Chapter 1
Optimization Methods Applied
to Power Systems: Current
Practices and Challenges
Jeremy Lin*, Fernando Magnago† and Juan Manuel Alemany†
*
DNV-GL Energy Advisory Group, Dallas, TX, United States, †National University of Rio Cuarto,
Rio Cuarto, Argentina
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INTRODUCTION
Planning and decision-making in energy, in general, involve the allocation of a
large amount of resources which significantly impacts all the actors in an economy. The main objective of planning in power systems is to define an appropriate strategy with regards to generation and transmission to efficiently
utilize the system. In a general framework for energy efficiency, optimization
applied to the efficient use of electric resources is essential. In the case of power
systems, applied optimization is related to the planning and scheduling of the
resources to aid in the system operation [1].
In this chapter, we will begin by outlining the key problems in power systems that are naturally suitable for applications of classical optimization
methods to solve these problems. For example, unit commitment (UC), economic dispatch (ED), and optimal power flow (OPF) are three key problems
which are critical to the economically efficient operation of the power system.
UC is a large-scale, nonconvex, mixed-integer programming problem which
requires effective and efficient numerical methods to obtain its solution. ED is a
step that needs to be done after completing the UC process, while the OPF
solves the ED by adding additional constraints to the optimization problem [2].
In all these activities, the optimization methods emanating from the operations research field play a significant role in the decision-making process associated with UC, ED, and OPF. After outlining these problems, the optimization
methods that are currently employed in solving these problems are presented.
Mathematically, the optimization methods employed in these formulations are
linear programming, mixed-integer programming, decomposition methods and
Classical and Recent Aspects of Power System Optimization. https://doi.org/10.1016/B978-0-12-812441-3.00001-X
© 2018 Elsevier Inc. All rights reserved.
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Classical and Recent Aspects of Power System Optimization
stochastic approaches. These methods constitute a core set of optimization
methods that are currently applied [3].
After presenting these optimization problems and solution methods, we will
attempt to present some of the emerging challenges we face in current and
future power systems. It is believed that the future grid will face some challenging developments such as the smart grid, microgrid, distributed energy
resources (DER), and massive amounts of renewable energy sources (RES),
to name just a few. These new developments will pose very different kinds
of problems to the existing power systems.
In the final section of the chapter, we will present some of these challenges
in detail and discuss the optimization methods currently being used to solve
these problems. The problems described will be focused on practical applications to the real large-scale systems, including time-coupling constraints, network constraints, size and variation of the control variables, or stochastic
nature of renewable resources. Because these are the frontier areas of research
with uncertain results, we will also provide some speculations in terms of applications of potential optimization methods. Some innovations that incorporate
energy efficiency in to the planning and operation of power systems with high
renewables penetration will also be described. New concepts and techniques
necessary to manage uncertainty in the models and algorithms that are commonly developed for deterministic conditions will be included as well.
2 KEY SCHEDULING PROBLEMS IN POWER SYSTEM
OPERATION
From a general perspective, the power system economic operation involves the
operation of generation, transmission, and distribution subsystems with the
objective of operating these subsystems efficiently and fulfilling the requirement of maintaining the balance between generation and load at all times. Moreover, the voltage at each system node and the system frequency must remain
within prespecified limits. This combination of economic and technical requirements for system operation is the most important task for the system operators.
The system operator needs to execute three essential calculations: the UC, the
ED, and the OPF. These major decision-making problems are described in Fig.
1 from a temporal perspective.
The next sections will cover these three problems.
2.1 Unit Commitment
The UC problem can be defined as the process of predispatching generation
units over a time horizon to satisfy the electricity demand and operational constraints. Typically, a solution to the UC problem is achieved by formulating and
solving a specific objective such as production cost minimization [4].
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Long term planning
5–15 years
Generation and transmission
Mid-term planning
Maintenance
Fuel scheduling
Optimal operation cost
Emission rights
1 year
Short-term operation
Security constrained unit commitment
Security constrained optimal power flow
1 day to 1 week
Real-time operation
Economic dispatch
Contingency analysis
5 minutes
FIG. 1 Temporal perspective—major decision-making problems.
Security-constrained unit Commitment (SCUC) is an extension of the
generic UC in which the network constraints are included in the UC problem
to ensure that the system is secure after completing this important step. In SCUC
formulations, the set of constraints generally include: generation capacity
limits; minimum run and down times; ramp up and down rates; operating, regulation, and spinning reserve requirements for multiple zones; fuel constraints
by generator, area or system; environmental constraints, such as SOx and NOx
emissions; interruptible load contracts; hydro reservoir volume limits and targets; discharge limits; natural inflows; maximum spillage; delay times; forbidden zones; energy limits; maintenance schedules; inter-area and inter-zone flow
limits, etc. [5].
Nowadays, the optimization techniques applied to these types of problems
have been experimental and evolutionary, and as a consequence, become
important players in modeling and execution of UC problems.
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Classical and Recent Aspects of Power System Optimization
As a general example, a UC problem can be mathematically represented as a
nonlinear optimization formulation as follows:
XX
ug, t Cpg, t + sg, t Cupg, t + zg, t Cdowng, t
min F ¼
u, p
t2T g2G
Subject to:
Global constraints
Power balance equation:
X
ug, t pg, t ¼ Dt 8t ¼ 1, …,T
g2G
Reserve capacity:
X
ug, t Pg Dt + Rt 8t ¼ 1, …, T
g2G
Local constraints
Unit generation limits:
Pg pg, t Pg 8t ¼ 1,…, T
Ramp limits:
pg, t pg, t1 URg 8t ¼ 1, …,T
pg, t1 pg, t1 DRg 8t ¼ 1, …,T
Minimum on/off limits:
xg, t1 MUT g ug, t1 ug, t 0 8t ¼ 1, …,T
xg, t1 + MDT g ug, t ug, t1 0 8t ¼ 1,…, T
where, Dt is the total system load at period t [MW]; Rt is the total system reserve
at period t [MW]; Cpg, t is the generation cost [$] of unit g at period t; Cupg, t is
the startup cost [$] of unit g at period t; Cdowng, t is the shutdown cost [$] of unit
g at period t; ug,t is the status variable that indicates if unit g is on (1) or off (0) at
period t; sg, t is the startup status that establishes if unit g goes into service at
period t; zg, t is the shutdown status that tells if unit g changes from on to off
at period t; xg, t is a positive/negative variable that sets the number of periods
the unit is on/off at period t; G is the number of generators; T is the total simulation time; pg, t is the power generation of unit g at period t [MW]; Pg is the
minimum power limit of unit g; Pg is the maximum power limit of unit g; MUTg
is the minimum time that unit g must remain on when it is set to on; MDTg is the
minimum time that unit g must remain off when is set to off; URg is the maximum ramp up limit of unit g and DRg is the maximum ramp down of unit g [6].
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The output from solving this mathematical formulation represents the set of
generators that will be set to on for each period. This information is given by the
value of binary variable ug,t.
2.2 Economic Dispatch
Once the UC process is completed, the next step is to solve the ED problem, the
solution of which is also used for day-ahead, rescheduling, intraday, and realtime markets. The objective of ED problem is to minimize the total operational
cost by taking into consideration the constraints related to the power balance
and the unit limits. Mathematically, the ED problem can be described as
follows:
X
Cpg, t 8t ¼ 1, …,T
min F ¼
p
Subject to:
Power balance equation:
X
g2UC
pg, t ¼ Dt 8t ¼ 1, …,T
g2UC
Unit generation limits:
Pg pg, t Pg 8t ¼ 1, …, T
where UC is a subset of G, representing the number of units that the prior UC
problem established as committed [7].
2.3 Optimal Power Flow
The OPF solves the ED problem by adding additional constraints to the optimization problem. The key addition to the ED problem is the inclusion of network
constraints. The OPF problem calculates the control and state variables which
optimize a selected objective function and guarantee the secure operation of the
system. Nowadays, with advances in optimization tools applied to power system problems, several new objectives and constraints are included into a traditional OPF problem. Mathematically, an OPF problem can be formulated as
follows:
Subject to :
Min
f ðx, uÞ
gðx, u, pÞ ¼ 0 , λ
hðx, u, pÞ 0 , μ
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Classical and Recent Aspects of Power System Optimization
where x is the state variable vector, u is the control variable vector, p is the
parameter vector, f is the objective function, g are power balance equations,
h are the unit and system constraints, λ is the Lagrange multipliers vector
related to the power balance equation, and μ is the Lagrange multipliers vector
related to the units and system constraints [8].
2.4 Fuel Scheduling
In general, mainly for thermal units, the cost of fuel represents the majority of
operating costs. Restrictions on fuel contracts or the transport of fuels generally
mean that the UC must be calculated after considering very restrictive
conditions.
Therefore, it is essential to formulate a fuel allocation strategy that allows
for the use of long-term fuel consumption constraints within short-term fuel
consumption targets in which the objective is to minimize the operational cost.
Also, it is important to dispatch generation while optimizing the fuel consumption by considering constraints related to generation, transport, and availability
of fuel [9].
In all these activities, the optimization methods from the operations research
field play a significant role in the decision-making process associated with ED,
UC, OPF, and even fuel scheduling.
3 OPTIMIZATION METHODS
Several optimization techniques have been applied to solve the types of problems described in the previous sections. These methods comprise a broad range
of mathematical approaches, including the use of mathematical programming
algorithms such as linear and nonlinear programming, dynamic programming,
and interior-point methods. Other techniques include artificial intelligence (AI)
methods, such as neural networks and fuzzy systems, and evolutionary
methods, such as genetic algorithms and the simulated annealing [10].
Fig. 2 illustrates the relationship between the major decision-making tools in
power system and the classical optimization problems.
The methods considered in this chapter can be classified as follows:
l
l
l
l
l
Linear programming (LP).
Mixed-integer programming (MIP).
Decomposition methods.
Stochastic programming (SLP).
Artificial intelligence methods (AI).
These optimization methods can be classified into three main groups: (1) Deterministic methods, (2) stochastic methods, and (3) heuristic and artificial intelligence methods.
Optimization Methods Applied to Power Systems Chapter
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(mostly)
Long-term
Generation expansion
Network expansion
Demand forecast
Variant
LP
MIP
Real-time
control
NLP
MINLP
Operation
Hydro plan scheduling
Thermal plant scheduling
Demand forecast
Renewable plant forecast
AI
Very large LP
SLP
FIG. 2 PES industry and classical optimization problem relationship.
Deterministic methods include linear programming, mixed-integer programming, decomposition methods, and Lagrange relaxation techniques. Stochastic methods include Monte Carlo simulation, chance-constrained
programming, Pareto curves, and risk management approaches. Finally, the
genetic algorithm, particle swarm optimization method, and evolutionary
methods are part of the artificial intelligence methods. Fig. 3 summarizes these
methods. Although there are more methods, the most common ones are presented as examples to highlight the complexity of the problems. Several of these
methods are further described in the next section.
Linear programming (LP)
Convex
Nonlinear programming (NLP)
Deterministic
Continuous (NLP)
Nonconvex
Integer
General
optimization
methodologies
Discrete
Mixed integer linear programming
(MILP)
Integer programming (IP)
Mixed integer nonlinear programming
(MINLP)
Decision tree models
Stochastic
Game theory
Stochastic process
Monte Carlo simulation
Artificial intelligence
Neural networks
Support vector machines
Genetic algorithms
Ant colony
Particle swarm optimization
Simulated annealing
FIG. 3 Optimization classification.
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Classical and Recent Aspects of Power System Optimization
3.1 Linear Programming
In general, the objective functions of optimization problems related to power
systems are mainly quadratic. However, they can be linearized around an operating point to make the problem amenable for solving using linear programming
techniques. The main advantage of a linear programming approach is the guaranteed solution of a well-formulated problem. Besides, it is possible to solve a
large-scale problem using linear programming in a reasonable time. Also, it is
the natural way to start solving a complex optimization problem. Finally, most
of the commercially available optimization software, particularly those applied
to solving power system problems, includes robust and powerful linear programming algorithms. The main algorithms used in the linear programming
techniques are the simplex methodology, revised simplex method, sequential
linear programming, and interior point techniques [8].
3.2 Mixed-Integer Programming
The implementation of algorithmic solutions based on mixed-integer programming (MIP) began in the early 1960s with the development of two classical
methods: (1) Cutting Planes algorithm and (2) branch and bound (B&B) methodology. However, significant innovations related to this method have emerged
only during the last 20 years. These innovations and developments have allowed
for the application of these methods in real-scale problems. Most of these
advances have already been implemented in commercially available programs.
Some of the latest advancements are: (1) Significant progress in linear programming algorithms; (2) development of numerical methods for scattered data systems; (3) different types of cuts; (4) preresolution of the problem and nodes; (5)
flexible selection of variables, and (6) heuristic techniques [11].
3.3 Decomposition Methods
The decomposition methods were proposed to efficiently solve large-scale optimization problems, such as SCUC. These methods take advantage of the special
structure of the problem, by iteratively solving small-scale problems. While
these methods are general, their application depends mainly on the particular
problem. The decomposition methods can be classified into two forms: (1)
when variables are decomposed, or (2) when constraints are decomposed.
The Benders (decomposition) algorithm is usually applied when the restrictions (aka constraints) are decomposed. Hence, the Benders method is mostly
used for SCUC-type problems. The Benders method decomposes the SCUC
problem into a master problem and a subproblem. The master problem, such
as the conventional UC, is represented at the first level of optimization and
the signals from the network, commonly referred to as slices, are derived from
the subproblems at the second level of optimization [12].
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There are two primary applications for the Benders method: (1) when variables make the problem much more complex, and (2) when the master problem
and the subproblem are of a different nature. For example, a general bi-level
optimization problem can be formulated mathematically as follows:
min c1 T x1 + cT2 x2
x1 , x2
A1 x1
¼ b1
B1 x1 + A2 x2 ¼ b2
x1
, x2 0
where x1 are the variables considered at the first level and x2 are the variables
considered at the second level. Then, the problem can be expressed as follows:
min cT x1 + θ2 ðx1 Þ
x 1 , θ 2 ð x1 Þ
A1 x 1 ¼ b 1
x1 0
the function θ2(x1) represents the objective function of the following
subproblem:
θ2 ðx1 Þ ¼ min cT2 x2
x2
A 2 x 2 ¼ b 2 B1 x 1 : π 2
x2 0
where π 2 represents the dual variables of the constraints (known as shadow
prices).
The dual formulation is:
θ2 ðx1 Þ ¼ max ðb2 B1 x1 ÞT π 2
AT2 π 2 c2
π2
The decomposition method above is presented as an example. For the ED
problem, other decomposition techniques exist that have been successfully
applied and we encourage the readers to investigate them. For example, dual
decomposition, alternating direction method of multipliers, Dantzig-Wolfe
reformulation, etc.
3.4 Stochastic Approaches
The stochasticity or uncertainty appears in all the power system problems
described above. However, until recently, it was not possible to solve the optimization problems of large-scale systems considering this stochasticity explicitly. In this context, the uncertainty may be due to a lack of reliable data, or
information about the future state of variables, such as the future demand, future
prices, and the renewable energy contributions. Mathematically, as an example,
a linear multistage stochastic problem can be represented as follows:
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Classical and Recent Aspects of Power System Optimization
min
F¼
wp
xp
P X
X
p¼1 wp 2Ωp
wT
pwp p cp p xwp p
Subject to:
w
w
p
p1
Bp1
xp1
+ Awp p xwp p ¼ bwp p p ¼ 1,⋯,P
xwp p 0
Bw0 1 0
where w represents each scenario with probability p [13].
3.5 Artificial Intelligence Methods
Among the artificial intelligence (AI) techniques, the main algorithms applied
in power systems are: artificial neural networks, fuzzy logic systems, genetic
algorithm, particle swarm optimization, colony optimization, simulated annealing, and evolutionary computing. Some of the distinctive properties of AI
methods are: (1) the ability to remember past findings; (2) the methods learn
and adapt in their subsequent performances; and (3) the methods can plan their
path forward and act intelligently by mimicking human or social intelligence.
Improved or hybrid AI methods have been developed by combining the advantages of various search methods [14].
4 APPLICATIONS OF OPTIMIZATION METHODS
ON EMERGING CHALLENGES
4.1 Smart Grid Applications
For the optimization problem of a traditional power system, the model is typically based on the estimate of the network’s state. However, the optimization
problem of a Smart Grid (SG) scenario, in general, will rely on real-time information thanks to the developments of advanced metering infrastructure (AMI)
and two-way communications system [15].
One important feature of the Smart Grid is the widespread installation of
distributed generations (DG) which include renewable energy sources, microturbines, fuel cells, etc. From the optimization perspective, DGs introduce additional uncertainties into the model. This additional set of uncertainties makes
resource scheduling decisions very challenging. For example, it is generally difficult to predict, with any accuracy, near-term or real-time wind speed and wind
availability.
The potential deployment of massive DGs suggests that additional optimization areas need to be explored to solve the following problems associated with
the strategic placement of DGs in the network: (1) to improve the grid reinforcement, (2) to reduce the losses and the on-peak costs, (3) to improve the voltage
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profiles, and (4) to improve the security, reliability, and efficiency of the
system.
With regards to the model implementation, it is critical to assess the computational performance in solving the power flow problem in SGs. For example,
the new communication networks provide faster updates of network information, and the incorporation of phasor measurement units (PMUs) require new
analytic tools with more rapid decision-making capabilities.
Volt/Var optimization tools have become popular within the SGs infrastructure. Incorporating variables and constraints related to the reactive part of the
model adds another level of complexity. One approach to simplify this problem
is to implement the concept of conservative voltage reduction (CVR). The purpose of CVR is to lower the voltage utilization at the end-use consumers such
that their energy consumption decreases. Among the potential benefits made
possible by this scheme, the most important is peak load reduction, and consequently, the reduction in power supply cost.
New challenges related to the discrete optimization problems arise with
Smart Grid networks, particularly related to topological optimization. The
reconfiguration of distribution feeders may reduce the system losses, the system
cost, and the network congestion. The primary purpose of the feeder reconfiguration is to change the topological structures of distribution feeders by changing the open/closed states of the devices. The key emerging developments in
this area are to evaluate how to efficiently solve the underlying discrete optimization problem. Also, load forecast techniques need to be reevaluated in this
area, mainly due to their granularity needs. SGs require load forecasts at feeder
and housing level to be as accurate as possible.
In summary, it is important to remark that grid optimization applied to SGs
strongly relies on the up-to-date technologies in the areas of telecommunication, sensing, and measurements.
4.2 Distributed Energy Resources and Massive Amount
of Renewable Energy Resources
Related to the Smart Grid scenario, the increasing penetration of DER will
cause significant transformations to power generation systems. The impacts
of DER on SGs at the distribution domain are mostly due to the
bi-directional power flows because of the installation of small DER units at
medium or low voltage levels of the network [16].
Another challenging scenario related to the integration of renewable
resources, such as wind, solar, biomass, hydro, etc., is the locations at which
these resources will be stored. Most often, the location of these resources is
far away from the consumption sites. Thus, it becomes necessary to evaluate
and optimize the network facilities, which is extremely expensive.
Development of efficient energy storage devices is another significant challenge. The principal task of the storage devices is to compensate the temporal
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Classical and Recent Aspects of Power System Optimization
mismatches between the typically volatile renewable generation, and the energy
consumption. With the development of energy storage, it becomes critical to
develop optimization tools that address their efficient charge and discharge
logic, which can prolong their useful life.
Among the energy storage devices, the electric vehicle (EV) is a unique
resource that can be considered as either a distributed generator or a dispersed
energy storage device. If the EVs are used appropriately, they can improve electrical power management and increase transportation efficiency. From the electrical power perspective, the EV inclusion in the grid can improve the voltage
profiles or reduce peak load. In addition to optimal scheduling, the presence of
EVs into the electricity market will require new models to be developed and
incorporated into the optimization formulations. For example, such models
can include centralized management of charging stations, and handle the inclusion of simultaneous EV chargers.
The optimization tools also need to be ready to incorporate intermittent
energy, and be flexible enough for scalability, advanced information management, and improved tools related to predictions. Optimizing resource usage at
different locations to maximize revenue is a huge challenge. Besides, Because
the amount of data will increase considerably, the incorporation of intelligent
big data analytics tools is imperative. Moreover, it is essential to develop
sophisticated techniques which will take automated decision making in realtime into consideration when working with large-scale data storage.
4.3 Energy Efficiency
The primary objective of planning in power systems is to define an appropriate
strategy for the generation, transmission, and distribution of energy in order to
efficiently exploit the system. Nowadays, the level of global investment in
energy efficiency is comparable to those in renewable and conventional
resources. Global investment in energy efficiency is so important that it has
come to be known as the primary fuel or hidden fuel. This statement suggests
that energy efficiency is not only a hidden fuel but the first fuel in the world.
Therefore, developments regarding energy efficiency play a fundamental role
in every energy policy in any country. In a general framework for energy efficiency, optimization applied to the exploitation of electric resources is critical.
Specifically, in the power systems area, the energy efficiency can be improved
with the addition of renewable generation because the power supply is increased
without relying on fuels. Consequently, exploitation of oil resources is reduced,
emissions are reduced, and in some countries, the imports of fossil fuels are
reduced [17].
In this context, developing innovations that contribute to energy efficiency
in the planning and operation of power systems with high wind power penetration is a major challenge. Moreover, new concepts and techniques are needed to
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manage uncertainty in models and algorithms that are commonly developed for
deterministic conditions.
The incorporation of wind power into power systems requires the application of more advanced methodologies to the operation programming problem.
These methods include, for example, the dynamic backup reserve allocation and
the use of stochastic optimization techniques. Stochastic models present significant computational challenges due to the large scale of the problem. This fact
suggests that it is necessary to use decomposition techniques in stochastic problems. Furthermore, it can be useful to take advantage of the units’ ability to
reschedule generation for reserve allocation.
Therefore, several research proposals are put forward to improve the decomposition methodology that allows the iterative solution of two problems: (1) the
first stage problem where generation is scheduled conventionally; and (2) given
these results, a second stage problem where generation rescheduling mitigates
sudden changes of wind generation.
Decomposition combined with rescheduling is a significant development
because it would allow the development of a resolution methodology for an
optimization problem which is stochastic mixed-integer and which would otherwise be unsolvable in large-scale power systems.
5 CURRENT CHALLENGES FOR OPTIMIZATION METHODS
APPLICABLE TO POWER SYSTEMS
5.1 Time-Coupling Constraints
In general, the time constraints are modeled in a UC problem in which the
periods are coupled by unit constraints such as ramp up/down, or minimum
on/off restrictions. Other problems, such as the ED problem, are frequently formulated by assuming instantaneous loading for generation units, meaning there
are no ramp up/down constraints. Nowadays, the emerging developments in
power systems require that even problems previously expressed as static problems need to be formulated as dynamic problems [18].
As an illustration, let us consider the problem of the increasing requirement
in operational planning for advanced voltage/VAr management solutions.
These new requirements establish more automated scheduling of voltagesVArs, over a near-future time frame which can range from one interval (i.e.,
1 h) to several intervals (i.e., 24 or 168 h). That need is exacerbated by the higher
uncertainty, such as the penetration of renewable resources. These dispatch
solutions are characterized by a multistage problem that includes inter-temporal
constraints to secure a soft and realistic operational trajectory over time. As
we can see, these complex optimization problems require sophisticated solution techniques. In this particular scenario, the challenge is to implement a
sequence of security-constrained optimal power flow (SCOPF) calculations
with full AC network models. As a result, the optimal solution needs to provide
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Classical and Recent Aspects of Power System Optimization
forward-looking dispatch and control variable schedules over the entire simulation time frame, without violating any network and operational constraints.
Another important feature that needs to be evaluated is to allow the execution
of a SCOPF for each period in sequence with a definable objective, ramping
limits, and tunable movement-inhibiting penalties on specified controls (generator voltages, taps, shunts). It is also necessary to coordinate each SCOPF
solution over a time horizon by enforcing time-coupling constraints, while
optimizing the equipment settings over a sequence of hours for generator voltage set points by dispatchable reactive power, transformer taps, and shunts.
Concerning the objective functions, new tools must be able to handle different types of objectives. For example, one such objective would be minimizing
the movement of control variables from their target values, or minimizing the
losses considering the following inter-temporal constraints as a function of the
control variables: (1) ramping up and down, (2) minimum on/off time, (3) number of movements at each period, and (4) number of changes over the study
horizon.
From the aforementioned challenges, it can be inferred that large-scale problems that include time-coupling constraints are complex, multiperiod, mixedinteger programming (MIP) optimization problems. The solution for this type
of complex problem can be found by a decomposition process comprising a sublevel problem, and a master level problem.
In the sublevel problem, the calculation of the optimal network-constrained
control variable dispatch for each period can be performed. Then, a mixedinteger problem can be formulated to couple all periods by including constraints
to limit control actions between the periods and the network model constraints
not only for the base case but also for the contingency case. This problem can be
formulated as the master level problem.
In general, this type of problem can potentially have a huge set of critical
network constraints (i.e., potentially binding constraints). One of the mostwidely used techniques to solve this type of problem is the Benders Decomposition, which can enforce these restrictions efficiently at the master level.
5.2 Network Constraints
Typical network constraints for pre- and all postcontingency scenarios include
the bus voltage magnitude limits, the net MVAr interchanges of zones or groups
of zones, MVAr reserves of any designated generator groups, weighted sums of
the MVAr flows in designated sets of branch groups, voltage magnitude differences between designated pairs of buses, changes in bus voltage magnitudes
between the base-case and contingency cases, and branch MVA flows [19].
Considering these constraints in the backdrop of emerging trends in power
systems, it becomes necessary to develop more rigorous network optimization
methods. Accurate AC network models, both in the pre- and postcontingency
states must be included in the problem formulation. Most of the current
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optimization methods employ only DC network model formulation due to the
performance and size issues. In addition, the increasing need to include variables and constraints related to the reactive power into the problem hastens
the need to incorporate AC network models, because approximated DC network
models for reactive-power scheduling can give incorrect results due to their
inherent inaccuracy. Furthermore, they require further work to improve handling contingencies, such as an automatic recursive contingency analysis which
allows capturing all system insecurities. This is because correcting existing constraint violations often creates new violations which, if ignored, can produce
even worse solutions. Also, additional functions need to be added to handle
solution infeasibilities. Solution infeasibility is the most difficult challenge
among all other usable engineering results when not all violations are enforceable, because it requires sophisticated analytical and heuristic techniques if
practical results are to be obtained. The solutions to this unique problem
may include constraint relaxation techniques or incorporation of priority levels
for constraints and controls.
From the mathematical point of view, the challenges are related to the nonconvexity nature of the problem caused by nonlinearities of power flow. In particular, these challenges are caused by nonconvexities due to the minimum
voltage level and branch flow limits. An additional major challenge comes from
the discrete control decisions which make the problem nonconvex.
5.3 Size and Variation of Control Variables
As the size of the power system grows, the number of variables in the optimization problem associated with that system are multiplied. For example, the
number of generators in a large-scale real system may be in the thousands
and the number of branch constraints modeled in the problem can be in the hundreds or even thousands. Due to the smart-grid development, along with the
growing number of DER, future solution methods need to be able to model
and find optimal solutions for problems with different types of constraints
related to carbon emissions, storage resources, electric vehicles, and renewable
energy sources, such as wind farms and PV installations (photovoltaic or solar
power system) [20].
Multiplicity and variation of these constraints, as well as those of the state
and control variables, can make the original problem too complex and too big to
solve. State variables in the power system include bus voltage magnitudes, relative bus voltage angle difference with respect to the reference angle while system control variables are active and reactive power, generator voltage
magnitudes, transformer taps, and phase shifters. Simply having too many generators in the system, even if they are small, will easily multiply the number of
control and state variables in the optimization problem formulation which will
make it very difficult, if not impossible, to solve. These types of models may
require modeling constraints as quadratic, which will compound the already
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Classical and Recent Aspects of Power System Optimization
complex problem. In addition, it is necessary to incorporate into the model
unexpected events associated with changes in system interchanges, generation
failures, or device outages; these events are associated with discrete disturbances. Therefore, it becomes necessary to develop computational methods
which can handle a huge amount of control variables.
5.4 Stochastic Nature of Renewable Resources
One of the main challenges raised in this chapter is related to the difficulty associated with accurately predicting wind and solar generation. This key challenge
suggests the application of a stochastic approach in operating the power system
without drastically changing the planning or operation scheme [21].
The main reason for this challenge is that the weather parameters that influence the operation, and hence the output of renewable resources, are difficult to
predict. Consequently, the variable and uncertain nature of the wind and solar
generation output make it harder to solve the problems related to the optimal
operation of the power systems.
One of the solutions to deal with the challenge associated with this uncertain
generation is to properly and accurately characterize the uncertainties. Normally, the uncertainty quantities are determined or estimated based on the Probability Density Functions (PDFs) because PDFs portray the most granular form
of uncertainty representation. Because PDFs are based on sampling methods, it
is necessary to develop techniques that efficiently handle large quantities of
sampled data. Typically, the Monte Carlo method is the method that is mostwidely used to deal with that problem.
Scenario tree models are another way of solving these problems. Depending
on the solution methodology, this type of modeling turns the original problem
into different problem formulations among which the most popular are stochastic linear programming, stochastic dynamic programming, stochastic programming with recourse, and stochastic dual dynamic programming. All these
techniques are variations of a mathematical method known as multiperiod
models with recourse. For this reason, the main challenge is to improve this type
of methodology in order to develop better and more accurate modeling of
uncertainties.
Pareto curve is another tool that can be employed for these types of problems. Pareto curve does not provide additional information that is helpful for
making decisions. Instead, it drives the decision-maker to choose among the
best solutions for a given set of data having a single variable without
compromising the decision related to another variable. Some companies began
to use the Pareto approach to study balancing reserves with the inclusion of a
large amount of wind power on the system.
A new development within this type of formulations, which use probabilistic
models, makes it possible to explicitly consider the risk levels that were previously assumed. The key challenge that begs further investigation is to ensure an
Optimization Methods Applied to Power Systems Chapter
1
17
optimal solution can be obtained. With current probabilistic methods, the optimal solutions may not be easily found. As alternatives, robust heuristics and
approximation methodologies can be applied to overcome these types of
problems.
6
SUMMARY
This chapter summarizes the state-of-the-art related to the application of optimization methods and tools in the planning and operation of power systems to
aid in their efficiency. Numerous algorithms used in these optimization methods
for solving different power system problems have been described. Several areas
that deserve attention and require the utilization of new optimization methods
were also mentioned. A significant challenge associated with gradually increasing system sizes tests the limits of the existing algorithms, particularly in realtime applications. Furthermore, the growing penetration of renewable energy
resources in the system brings uncertainties to the current problem
formulations.
The use of new technologies in dealing with those issues has also been
highlighted in the chapter.
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