find the optimal dispatch
and total cost in $/hr for the three generators and the given load demand.
$
𝑀𝑊ℎ𝑟
$
"
𝐶" = 400 + 5.5𝑃" + 0.006𝑃"
𝑀𝑊ℎ𝑟
$
"
𝐶# = 200 + 5.8𝑃# + 0.009𝑃#
𝑀𝑊ℎ𝑟
𝐶! = 500 + 5.3𝑃! + 0.004𝑃!"
𝑃$%&'() = 800𝑀𝑊
Example
𝜆=
%*"% +'
3!"#$%& 4∑'()
,-'
%*"% )
∑'()
,-'
5.3
5.5
5.8
800 +
+
+
$
0.008
0.012
0.018
=
= 8.5
1
1
1
𝑀𝑊ℎ𝑟
+
+
0.008 0.012 0.018
Example
. /0$
𝑃* = 12
$
→
8.5−5.3
𝑃, = 1(7.779) = 400 𝑀𝑊
8.5−5.5
𝑃1 =
= 250 𝑀𝑊
2(0.006)
8.5−5.8
𝑃; =
= 150 𝑀𝑊
2(0.009)
𝑃%&'#() = 800 𝑀𝑊 = 400 + 250 + 150
Example
Energy supply chain
• The energy supply chain is defined as a complicated network of
production, supply, transport and storage interconnected through
physical and financial infrastructure, information sharing and
transmission.
Source: Zhang, L.; Fu, S.; Tian, J.; Peng, J. A Review of Energy Industry Chain and
Energy Supply Chain. Energies 2022, 15, 9246. https://doi.org/10.3390/en15239246
• The interconnected network of businesses that supports utilities and
conventional and renewable energy producers and eventually
provides energy products and services to end users is known as the
energy supply chain.
• The suppliers operate the power plants.
A basic energy
supply chain
• Then, the electricity is transported through transmission lines at a very high voltage (in
order to decrease the losses) by the Transmission System Operator (TSO), and then through
distribution lines (at high, medium or low voltage) to the final users (homes, offices and
factories) by the Distribution System Operator (DSO).
• Between power plants, transmission, distribution, and final users, we have substations that
are responsible for converting the voltage and connecting the different layers, acting
therefore as infrastructures that provide safety and security to the operation of the grid.
• Finally, the electricity is sold to the customers by retailers.
Energy supply chain
• Optimization methods
• Simulation
• Hybrid (optimization + simulation)
Example
Powerco has three electric power plants that supply the needs of four
cities. Each power plant can supply the following numbers of kWh of
electricity: plant 1, 35 million; plant 2, 50 million; and plant 3, 40
million. The peak power demands in these cities as follows (in kWh):
city 1, 45 million; city 2, 20 million; city 3, 30 million; city 4, 30 million.
The costs of sending 1 million kWh of electricity from plant to city is
given in the table below. To minimize the cost of meeting each city’s
peak power demand, formulate a mathematical model.
Source: Winston, Wayne L. Operations research:
applications and algorithm. Thomson Learning, Inc., 2004.
Example
• Powerco problem can be formulated as a transportation problem (LP
model)
• Decision variable
𝑥*+ : number of (million) kwh produced at plant i and sent to city j.
• Objective function
Minimizing the cost of meeting each city’s peak power demand
• Constraints
• Demand
• Supply
• Total supply & total demand both equal 125: “balanced transportation
problem”
35 + 50 + 40 = 45 + 20 + 30 + 30
Example
min 𝑧 = 8𝑥!! + 6𝑥!" + 10𝑥!# + 9𝑥!$
+9𝑥"! + 12𝑥"" + 13𝑥"# + 7𝑥"$
+14𝑥#! + 9𝑥#" + 16𝑥## + 5𝑥#$
Example
s.t.
𝑥!! + 𝑥!" + 𝑥!# + 𝑥!, ≤ 35
𝑥"! + 𝑥"" + 𝑥"# + 𝑥", ≤ 50
𝑥#! + 𝑥#" + 𝑥## + 𝑥#, ≤ 40
𝑥!! + 𝑥"! + 𝑥#! ≥ 45
𝑥!" + 𝑥"" + 𝑥#" ≥ 20
𝑥!# + 𝑥"# + 𝑥## ≥ 30
𝑥!, + 𝑥", + 𝑥#, ≥ 30
𝑥*+ ≥ 0 ∀i, j
Supply constraints
Demand constraints
Unit commitment problem
• The total load on the system will generally be higher during the daytime and early
evening when industrial loads are high, lights are on, and so forth and lower
during the late evening and early morning when most of the population is asleep.
• The use of electric power has a weekly cycle, the load being lower over weekend
days than weekdays.
Why is this a problem in the operation of an electric power system?
Why not just simply commit enough units to cover the maximum system load
and leave them running?
Unit commitment problem
• to “commit” a generating unit is to “turn it on,” that is, to bring the unit up to
speed, synchronize it to the system, and connect it so it can deliver power to the
network.
• The problem with “commit enough units and leave them on-line” is one of
economics.
• it is quite expensive to run too many generating units. A great deal of money can
be saved by turning units off (decommitting them) when they are not needed.
Unit commitment vs Economic dispatch
• The economic dispatch problem assumes that there are Ngen units already
connected to the system.
• The purpose of the economic dispatch problem is to find the optimum operating
policy for these Ngen units.
• The unit commitment problem, on the other hand, is more complex. We may
assume that we have Ngen units available to us and that we have a forecast of the
demand to be served.
Given that there are a number of subsets of the complete set of Ngen generating
units that would satisfy the expected demand, which of these subsets should be
used in order to provide the minimum operating cost?
Unit commitment vs Economic dispatch
• This unit commitment problem may be extended over some period of time,
such as the 24 h of a day or the 168 h of a week.
• The unit commitment problem is a much more difficult problem to solve.
• The solution procedures involve the economic dispatch problem as a
subproblem.
• That is, for each of the subsets of the total number of units that are to be
tested, for any given set of them connected to the load, the particular
subset should be operated in optimum economic fashion.
• This will permit finding the minimum operating cost for that subset, but it
does not establish which of the subsets is in fact the one that will give
minimum cost over a period of time.
A simple unit
commitment problem
• Suppose one had the three units given here:
• If we are to supply a load of 550 MW, what
unit or combination of units should be used
to supply this load most economically?
• To solve this problem, simply try all
combinations of the three units.
• Some combinations will be infeasible if the
sum of all maximum MW for the units
committed is less than the load or if the sum
of all minimum MW for the units committed
is greater than the load.
A simple unit commitment problem
• For each feasible combination, the economic dispatch problem should be solved.
• the least expensive way to supply the generation is not with all three units running or even any combination
involving two units. Rather, the optimum commitment is to only run unit 1, the most economic unit.
𝐹" = 510 + 7.2 550×1.1 + 0.00142 550×1.1 #
𝐹" + 𝐹# = 510 + 7.2 295×1.1 + 0.00142 295×1.1 # + 310 + 7.85 255×1 + 0.00194× 255×1 #
A simple unit commitment problem
• Suppose the load follows a simple “peak-valley” pattern.
• If the operation of the system is to be optimized, units must be shut down as the load
goes down and then recommitted as it goes back up.
• We would like to know which units to drop and when.This problem is far from trivial
when real generating units are considered.
A simple unit commitment problem cont’d
• Suppose we wish to know which units to drop as a
function of system load.
• The load varying from a peak of 1200 MW to a valley of
500 MW.
• To obtain a “shutdown rule,” simply use a brute-force
technique wherein all combinations of units will be
tried for each load value taken in steps of 50 MW from
1200 to 500. The results of applying this brute-force
technique are given in table.
• The shutdown rule is quite simple: When load is above
1000 MW, run all three units; between 1000 and 600
MW, run units 1 and 2; below 600 MW, run only unit
1.
A simple unit commitment problem cont’d
• One simple constraint: Enough units will be committed to supply the load.
Constraints
• Spinning reserve
• Thermal unit
• Minimum uptime
• Minimum downtime
• Crew constraints
• Other constraints
• Must run
• Fuel
• Hydro
Spinning reserve
• Spinning reserve is the term used to describe the total amount of
generation available from all units synchronized (i.e., spinning) on the
system, minus the present load and losses being supplied.
• Spinning reserve must be carried so that the loss of one or more units
does not cause too far a drop in system frequency. Quite simply, if
one unit is lost, there must be ample reserve on the other units to
make up for the loss in a specified time period.
Spinning reserve
• Spinning reserve must be allocated to obey certain rules, usually set by regional reliability councils that (specify how the) reserve is to be allocated to various units.
• Typical rules specify that reserve must be a given percentage of forecasted peak
demand or that reserve must be capable of making up the loss of the most heavily
loaded unit in a given period of time.
• Others calculate reserve requirements as a function of the probability of not having
sufficient generation to meet the load.
• Not only must the reserve be sufficient to make up for a generating unit failure, but the
reserves must be allocated among fast-responding units and slow-responding units. This
allows the automatic generation control system to restore frequency and interchange
quickly in the event of a generating unit outage.
Spinning reserve
• Beyond spinning reserve, the unit commitment problem may involve
various classes of “scheduled reserves” or “off-line” reserves. These
include quick-start diesel or gas-turbine units as well as most hydrounits and pumped-storage hydro-units that can be brought online,
synchronized, and brought up to full capacity quickly. As such, these
units can be “counted” in the overall reserve assessment, as long as
their time to come up to full capacity is taken into account.
• Reserves, finally, must be spread around the power system to avoid
transmission system limitations (often called “bottling” of reserves)
and to allow various parts of the system to run as “islands,” should
they become electrically disconnected.
Example
Suppose a power system consisted of
two isolated regions: a western region
and an eastern region. Five units, as
shown in Figure, have been committed
to supply 3090 MW. The two regions are
separated by transmission tie lines that
can together transfer a maximum of
550MW in either direction. This is also
shown in Figure. What can we say about
the allocation of spinning reserve in this
system?
Example
• With the exception of unit 4, the loss of any
unit on this system can be covered by the
spinning reserve on the remaining units. Unit
4 presents a problem, however.
• If unit 4 were to be lost and unit 5 were to be
run to its maximum of 600 MW, the eastern
region would still need 590 MW to cover the
load in that region. The 590 MW would have
to be transmitted over the tie lines from the
western region, which can easily supply 590
MW from its reserves. However, the tie
capacity of only 550 MW limits the transfer.
Therefore, the loss of unit 4 cannot be covered
even though the entire system has ample
reserves. The only solution to this problem is
to commit more units to operate in the
eastern region.
Thermal unit constraints
• Thermal units usually require a crew to operate them, especially when
turned on and turned off. A thermal unit can undergo only gradual
temperature changes, and this translates into a time period of some hours
required to bring the unit on-line. As a result of such restrictions in the
operation of a thermal plant, various constraints arise, such as:
• Minimum uptime: once the unit is running, it should not be turned off
immediately.
• Minimum downtime: once the unit is decommitted, there is a
minimum time before it can be recommitted.
• Crew constraints: if a plant consists of two or more units, they cannot
both be turned on at the same time since there are not enough crew
members to attend both units while starting up.
Thermal unit constraints
• Because the temperature and pressure of the thermal unit must be moved slowly, a
certain amount of energy must be expended to bring the unit online. This energy does
not result in any MW generation from the unit and is brought into the unit commitment
problem as a start-up cost.
Other constraints
• Must Run. Some units are given a must-run status during certain times of
the year for reason of voltage support on the transmission network or for
such purposes as supply of steam for uses outside the steam plant itself.
• Fuel Constraints. A system in which some units have limited fuel, or else
have constraints that require them to burn a specified amount of fuel in a
given time, presents a most challenging unit commitment problem.
• Hydro-Constraints. Unit commitment cannot be completely separated
from the scheduling of hydro-units.
Unit commitment solution methods
• The commitment problem can be very difficult.
• We must establish a loading pattern for M periods.
• We have Ngen units to commit and dispatch.
• The M load levels and operating limits on the Ngen units are such that any one
unit can supply the individual loads and that any combination of units can
also supply the loads.
Unit commitment solution methods
Assume we are going to establish the commitment by enumeration (brute force).
The total number of combinations we need to try each hour is
where C(Ngen, j) is the combination of Ngen items taken j at a time. That is,
For the total period of M intervals, the maximum number of possible combinations
is (2Ngen − 1)M, which can become a horrid number to think about.
Unit commitment solution methods
• For example, take a 24-h period (e.g., 24 one-hour
intervals) and consider systems with 5, 10, 20, and 40
units. The value of (2Ngen − 1)24 becomes the following.
• These very large numbers are the upper bounds for the
number of enumerations required. Fortunately, the
constraints on the units and the load–capacity
relationships of typical utility systems are such that we do
not approach these large numbers. Nevertheless, the real
practical barrier in the optimized unit commitment
problem is the high dimensionality of the possible solution
space.
Unit commitment solution methods
• The most talked-about techniques for the solution of the unit
commitment problem are:
• Priority-list schemes
• Dynamic programming (DP)
• Lagrange relaxation (LR)
• Mixed integer linear programming (MILP)
Priority list method
• The simplest unit commitment solution method consists of creating a
priority list of units.
• A simple shutdown rule or priority-list scheme could be obtained
after an exhaustive enumeration of all unit combinations at each load
level.
Priority list method
Most priority-list schemes are built around a simple shutdown algorithm that might operate as
follows:
• At each hour when load is dropping, determine whether dropping the next unit on the priority list
will leave sufficient generation to supply the load plus spinning reserve requirements. If not,
continue operating as is; if yes, go on to the next step.
• Determine the number of hours, H, before the unit will be needed again, that is, assuming that
the load is dropping and will then go back up some hours later.
• If H is less than the minimum shutdown time for the unit, keep the commitment as is and go to
the last step; if not, go to the next step.
• Calculate two costs. The first is the sum of the hourly production costs for the next H hours with
the unit up. Then recalculate the same sum for the unit down and add in the start-up cost for
either cooling the unit or banking it, whichever is less expensive. If there is sufficient savings from
shutting down the unit, it should be shut down; otherwise, keep it on.
• Repeat this entire procedure for the next unit on the priority list. If it is also dropped, go to the
next and so forth.
Example
• First, the full-load average production cost will be calculated:
• A strict priority order for these units, based on the average
production cost, would order them as follows:
8
Example
• And the commitment scheme would (ignoring min up-/downtime,
start-up costs, etc.) simply use only the following combinations.
where unit 2 was shut down at 600 MW leaving unit 1:
With the priority-list scheme, both units would be held on until
load reached 400 MW, then unit 1 would be dropped.
Mixed integer linear programming
Pit: The power output of unit i during period
t (Variable)
Ploadt: The demand during period t
(Parameter)
Pimin, Pimax: The minimum and maximum
generation limits of the real power output of
unit i (Parameter)
Mixed integer linear programming
Mixed integer linear programming
Other constraints can also be formulated such as:
• Unit minimum up and downtime constraints
• Transmission security constraints
• Spinning-reserve constraints
• Generator fuel limit constraints
• and system air quality constraints in the form of limits on emissions from fossil-fired
plants
References
• Winston, Wayne L. Operations research: applications and algorithm. Thomson
Learning, Inc., 2004.
• Yahia Baghzouz, EE 740 Economic Dispatch, 2013. avaliable at:
http://www.egr.unlv.edu/~eebag/Economic%20Generator%20Dispatch%20740.p
df
• Carlos Santos Silva & Fernanda Margarido, Energy Management Lecture Notes,
2020.