Content uploaded by Mohamed EL-Shimy
Author content
All content in this area was uploaded by Mohamed EL-Shimy on Nov 10, 2019
Content may be subject to copyright.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
137
Chapter 6: Static Security Constrained Optimal
Economic Dispatch (SSCOED) – An Overview
M. EL-Shimy
Professor of electric power systems. Electric power and machines
department, faculty of engineering, Ain Shams university, Cairo, Egypt.
Emails: shimymb@yahoo.com; Mohamed_bekhet@eng.asu.edu.eg; Phone:
+201005639589
Abstract
Avoiding power interruption is among the main requirements of sustainable
energy sources. The chapter presents a brief summary of the security
requirements, and operational states of power systems. In addition, a
detailed comparison between the optimization formulations of the classical
OED, OPF, and SSCOED is presented. Linear programming is used for
solving the linearized formulation of the SSCOED. The generator scheduling
of the IEEE 30 bus system using the SSCOED shows the capability of the
linear programming of solving the considered problems.
Keywords
Optimal economic dispatch (OED); power system security; security
programs; contingency analysis; static security constrained optimal
economic dispatch (SSCOED)
1. Introduction
Power system security may be defined as173,174 the continued ability of
the power system to keep all the system limits not violated with minimum
interruption to the supplied loads. The main target of the power system
security is to keep the system intact under normal and disturbed conditions.
Therefore, the successful security system should minimize the impact of
disturbances on the operation, economics, and power quality of power
systems. In addition, an acceptable system security level guarantees the
173 Wood AJ, Wollenberg BF. Power generation, operation, and control. Canada: John
Wiley & Sons; 2012.
174 Mohamed EL-Shimy: Dynamic Security of Interconnected Electric Power Systems -
Volume 1. 05/2015; Lap Lambert Academic Publishing / Omniscriptum Gmbh &
Company Kg; Germany., ISBN: 978-3-659-71372-9, DOI:10.13140/RG.2.2.19425.71520
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
138
immunity of power system to disturbances and makes the system to be
defensive. Therefore, secure operation of power systems requires the
integration of all practices designed for keeping acceptable system operation
when components fail.
Power system security covers both static and dynamic phenomena.
Therefore, the security analysis is usually categorized into static (or
adequacy) and dynamic security175,176. The static security considers the
impact of static or slow changes in the system limits while the dynamic
security considers the dynamical impacts of disturbances (or contingencies)
on the system. It is known that the static security studies are performed by
neglecting the dynamical impacts of changes or disturbances. Consequently,
it is assumed that the transients associated with the motion of the operating
conditions from one state to another are neglected. Therefore, the static
security may be considered as a sub-study of the dynamic security. It is
assumed the system transients associated with disturbances are stable, while
the post-disturance steady state operating conditions are evaluated in the
static security studies where the focus on the loading levels, and voltage
magnitudes.
Fig. 1: Operational constraints
The core definition of the dynamic security and stability is the same,
but the security is a wider term than stability. The stability is defined as177
“the ability of an electric power system, for a given initial operating
condition, to regain a state of operating equilibrium after being subjected to a
physical disturbance, with most system variables bounded so that practically
175 Eremia M, and Shahidehpour M, (eds). Handbook of Electrical Power System
Dynamics: Modeling, Stability, and Control, John Wiley & Sons, Inc., Hoboken, NJ,
USA. 2013.
176 Mohamed EL-Shimy: Dynamic Security of Interconnected Electric Power Systems -
Volume 2: Dynamics and stability of conventional and renewable energy systems.
11/2015; Lap Lambert Academic Publishing / Omniscriptum Gmbh & Company Kg;
Germany., ISBN: 978-3-659-80714-5, DOI:10.13140/RG.2.2.36832.07683
177 Kundur P, Paserba J, Ajjarapu V, Andersson G, Bose A, Canizares C, et al. Definition
and classification of power system stability IEEE/CIGRE joint task force on stability
terms and definitions. Power Systems, IEEE Transactions on. 2004;19(3):1387-401.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
139
the entire system remains intact”; however, “Security not only includes
stability, but also encompasses the integrity of a power system and
assessment of the equilibrium state from the point of view of overloads,
under- or overvoltages and underfrequency”178.
The system limits define the normal operation of power systems.
These limits or constraints can be classified into two categories; the equality
and inequality constraints as illustrated in Fig. 1, while the definitions of the
input (or independent) and state (or dependent) variables are shown in Fig. 2.
In addition, the system limits may be classified according to their origin into
intrinsic and operating range limits. The equality constraints basically
represent the load flow equations while the inequality constraints represent
the allowable range of acceptable operation of various components in the
system.
In fact, the intrinsic and operating range limits elaborate the inequality
constraints associated with a specific component. The intrinsic limits of an
equipment are determined basically from the design and characteristics of the
equipment. The operating range limits are generally less than the intrinsic
limits and they are limited by the fulfillment of the overall operational
requirements of the system.
For example, consider a simple hypothetical system where an off-grid
generating plant supplies a load center via a short transmission line with
negligible impedance. The generator is capable of producing a voltage
magnitude at its terminal in the range 85% - 115%, while the load requires a
voltage magnitude in the range 95% - 105%. In this case, the generator
voltage limits present the intrinsic limits of the generator and they are mostly
related to its design. Successful operation requires that the voltage magnitude
at the load bus should not be violated. Therefore, the operating range limits
of the generator bus-voltage magnitude become equal to the load
requirements (i.e. 95% - 105%). It is worth mentioning that the 95% - 105%
voltage limits present an intrinsic limit as viewed from the load perspective.
It is also important to know that the operating range limits should not violate
the intrinsic limits of any component within a system. Otherwise, the system
will be incapable of fulfilling the operational requirements.
Both intrinsic and operating range limits are not absolute constants.
The intrinsic capability limits usually decline with time due to degradation of
the equipment. For example, the annual output degradation rate179 of specific
technologies of solar-PV systems is about 0.7%/year. The degradation may
be attributed to the aging, operational stresses, and maintenance quality. The
operating range limits are also variable. For example, the ampacity (or
178 Machowski J, Bialek J, Bumby J. Power system dynamics: stability and control: John
Wiley & Sons; 2011.
179 EL-Shimy M. Analysis of Levelized Cost of Energy (LCOE) and grid parity for utility-
scale photovoltaic generation systems. 15th International Middle East Power Systems
Conference (MEPCON’12), Dec. 23-25, 2012, Alexandria, Egypt, pp. 1- 7
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
140
ampere capacity or current limits) of a cable are highly dependent on the
temperature of its surroundings. The ampacity limits are usually increased
during the winter and decreased during the summer. This is for avoiding
over-temperature of the cable insulation.
Recalling that in the normal operation of a power system, all the
inequality and equality constraints as well as the security constraints of the
system are satisfied. For example, the system security requires a minimum
available, reserve margin (see Fig. 3).
(a)
(b)
Fig. 2: Definition of variables. (a) Input variables; (b) state variables
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
141
Fig. 3: chronological load curve showing daily load fluctuations, energy mix
requirements, and the generation reserve180
Again, power system security may be also defined as the ability of the
system to withstand credible contingencies without violating the normal
operation limits. A system operating under normal conditions is also said to
operate in the normal state. The security strength of the system is usually
defined by the maximum number of time-independent, and simultaneous
disconnection of major system components (such as generators, transformers,
and line) without affecting the normal operation of the system. Defining N as
the number of available components (generation, transmission, and
distribution) to supply and can supply the system peak load (see Fig. 3). A
system with an N-k security criterion is a system in which k random
components may be simultaneously disconnected and the system will be able
to fulfil the normal state requirements in the post-contingencies. Due to
investment constraints, power systems are usually designed according to the
N-1 security criterion. The normal state is a secure state and a system
operating in the normal state is said to be intact.
180 Mohamed EL-Shimy: Dynamic Security of Interconnected Electric Power Systems -
Volume 1. 05/2015; Lap Lambert Academic Publishing / Omniscriptum Gmbh &
Company Kg; Germany., ISBN: 978-3-659-71372-9, DOI:10.13140/RG.2.2.19425.71520
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
142
Fig. 4: State transition diagram
Deviations from the normal state requirements cause the system
operation to move to insecure operating states. These deviations are mainly
caused by contingencies which are stochastic and unexpected events;
however, the rate of contingencies may be reduced for example by proper
maintenance of components. Four insecure operating states can be
realized181. These states are the alert, emergency, extreme (or collapse), and
restoration states. Fig. 4 illustrates the main operational characteristics of
these states, and the interrelations between them. This figure is usually called
the state transition diagram. Table 1 summarizes the characteristics of
various states, some causes of state transitions, and examples of the
corrective actions for each state. The nomenclature used in the table is
illustrated in Fig. 4.
It is worth mentioning here that any intact system is capable of
providing power balance; however, not all intact systems are secure. If the
power balance could not be achieved, then the system becomes not intact.
Consequently, the synchronization of generators upsets. Therefore, the
181 Fink LH, Carlsen K. Operating under stress and strain. IEEE Spectrum;(United States). 1978;15(3).
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
143
system frequency protection devices will split the system into parts or
islands; the situation is called islanding and it is within the extreme state.
The frequency and power balance conditions in each island are
different and abnormal. Therefore, system blackout or unintentional
brownout is usually detected. Delay is activating the possible corrective
actions while the system is in the emergency state may be the main cause of
the transition to the extreme state.
Table 1: Summary of operating states and state transitions
State
E
I
N-
1
N
Intact
Causes of
transition from
normal state
Corrective
actions
Normal
-
-
Alert
Constraints are
near their limits.
Examples,
reduction in the
reserve margin
or bus voltage
close to the
limits.
Preventive control.
Examples, startup of
non-spinning reserve
or switching on
compensators
respectively.
Emergency
Severe
disturbances.
Example, short-
circuit faults or
cascaded
outages.
Emergency control
actions (heroic
measures).
Example: fast fault
isolation or operation
of reclosers.
Extreme
Delayed or
unsuccessful
emergency
control actions.
Severe power
imbalance.
Heroic and remedial
actions such as load
shedding, generator
trip, or intentional
islanding for keeping
power balance.
Restoration
Attempt of
restoring the
system to the
normal state or
at least to the
alert state.
Manual or automated
reinsertion of
generators and loads.
The inequality
constraints should be
kept satisfied during
the entire restoration
process.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
144
Generally, delayed or unsuccessful corrective actions during the
operation in any state may lead to severe consequences. Therefore, any
security program (Fig. 5) includes a contingency analysis block.
The contingency analysis is an investigative simulation of
hypothesized contingency for evaluating their impact on the system security.
On the other hand, the corrective action analysis is the process of figuring the
possible actions that may be taken for overcoming the consequences of
security upsetting contingencies.
Fig. 5: Main functions of a security program
The corrective action analysis works in two distinct modes. The first
mode operates for solving the problems found by the contingency analysis.
Therefore, this mode is offline while the second mode operates in real time
operation for securing the system during its real-time operation. The
contingency analysis and the corrective action analysis require the simulation
of the system. Therefore, an accurate system model should be available. In
addition, the results obtained from these analyses are highly dependent on the
accuracy of the system model.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
145
Real-time models of a power system require centralized real-time data
collection available from local measuring and monitoring devices at each
system component. Therefore, telemetry is required for communication
within the system and for estimating its state.
In the conventional static security studies, the post-contingency state is
related to the pre-contingency (or the base case) through linear sensitivity
factors (LSF)182,183. These are the generation shift factors, and line sensitivity
factors shown in Fig. 6.
Fig. 6: Linear sensitivity factors for contingency analysis
where al,i represents the generation shift factor that relates the change in the
flow on line l (fl) to the change at the generation at bus i (Pi). The line
outage distribution factor dl,k relates the change in the flow on line l due to
the outage of line k. This is also illustrated in Fig. 7. The LSFs are based on
the DC power flow model where all the resistances are neglected, and the
voltage magnitudes are assumed constant. Therefore, this approach only
accounts for the active power loadings, while the reactive power effects
cannot be accurately calculated. In this figure, Xij is the element (i, j) in the X
matrix of the network, while xn is the reactance of line n.
182 Wood AJ, Wollenberg BF. Power generation, operation, and control. Canada: John Wiley & Sons; 2012.
183 EL-Shimy M. Improved Evaluation of ATC with Line-thermal Limits and Bus-voltage Quality Constraints. Scientific
Bulletin - Faculty of Engineering - Ain Shams Uni. 2005;40(1):619 - 30.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
146
(a)
(b)
Fig. 7: Definition of the LSF. (a) Line outage; (b) Generation shift
As shown in Fig. 6, the post-contingency flow
can be determined
based on the initial base flow (fo) on any line using the LSF. The post-
contingency flow on a line are then compared with the rated flow of the line
for determining the condition of the contingency. If all the flows are less than
or equal to the rated values
, the contingency is safe. Otherwise
, the contingency is unsafe and the contingency analysis
module transfer the found problem(s) to the corrective action analysis for
finding proper corrective actions. It should be noted that if
is very close to
, then the contingency analysis provides an alert of insecure condition.
For N-1 static security analysis, the contingency analysis id performed
according to the flowchart shown in Fig. 8.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
147
Fig. 8: Contingency analysis algorithm
The optimal economic dispatch (OED) is essential for real time
control of power system operation. It is a computational process whereby
the total generation required to be allocated among the available generating
units so that the constraints imposed are satisfied and the cost of energy
requirements to be minimized; however, the standard OED does not
consider the security requirements of power systems.
Security of power system184,185 operation has a considerable influence
on the secure economic dispatch problem. The system operator is
responsible for the minute-by-minute control of the system where several
problems arise, such as load variations, and component outages. These
include outage of generation, and network components. If the new system
state has any overloaded transmission lines, the system operator must take
appropriate actions to alleviate this problem by, for example, redispatching
184 Mohamed EL-Shimy: Dynamic Security of Interconnected Electric Power Systems -
Volume 1. 05/2015; Lap Lambert Academic Publishing / Omniscriptum Gmbh &
Company Kg; Germany., ISBN: 978-3-659-71372-9, DOI:10.13140/RG.2.2.19425.71520
185 El-Shimy M. Dynamic Security of Interconnected Electric Power Systems-Volume 2:
Dynamics and stability of conventional and renewable energy systems. Lap Lambert
Academic Publishing/Omniscriptum Gmbh & Company Kg. 2015 Nov.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
148
the real power generation in the system. The nonlinear load flow provides
power flows and voltages for a new power system state; however, the
nonlinear iterative load flow techniques are relatively slow and prone to
convergence problems. Sensitivity analysis186 is developed to compute the
power flow in each line, the load voltage and the power generation
redispatch as well as a quick solution of the new power system state without
the need of recalculations of load flow and power generation dispatch
techniques; see Fig. 6, and 7. System contingencies should also be
considered in the generation scheduling (or dispatch) for satisfying various
secure operational constraints. A simple example that illustrates the OED
and the static security constrained OED (SSCOED) is shown in Fig. 9.
This chapter presents the fundaments of SSCOED for educational purposes.
2. Effect of load Uncertainty on Power System Operation
Since the basic requirement of a power system is to supply electric
power and energy in a secure and economic manner, load forecasting
methods187 have been used to determine the generation capacity
requirements in advance. If the hourly variation in load demand is not
accurately predicted, the actual demand may be either greater or less than
the predicted values. For a greater demand than its predicted value, the new
system state exists which may include overloaded components, such as
generators or lines. In addition, there are possibilities of voltage magnitude
problems, such as over-voltage, or under-voltage as well as under-frequency
problems. Therefore, there is a necessity of generation reserve to be in
service quickly for avoiding service interruption.
The reserve in this case in very expensive and its amount is limited.
On the other hand, if the demand value is than its predicted value, there will
be a surplus of power reserve which results in an increase in the operating
cost, and possible over-frequency problems. As a result, the capability of
accurately predicating the system load several hours in advance is essential
for secure operation, and correct economic scheduling of power generation.
A precise short term load forecasting is essential for monitoring and
controlling power system operation. Short-term load forecasting deals with
the hourly load forecast for few hours in advance. It can be mainly divided
according to the time frame into:
186 Wood AJ, Wollenberg BF, Sheblé GB. Power generation, operation, and control. John Wiley & Sons; 2013 Dec 18.
187 Hahn H, Meyer-Nieberg S, Pickl S. Electric load forecasting methods: Tools for decision making. European journal
of operational research. 2009 Dec 16;199(3):902-7.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
149
(a)
(b)
Fig. 9: Comparison between the impact of contingencies on the traditional
OED, and the SSCOED.
1. Hourly load forecast: the hourly forecast of load demand is necessary
for on-line operation and control of the power system.
2. Four hour load forecast: Forecasting the system load 4-hours ahead or
less is needed for the economical load dispatch and for real time state
monitoring and security programs.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
150
3. One day ahead load forecast: the 24-hours ahead load forecasting is
desired for the system generation policy during the base and peak
load intervals and for the provisions of power reserve.
Decreasing the loss of the economy is of the most important functions
of electric utilities. This depends mainly on the accuracy of load
forecasting models and accurate state monitoring. In addition, the errors
in the forecasted loads specially underestimating the actual load may
result in a breach of system security function since the system risk may
exceed the specific value agreed upon for secure operation. Since,
economy, security and reliability are affected by the load forecast
uncertainties, the economical reliability level is also affected by the
accuracy of load forecasting.
3. Spinning Reserve Requirements
As shown in Fig. 3, it is required that a power system at all times
operates an amount of generating capability in excess of its actual demand,
which is available within a specified interval of time sufficient to cover the
unscheduled loss of generating equipment, or any unexpected deviation of
this demand from the anticipated levels. The power reserve can be
classified188, as shown in Fig. 10 and Table 2.
Fig. 10: Classification of power reserve
188 Towne HW, inventor; AT&T Corp, assignee. Alternating current power reserve system. United States patent US
1,951,482. 1934 Mar 20.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
151
Fig. 10 shows the reserve application procedure after an outage of a
generating unit. The power deficiency for the first few minutes after an outage is
covered by the momentary reserve of all partners of the interconnection system.
This is provided by the primary governor's action of all operating units. Within a
short time (1-10) minutes it is needed to cover the power deficiency by the fast
reserve. So, a fast reserve with a mean access time of (1-10) minutes should be
available. The previous types of reserve should be replaced by the fast reserve as
soon as possible (on the average this takes about 0.5 to 8 hours). This because
the operating cost of the fast reserve is significantly expensive with respect to
the slow reserve. In addition, the fast reserve has a limited capacity and it should
also be available for any upcoming random contingencies that cause power
deficiencies.
Table 2: Spinning reserve classification
Reserve
type
Duration
Description
Momentary
Mean access time to
fast reserve (EZ1= 1
to10 min).
The momentary power reserve is the
result capacity of primary governor's
action of all interconnected units of all
partner systems.
Fast
Mean access time to
the slow reserve (EZ2
= 0.5 to 8 hours).
It is the total power of a number of
generating unite operated at minimum
or on-load conditions. These units are
mainly gas turbine, pumped storage
and hydro units.
Slow
Operates till the end
of the repair time.
The mean repair time
(ET) equals 30 to 250
hours.
It is the total power of a number of
generating units operated when
required. These units are mainly
thermal units.
4. Formulation of the Static Security Constrained Optimal Economic
Dispatch (SSCOED)
The popular formulation of the Optimal Economic Dispatch (OED) is
given as:
Minimize
n
iii PCC
1
)(
Objective function (1)
Subjected to:
0)(
1
iLD
n
iiPPPP
Equality (or power balance) constraint (2)
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
152
maxmin iii PPP
Inequality (or generator limits) constraints (3)
where C is the total cost function of the committed generating units in $/h,
Ci(Pi) of the cost function of the generating unit number i, n is the number of
committed generating units, Pi is the net power output of unit I in MW, PD is
the power demand in MW, PL(Pi) is the power loss in MW, Pimax is the
maximum net output power of generating unit i, and Pimin is the minimum net
output power of generating unit i.
This classical formulation did not take into account the operational
and security requirements (see Fig. 1, 2, Table1, and Fig. 10). Therefore, it
cannot provide generation allocation (or scheduling) that ensures secure
operation of the power system; however, it provides the allocation of
generators for minimum costs189. An extension to this formulation can take
into consideration the intrinsic as well as the operational limits, and
minimization of the GHG emissions190, but again the security requirements
cannot be guaranteed.
The operation cost function of conventional units (thermal, and hydro)
takes the approximate quadratic function form191,
Ci (Pi) = i + i Pi + i Pi2 $/hr (4)
Therefore, the initial operating point (Pio) obtained from the classical OED
results in the initial cost,
This operating point has to be modified for considering the operational and
security requirements of the system i.e. the solution obtained from the
classical OED is considered as the initial guess for the SSCOED. For
simplification, the common factor cost function is used in the following
formulations for the SSCOED.
189 Wood AJ, Wollenberg BF. Power generation, operation, and control. Canada: John
Wiley & Sons; 2012
190 A. N. Afandi, A. P. Wibawa, SyaaPadmantara, Goro Fujita, W. Triyana, Yunis
Sulistyorini, H. Miyauchi, Nedim Tutkun, M. EL-Shimy Mahmoud, X. Z. Gao, "Designed
Operating Approach of Economic Dispatch for Java Bali Power Grid Areas Considered
Wind Energy and Pollutant Emission Optimized Using Thunderstorm Algorithm Based on
Forward Cloud Charge Mechanism", International Review of Electrical Engineering
(IREE), Vol 13, No 1 (2018). DOI: https://doi.org/10.15866/iree.v13i1.14687
191 Stevenson WD. Elements of power system analysis. New York: Mcgraw-hill; 1982
Jan.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
153
or
where
This form of the cost function is linearized because the SSCOED constitutes
a nonlinear programming problem with convergence difficulties if the
nonlinear cost functions are adopted.
Static Security Constraints Optimal Economical Dispatch (SSCOED)
is a computational process whereby the total generation required is allocated
among the available generating units so that the constraints imposed are
satisfied and the objective cost function is minimized i.e. its formulation
takes the form of equations (8) to (14) simultaneously.
The system total power generation should meet the system load demand
and network transmission losses
The generator output power must be within its maximum and minimum
generation limits
The generating units cannot ramp their output instantaneously, but they
take a specified time () to meet the load demand. Therefore, the
ramping capabilities of generating units are constraints that have to be
met in following the load changes. The range of actual operation of
online generating unit is restricted by its ramp rate limits192 as shown in
192 Benhamida F, Ziane I, Souag S, Salhi Y, Dehiba B. A quadratic programming
optimization for dynamic economic load dispatch: Comparison with GAMS. In3rd
International Conference on Systems and Control 2013 Oct 29 (pp. 625-630). IEEE.
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
154
equation (11).
(a)
(b)
Fig. 11: Ramp rate limit constrains. (a) Increased demand; (b) Reduced
demand
These limits have impacts on the operation and maneuverability of
generating units. In other words, the power schedule of a given unit in a
given hour is affected by the previous hour schedule of the same unit and
affects the upcoming hour schedule. This is due to the ramp rate limit.
where PDi is the ramp rate limit of generating unit i to meet load reduction;
PUi is the ramp rate limit of generating unit i to meet load increase; Pi(t) is
the output power of generating unit i at time step t.
This constraint is explained in Fig. 11 considering linearized ramp rate
characteristics, which is valid for small deviations of the output power.
Among the main constraints is to maintain the power flow in each line of
the transmission network to be within specified limits. Therefore, the
active power flow on a transmission link k is PTk, and then
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
155
It can be seen from Fig. 6 that the LSF can be used to linearly link the
change in the flow on a specific line l with either the change in the power
generation at bus i, or the change of the power flow on another line. These
LSF will be used in evaluating the impacts of the change of the power
generation schedule of the initial values (obtained from the OED) for
security reasons. Till this point, the classical OED is now formulated as an
optimal power flow (OPF) problem. The following additional constraints
present the spinning reserve constraints for security requirements. The
spinning reserve constraints can be written as,
where
the minimum allowable value of total required spinning reserve
;
the maximum available spinning reserve that can be contributed from
unit i in less than or equal to 10 minutes when the system operates in the
emergency state; Yi is the amount of spare capacity of the unit i that is
unavailable due to, for example, bad weather conditions, or low state of
health.
5. Solution of the SSCOED
Fig. 12 shows of the linear programming iterative approach for
solving the considered problem. The presented example is not a complete
solution of the SSCOED formulation and its objective is only the
demonstration of the differences between OED, OPF, and SSCOED
problems. For simplification and due to the incomplete required data, the
ramp rate limits of equation (11) are neglected, while the output power limits
are considered i.e. . The consideration of the
constraints of the equation (11) converts the OED, OPF, and SSCOED to
dynamic problems as the time is included.
The presented algorithm193 is applied to the IEEE 30 bus system
shown in Fig. 13 with a total load of 296.28 MW, and the losses are
neglected for the purpose of demonstrating the method. The results are
shown in Table 3. The change in the power schedule due to the application
of the SSCOED w.r.t the OPF is shown in Fig. 14. The load flow results
ensure the satisfaction of the operational and intrinsic limits, while the
contingency analysis ensures the satisfaction of the spinning reserve
193 http://www.fglongatt.org/Test_Systems/IEEE_30bus.html
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
156
requirements and security constraints.
Fig. 12: Solution algorithm of the SSCOED
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
157
Fig. 13: IEEE 30 bus test system
Table 3: OED, OPF, and SSCOED of the IEEE 30 bus system
i
Initial OED
SSCOED iterations
1
2
3
1
138.28
152.48
152.48
152.48
2
56.56
65.56
49.56
65.56
3
24.1
17.06
32.06
17.06
4
35
35.00
31.41
31.41
5
26.15
13.43
13.43
17.77
6
16.19
12.75
17.34
12.00
Total Gen. (MW)
296.28
296.28
296.28
296.28
Total cost ($/h)
822.12
832.80
824.47
Mohamed EL-Shimy (ed.). Sustainable Energy Technologies and Systems. LAP – Lambert Academic
Publishing, ISBN 978-620-2-05640-3, Aug. 2019.
158
Fig. 14: Change of the power schedule of the SSCOED w.r.t the OPF
6. Conclusions
It is essentially important for the system secure operation to not only
considers the operational and intrinsic limits for enhancing the scheduling
obtained from the classical OED problem, but security constraints must be
considered. This chapter presents a linearized SSCOED and the formulation
is solved using the linear programming.
The results show that efficiency of the method for fulfilling all the
constraints under normal and abnormal states of power systems.