1
Optimal Placement and Sizing of FACTS Devices
to Delay Transmission Expansion
arXiv:1608.04467v1 [math.OC] 16 Aug 2016
Vladimir Frolov, Priyanko Guha Thakurta, Scott Backhaus, Janusz Bialek Fellow, IEEE and
Michael Chertkov Senior Member, IEEE
Abstract—The Transmission System Operators (TSOs) plan to
reinforce their transmission grids based on the projected system
congestion caused by the increase in future system load, amongst
other factors. However, transmission expansion is severely limited
in many countries, and especially in Europe, due to many factors
such as clearance acquisitions. An alternative is to use Flexible
Alternating Current Transmission System (FACTS) devices, thus
allowing to delay or avoid much more expensive transmission
expansion. These devices are capable of utilizing the existing
transmission grid more flexibly. However, the locations and sizing
of future FACTS devices must be carefully determined during
the planning phase in order to avoid congestion under many
representative scenarios of the projected economic growth (of
loads). This paper proposes an optimization approach, based on
AC Power Flows, to determine suitable locations and sizes of
series and shunt FACTS devices. Non-linear, non-convex and
multiple-scenario based optimization is resolved via efficient
heuristics, consisting of a sequence of Quadratic Programming.
Efficiency and scalability of the approach is illustrated on IEEE
30-bus and 2383-bus Polish systems.
Index Terms—Non-convex Optimization, Optimal Investment
Planning, Optimal Power Grid Reinforcement, Series Compensation Devices, Static VAR Compensation Devices.
N OMENCLATURE
Parameters:
Nl
Nb
M
Number of power lines in operation
Number of buses in the system
Number of segments representing each
load duration curve
N
Number of scenarios representing each
segment
Minimum (maximum) loading level
α (α)
αi (αi )
Minimum (maximum) loading level for
segment i
pi = wi
Occurrence probability of a segment
Vector of initial line inductances
x 0 ∈ RN l
P G (P G ) ∈ RNb Vector of maximum (minimum) active
power generator outputs
QG (QG ) ∈ RNb Vector of maximum (minimum) reactive
power generator outputs
PD0 (QD0 ) ∈ RNb Vector of active (reactive) power demands
S ∈ R2Nl
Vector of line apparent power limits
V (V ) ∈ RNb
Vector of maximum (minimum) allowed
voltages
CSC ∈ R
Cost per Ohm of a series FACTS device
CSV C ∈ R
Cost per MVAr of a shunt FACTS device
Ny ∈ R
Service period of the system
Optimization variables:
V ∈ RN b
θ ∈ RN b
PG ∈ RNb
x ∈ RNl
∆x ∈ RNl
∆Q ∈ RNb
l0 ∈ RNb
Vector of bus voltage magnitudes
Vector of bus voltage angles
Vector of generator active power injections
Vector of line inductances modified by
SC devices
Vector of series FACTS capacities
Vector of shunt FACTS capacities
Vector of active and reactive loads for
the base configuration
Other variables:
Peij ∈ R2Nl
e ij ∈ R2Nl
Q
Vector of all active power injections into
lines
Vector of all reactive power injections
into lines
I. I NTRODUCTION
Power system planning and operation are terms which
encompass an entire range of activities performed by different
stakeholders. As shown schematically in Figure 1 the planning
activities fall under categories of long, medium and short term
investments ranging from 15 years to a season.
High penetration of variable renewable energy output coupled with significant increase in load uncertainty result in
frequent overloads of the system. To mitigate the overloads
TSOs have to plan for counter measures. Transmission expansion within the congested zones is one suitable option.
However, building new transmission lines is both expensive
and politically challenging. Moreover, some congestion management options like generation re-dispatch and demand side
management come at a significant extra cost to the TSOs in
a decentralized, e.g. European, environment. Hence, TSOs are
interested to find less expensive alternatives for their long term
planning strategies.
As argued in [2], [3] Flexible Alternating Current Transmission System (FACTS) is a technology capable to offer
the highly desirable and affordable alternative to traditional
network expansion. This strategic (grid reinforcement) use
of the FACTS devices comes in addition to many (more
traditional) operational benefits. This technology offers to
control power flows [4]–[8], improve voltage stability [9]–
[11], damp power system oscillations [12]–[15] and improve
2
Operational Planning
Planning
Operations
Real-Time Operation
Near Real-Time Operation
Short-Term
Miliseconds, Seconds
Minutes, Hours, Days
Months, Year (1)
Mid-Term
Long Term
Years (2-5)
Market Forces
Years (5-15)
Regulation, Legislation
Uncertainties in variable production and demand; External Forces; Vulnerabilities (physical and cyber)
Severe Emergency
Conditions requiring
Automatic and Self-Healing
Control and Protective
Actions
Fig. 1.
Non-Severe Emergency
Conditions requiring
Coordinated Operator
Assited Control and
Optimized Re-Dispatch and
Balancing
Steady State Operating
Conditions requiring
Preventive Control and
Power Flow Optimization
Resource Adequacy and
Grid Development Gaps
requiring Coordinated Grid
Planning Optimization
Long-term goals for
“green” resource adequacy
and grid development
optimization requiring
Coordinated Grid Planing
Time line for operations and planning [1]
small signal stability [16], [17]. Note that the (traditional)
operational use of the FACTS devices comes at a near-zero
cost to the TSOs, as many of those devices are already installed
and operated by them.
In a long term planning process TSOs make decisions of
investing in expanding/upgrading the existing transmission
infrastructure. The installation of FACTS devices can and
should be incorporated into this planning. This can be done
in a principal way in which the locations and sizing of the
new devices are resolved via optimization. In other words,
placement of the FACTS devices must be effective to manage
system congestion considering uncertainties that would arise
during the planned time period. The proposed approach in
this paper builds on a diverse set of prior attempts to pose and
resolve the problem of optimal FACTS placement and sizing.
Several research papers addressed the optimal location for
effective FACTS placement in a transmission system. Authors
of [18] have developed a FACTS placement toolbox to find
optimal locations and sizing parameters for multi-type FACTS
devices in large power systems based on a genetic algorithm.
It is concluded that the toolbox is effective and flexible
enough for analyzing a large number of scenarios with mixed
types of FACTS to be optimally sited at multiple locations
simultaneously. Wibowo et al. [19] have proposed a method
for optimal location of FACTS devices which is market-based
and as such allowing to account for annual cost and benefits
associated with FACTS installations. Models of FACTS shuntseries controllers along with a multi-objective optimization
methodology to find their optimal locations were developed
in [20]. Galloway et al. [21] have proposed two methods
for finding optimal location of FACTS devices considering
variations in demand and renewable generation output. These
authors proposed techniques based on the Monte Carlo simulation to determine locations of the highest benefit in terms of
the conventional generation cost savings. Placement of FACTS
devices was discussed in [22]–[25], while issues related to
FACTS operations in the context of the electricity market
and stochastic framework were also addressed in [26]–[31].
Van Hertem et al. [32] have discussed investment, planning,
scheduling and operations using already installed power flow
controlling devices (PFCs), including Phase Shifting Transformers (PSTs) and High Voltage Direct Current (HVDC), on
a realistic example of the Belgian grid.
To the best of our knowledge, no prior work has addressed
the problem of AC-aware and overload-free optimal placement
and sizing of FACTS devices to guide TSOs on possible
cheaper alternative to grid expansions.
In this paper, a method is proposed to find suitable locations
for placing FACTS devices in power systems and their required
installed capacities (sizing) in a robust way to future uncertainties due to load increase and integration of renewables.
Therefore, main contributions of this paper can be summarized
as follows:
1) Planning methodology to install new FACTS devices is
proposed. The methodology is developed to guide TSOs
on their long term strategies. In particular, the developed
methodology can be used to delay or avoid much more
expensive investments involving transmission expansion.
Note that the proposed framework is universal, and as
such it accounts for optimal operational (day ahead or
real time) use of the newly installed FACTS devices.
2) The proposed methodology is mathematically stated as
an optimization problem, the outputs of which are optimal FACTS placements and capacities accounting for
load growth and associated uncertainty. The uncertainty
is modeled via a scenario based approach. Full AC
power flow formulation is embedded into our optimization heuristics, consisting of a sequence of Quadratic
Programming (QP) steps.
3) It is observed that the developed methodology scales.
The developed approach is first analyzed on the IEEE
30-bus system and the scalability of our approach is
3
25,000
20,000 α
MW
illustrated on the Polish system consisting of 2383buses.
4) The resulting solutions are sparse, i.e. . The sparsity is
promoted by the l1 -norm regularized installation cost.
The paper is organized as follows. The basic optimization
problem is stated and discussed in Section II. The formulation is extended in Section III to account for load growth
and associated uncertainty. Our optimization heuristic scheme
(sequence of QPs) is described in Section IV. The complete
methodology is summarized in Section V. Section VI presents
results of our numerical experiments. Section VII finally draws
the conclusions of the paper.
i = 1...M ; pi = wi
15,000
10,000
α
5,000
0
II. O PTIMIZATION FRAMEWORK TO INCLUDE FACTS IN
LONG TERM PLANNING
In this paper, the prime focus lies on finding optimal locations and sizes to install series and/or shunt FACTS devices.
Multiple loading configurations/scenarios are accounted within
the proposed optimization framework. The objective function
in the optimization consists of the capital investment term
and the system operational costs depending on the planning
period (optimization horizon). Capacities and operational settings (different for different scenarios, but all under respective
capacities) of the newly installed Series Capacitors (SCs) and
Static Var Compensators (SVCs) are among the optimization
variables. (However, already installed series and shunt FACTS
devices can also be included within the framework.)
The optimization problem is stated as follows:
X
Ta ∗Ca (P (a) )
min CSC ∆x1 +CSV C ∆Q1 +Ny
4x,4Q
a=1..N
(1)
subject to:
(a)
x(a) = x0 + 4x(a)
(a)
PG
(a)
QG
(a)
P
(a)
Q
=
=
= f (V
= g(V
(a)
PG ≤
Q(a)
G
(a)
PD0 + P (a)
(a)
QD0 + Q(a)
(a) (a)
≤
,θ
(a)
,θ
− 4Q ≤ 4Q
V
(a)
≤V
(a)
≤V
(a)
(a)
[Peij ]T [Peij ]
+
75%
100%
Piece-wise step function approximation of the load duration curve.
(2) bounds actual line inductances, which are adjusted
according to the operational value of installed series compensation for each scenario, within their respective installed
capacities (represented by (9)). (3) and (4) represent active
and reactive power balances at each bus of the network. The
elements of vectors PG (QG ) and PD0 (QD0 ) are zeros for
buses containing no generators or loads respectively. (5) and
(6) represent the net active (P ∈ RNb ) and reactive (Q ∈ RNb )
power injections at the system buses. The term ∆Q expresses
shunt compensation by SVC adjusted to a scenario bounded by
respective installed capacities (represented by (10)). The limits
on active and reactive power generation by the generators are
represented by (7) and (8). Finally, voltage and thermal line
flow constraints are represented by (11) and (12).
∀a
(3)
III. ACCOUNTING FOR UNCERTAINTY
+ ∆Q
∀a
(4)
(a)
)
∀a
(5)
)
∀a
(6)
∀a
(7)
∀a
(8)
∀a
(9)
∀a
(10)
∀a
(11)
∀a
(12)
Uncertainty related to the system load growth over the
considered planning phase is dealt with through scenario
sampling. Standard power system load growth over a large
period of time is utilized via the load duration (LD) curve [35].
The LD curves are chosen to be piece-wise step functions.
Loading scenarios are sampled from the probability distributions associated with the yearly LD curves. Figure 2 illustrates
such a load duration curve and its approximation.
Each scenario is characterized by different loading configurations, occurrence probability, network topology and configuration of loads (active and reactive power consumption). The
scenarios may include sampled (typical) configurations and/or
contingency (rare) configurations projected for different levels
of loading. For the test cases and the projected level of stress
considered, ACOPF always remain feasible if line constraints
are ignored (there is enough generation capacity even for
the stressed cases). Depending on the sampled scenario, 3
situations may arise.
1) ACOPF is feasible and congestion price is zero (low
loading level).
2) ACOPF is feasible and congestion price is positive
(higher loading level representing peak conditions).
(a)
(a)
,x
≤ 4x
(a)
50%
(2)
(a)
(a)
PG ≤ P G
(a)
(a)
QG ≤ QG
(a)
− 4x ≤ 4x
25%
∀a
,x
(a)
Fig. 2.
0%
≤ 4Q
(a)
e (a) ]T [Q
e (a) ]
[Q
ij
ij
≤ (S
(a) 2
)
The objective function in (1) consists of three terms. The
first two, sparsity promoting [33], [34] terms, express the
capital investment costs of the installation of the two types
of FACTS devices. The third term stands for the operational
cost where the summation is over all the scenarios (∀a is
a shortcut for, ∀a = 1, · · · , N ) accounting for respective
occurrence probabilities multiplied by the number of years
(service period). The optimization stated is, therefore, an
operational aware planning.
4
TABLE I
I MPLEMENTATION OF THE LD CURVE SCHEME
IV. M ETHODOLOGY OF THE O PTIMIZATION H EURISTICS
i
wi
αi
σ
1
2
3
4
5
6
5,50
19,50
25,00
25,00
18,80
6,20
0,940
0,845
0,775
0,685
0,590
0,51
0,064
0,041
0,045
0,080
0,068
0,078
The optimization problem (1)-(12) is difficult due to nonlinearity of the thermal constraints (12) and non-linearity and
non-convexity of the power balance constraints (3) and (4). In
order to solve the optimization problem, an efficient linearization based heuristics is developed. Note that the linearization is
done analytically, which is critical for the algorithm scalability.
The linearization is illustrated for Eq. (12), which is replaced by:
3) ACOPF is infeasible mainly due to congestion of lines
but system has enough generation capacity. ACOPF
without apparent power limits on lines is feasible (overloaded conditions which are possible in the future).
The aim of planning installation of FACTS devices at the
right locations with their corresponding capacities is to reduce
generation cost for point 2, and to improve or extend feasibility
domain of the system for the point 3. Extra years of service
can hence be added to the existing grid by making it more
flexible, thereby allowing to delay investments on new lines
and generators.
A. Scenarios sampling for each segment
F (V, θ, x) = |S(V, θ, x)|2 ≤ F = S
2
(19)
The solution methodology consists in solving a sequence
of QPs, built by analytic linearization of Eqs. ((3),(4),(12))
around the current state:
∂F ∂F ∂F ?
(20)
,
,
](y ) × (y − y ? ) ≤ F
∂V ∂θ ∂x
where, y ? = (V ? , θ? , x? ) are the state variables representing
voltage magnitudes and phases at the buses connected to a
given line, and inductance of a line, respectively, for the
current state.
F (y ? ) + [
The loading level αi for a segment i is represented by:
αi + αi
(13)
2
Future loading configurations are obtained from the base case
by re-scaling all active and reactive loads by αi uniformly.
The resulting vector of loads for a segment is thus given by:
αi =
li0 = αi × l0
(14)
Loading configurations are generated, for each segment i
and each j = 1..N , through modification of initial li0 . It is
done by adding Gaussian correction to each load with zero
expected value and a respective standard deviation:
lij = li0 + N (0, σli0 )
(15)
pji
(16)
= wi /N
(probability of a given scenario)
where, σli0 is given by:
αi − αi
× li0
αi
= σ × li0
σli0 =
(17)
(18)
This scheme accounts for variations in the distribution of
loads, thereby simulating power system behavior during an
extended period of time in the future.
B. Implementation
In our simulations LD curves are modeled with M = 6
segments.
Table I shows parameters used to simulate future system
loading. In all simulations α = 1 corresponds to α. For the
first year α is taken to be 3% from AC-OPF infeasibility.
Yearly load growth (β) is in the 0.5% − 1.5% range.
V. O PTIMIZATION A LGORITHM
The sequence of QP optimizations with linearized constraints are solved using an iterative algorithm in order to find
a solution for optimization problem (1)-(12). The flowchart of
the algorithm is shown in Figure 3.
A. General description
If some of the constraints (1)-(12) are violated the initial
state of the system lies outside of the feasible domain (A)
defined by the constraints (2)-(12). At each iteration within
the sequence the nonlinear constraints are linearized around a
current state, thus resulting in the construction of the domain
(B).
CPLEX optimization solver [36] is used to evaluate the
resulting QP optimization with linear constraints over (B).
Replacement of non-convex and non-linear (A) by convex and
linear (polytope) (B) is achieved in a number of iterations,
needed to get from the initial state outside of the domain (A)
to a point which lies inside. The algorithm is terminated when
either the preset target precision or the maximum number of
iterations is reached.
B. Details of the algorithm
Flowchart of the algorithm is shown in Figure 3. Brief
description of the flowchart is as follows:
1) Each load configuration for each scenario is given.
2) OPF is solved for each scenario without considering
any line thermal limits. Solutions of the OPF define the
initial states for each scenarios.
3) Linearization of the thermal and power balance constraints over the current state for each scenario is applied
and QP is solved.
5
Given Load Configurations
OPFs without Thermal Limits
Define Initial States
Linearize Conditions
& Solve QP
Obtain and Update States
1
}
2
3
4
5
Next Iteration
Yes
tion is implemented if the objective was not improved
during the iteration.
2) As the operational cost increases with the system size
more investments can be made. In such cases linearization around current state becomes less accurate and solving AC-PF (back projection onto nonlinear equations)
may become infeasible. Controlling the step sizes either
leads to infeasibility of the QP step or infeasibility of the
AC-PF step via many small investments in the system.
Here, the practical solution is to adjust installation prices
in order to find the best solution (in terms of the
average operational cost). The investment cost is then
recalculated.
3) The challenge of visualizing a large system is met
by automatic visualization tools developed within this
framework.
VI. C ASE STUDIES
Violations?
No
Stop
Fig. 3.
Flowchart of the iterative algorithm
4) AC Power Flow (AC-PF) is solved to update the state
found at the previous step. This is done to prepare a
feasible solution for the next iteration/step.
5) Iterations are performed until no constraints remain
violated. Otherwise the iterations are terminated when
either target precision or allowed number of iterations
is reached.
The key idea of the algorithm is to maintain the physical
state of the system after each iteration. Note that the iterative
loop, including linearization and following solution of the
respective QP optimization and back projection onto nonlinear power balance equations, allows to maintain a feasible
solution at each step.
C. Scalability
One of the major contributions of this paper is in demonstrating the scalability of the scheme in practice, i.e. the ability
to resolve problems over realistic (consisting of thousands of
buses) power transmission systems efficiently. The challenges
faced to meet the scalability target and the way of addressing
them are summarized below:
1) All controllable parameters of the system being adjustable causes degeneracy of the linearized problem.
In case of a loopy network, one has the flexibility
in redistributing controllable voltages, active and reactive powers. In order to resolve the degeneracy, soft
controllable constraints for reactive power dispatch are
introduced at each QP step. The step size is reduced
adaptively as the algorithm advances. The Q-step reduc-
Developed approach is demonstrated on IEEE 30-bus and
2383-bus Polish systems, both available within the Matpower
[37] software package. (Winter peak configuration is considered as the base case for the Polish system.) The simulations
are performed on Intel i7 2600K PC with 12 GBs of system memory. Very powerful solvers are available for solving
convex optimization problems with linear constraints. Interior
point algorithm of the CPLEX solver [36] is used as a subroutine to solve the QP proposed in this paper.
A. IEEE 30-bus system
1) Scenario dependent system settings: At first, the performance of the multi-scenario solver is tested and illustrated
accounting (simultaneously) for 10 scenarios over the course
of a year. The scenarios are generated with α = 1.1 and
σ = 0.1. Notice that some of the scenarios are originally
OPF infeasible. In this experiment the scenarios are not
sampled from LD curve. The aim of this preparatory test is to
demonstrate novelty and flexibility of the developed approach
in terms of adjustment (and convenient visualization) of all
the controllable (optimization) degrees of freedom.
The optimization solution suggests to install an SVC device
at bus 8 and a SC device on the transmission line connecting
buses 6 and 8. These installations allow to resolve AC-OPF
infeasibility and reduce generation cost for the 10 generated scenarios. The optimal installed capacity of the SVC is
6.98 MVAr and one of the SC corresponds to the 51,90 % of
compensation. The resulting optimal placement of the devices
is sparse. The proposed investment cost is about 540k USD.
To illustrate the operational perspective of the optimal
placement Table II shows optimal settings of the two devices
for each of the 10 operational scenarios.
Dynamics of the algorithm is illustrated in Figure 4. The
left y-axis of the figure illustrates gradual (but generally nonmonotonic) reduction of the line overloads whereas the right
figure shows convergence of the total cost (value of the objective function). As clear from the Figure the multi-scenario
solver requires approximately 20 iterations to converge.
6
TABLE II
O PTIMAL SETTINGS OF THE INSTALLED FACTS FOR EACH SCENARIO
Scenario (+ for
ACOPF feas)
1 (-)
2 (+)
3 (+)
4 (-)
5 (-)
6 (+)
7 (-)
8 (-)
9 (-)
10 (+)
SC optimal settings
(%)
51,90
-1,52
11,91
51,90
51,90
15,40
47,03
14,86
16,40
7,23
SVC optimal settings
(MVAr)
6,98
6,98
6,98
6,98
6,98
6,98
6,98
6,98
6,98
6,98
16
6.4
14
6.3
6.2
Scn 1
Scn 2
Scn 3
Scn 4
Scn 5
Scn 6
Scn 7
Scn 8
Scn 9
Scn 10
Total cost
10
8
6
4
2
6.1
6
5.9
Total cost (M$)
Overload value (MVA)
12
5.8
5.7
0
2
Fig. 4.
4
6
8
10
12
14
Iterations
16
18
20
22
24
5.6
Fig. 5. Optimal solution for IEEE-30 bus system over 10 years planning
horizon. One SVC (solid dot) and one SC (dotted line) are installed.
TABLE III
C OMPUTATION TIMES
Illustration of the algorithm dynamics for 10 scenarios.
System
Based on this first test it can be concluded that the proposed
optimization approach is powerful and successful. An improvement is achieved not only through installation of FACTS
devices but also via smart use of the flexibility of these
devices combined with proper utilization of all other adjustable
equipments available within the network. This illustration
clearly indicates that the proposed methodology can be used by
the TSOs not only for planning but also to calculate optimal
settings of existing and/or new FACTS devices during dayahead operational planning and/or real-time operation.
2) Planning for multiple years: The planning period is set
to 10 years and the number of scenarios considered per year is
16. β is considered to be 1.5% per year. The resulting optimal
solution allows to resolve AC-OPF infeasiblility (observed
otherwise under some scenarios) and then to reduce system
congestion in the future. The optimal solution consists of an
installation of an SVC device at bus 8 with the capacity of
5.78 MVAr and also an installation of an SC device at the
line between buses 6 and 8 with a capacity of 1 %. The
proposed investment is 30k USD and the average economy
is 1.8 $/hour (0.43 %). Figure 5 shows the resulting optimal
investment solution.
B. 2383-bus Polish system
In the tour-de-force experiment with the realistic (in size)
Polish system, experimentation is performed with 1 year and
10 years of the planning horizon. The scenarios for the Polish
case are generated according to LD curve approach explained
in Section III.
30bus
30bus
Polish
Polish
No. of
scenarios
10
160
4
16
No. of
iterations
25
25
15
15
Time (sec)
28.6
260.2
591.6
10397.0
1) Planning for 1 year: Simulation is done for 4 ACOPF
feasible scenarios sampled from the top 2 segments of the
LD curve for the first year (see Table I, i = 1, 2). For lower
loading segments congestion price is 0 and no investments are
needed. The optimal investment found is shown in Figure 6a.
5 SCs are built to reduce congestion in the system. Average
congestion price is 1446 $/hour and the average generation
cost economy is 1504 $/hour.
2) Planning for 10 years: This simulation is done with
16 sampled scenarios (2 out of 16 are ACOPF infeasible)
for 10 years with assumed yearly loading growth equal to
0.5 %. The optimal investment found is shown in Figure 6b.
2 SCs and 1 SVC need to be built in order to reduce
congestion in the system and to improve feasibility domain for
future operations when loading increases. Average congestion
price is 5738 $/hour and average generation cost economy is
3369 $/hour.
Table III shows computational efforts of the proposed approach for the two considered systems.
VII. C ONCLUSIONS
In this manuscript, a new optimization framework for planning of placement and sizing of FACTS devices is proposed.
7
The approach takes into account non-linear power flow equations. The most important features of the newly developed
framework include, scalability, allowing to resolve congestion
over practical (thousands of buses) size transmission systems,
and also the ability to resolve multiple scenarios simultaneously. The optimization can be considered as generalizing
standard AC-OPF over multiple scenarios with an additionally
added cost of operation (over a time horizon) and the cost of
installation. The optimization accounts for both installation
and operations, allowing the installed devices to adjust operationally to a particular scenario within the bounds set by the
installed capacity. Many interesting cases are experimented.
In all the cases considered the output (optimal placement) is
spatially sparse also showing strong non-locality (in the sense
that placement of a device may resolve congestion in a distant
region of the grid). It is also observed that under highly loaded
conditions FACTS devices are beneficial in reducing the total
cost of generation. Optimal installation of the devices helps to
resolve infeasibilities that are projected to become even more
severe in the future.
Main technical achievement reported in this paper is the
development of efficient heuristics for solving the non-linear
and non-convex optimization. The developed methodology
builds a convergent sequence of convex optimizations with
linear constraints. Each constraint is represented explicitly
(a) 1 year horizon. Capacities of SCs from left to right: through exact analytical linearization of the original nonlinear
35.6 %, 24.4 %, 100 %, 9.24 %, 100 %.
constraints (e.g. representing power flows and apparent power
line limits) over all the degrees of freedom (including FACTS
corrections) around the current operational point. In order to
represent uncertainty in the projected growth of the system
(loads) a custom scenario sampling is introduced. Practicality
of our approach for resolving investment planning is illustrated
on the IEEE 30-bus and 2383-bus Polish systems. It is evident
from the experimental results that the approach is capable both
in improving the system economy (reduce congestion price
and generation cost) and resolving congestion by increasing
the feasibility domain. Development of convenient web visualization software within the project is also significant.
R EFERENCES
(b) 10 years horizon. Capacities of SCs from left to right: 14.4 %, 70.4 %.
SVC capacity: 3.30 MVAr.
Fig. 6.
Planning for 2383-bus Polish system. Yellow and black dots
represent loads and generators respectively. Blue long-dashed lines represent
compensated lines that are initially overloaded, green and red short-dashed
lines represent compensated and initially overloaded lines respectively. Large
green dot represents installed SVC compensator.
[1] P. Guha Thakurta, J. Maeght, R. Belmans, and D. Van Hertem, “Increasing Transmission Grid Flexibility by TSO Coordination to Integrate
More Wind Energy Sources while Maintaining System Security,” IEEE
Trans. Sustain. Energy, vol. 6, no. 3, pp. 1122–1130, July 2015.
[2] Y. Xiao, Y. Song, C. C. Liu, and Y. Sun, “Available transfer capability
enhancement using FACTS devices,” IEEE Trans. Power Syst., vol. 18,
no. 1, pp. 305–312, Feb 2003.
[3] J. Mutale and G. Strbac, “Transmission network reinforcement versus
FACTS: an economic assessment,” IEEE Trans. Power Syst., vol. 15,
no. 3, pp. 961–967, Aug 2000.
[4] D. J. Gotham and G. T. Heydt, “Power flow control and power flow
studies for systems with FACTS devices,” IEEE Trans. Power Syst.,
vol. 13, no. 1, pp. 60–65, Feb 1998.
[5] Y. Xiao, Y. Song, and Y. Sun, “Power flow control approach to power
systems with embedded FACTS devices,” IEEE Trans. Power Syst.,
vol. 17, no. 4, pp. 943–950, Nov 2002.
[6] N. Li, Y. Xu, and H. Chen, “FACTS-based power flow control in
interconnected power system,” IEEE Trans. Power Syst., vol. 15, no. 1,
pp. 257–262, Feb 2000.
[7] M. Noroozian, L. Angquist, M. Ghandhari, and G. Andersson, “Use of
UPFC for optimal power flow control,” IEEE Trans. Power Del., vol. 12,
no. 4, pp. 1629–1634, Oct 1997.
8
[8] N. G. Hingorani and L. Gyugyi, Understanding FACTS: Concepts and
Technology of Flexible AC Transmission Systems. New York: IEEE
Wiley Press, 1999.
[9] W. Shao and V. Vittal, “LP-Based OPF for Corrective FACTS Control
to Relieve Overloads and Voltage Violations,” IEEE Trans. Power Syst.,
vol. 21, no. 4, pp. 1832–1839, Nov 2006.
[10] N. Yorino, E. El-Araby, H. Sasaki, and S. Harada, “A new formulation
for FACTS allocation for security enhancement against voltage collapse,”
IEEE Trans. Power Syst., vol. 18, no. 1, pp. 3–10, Feb 2003.
[11] H. Feng Wang, H. Li, and H. Chen, “Coordinated secondary voltage
control to eliminate voltage violations in power system contingencies,”
IEEE Trans. Power Syst., vol. 18, no. 2, pp. 588–595, May 2003.
[12] Y. Li, C. Rehtanz, S. Ruberg, and L. Luo, “Wide-Area Robust Coordination Approach of HVDC and FACTS Controllers for Damping Multiple
Interarea Oscillations,” IEEE Trans. Power Del., vol. 27, no. 3, pp. 1096–
1105, July 2012.
[13] M. Zarghami, M. L. Crow, and S. Jagannathan, “Nonlinear Control
of FACTS Controllers for Damping Interarea Oscillations in Power
Systems,” IEEE Trans. Power Del., vol. 25, no. 4, pp. 3113–3121, Oct
2010.
[14] U. P. Mhaskar and A. M. Kulkarni, “Power oscillation damping using
FACTS devices: modal controllability, observability in local signals, and
location of transfer function zeros,” IEEE Trans. Power Syst., vol. 21,
no. 1, pp. 285–294, Feb 2006.
[15] B. K. Kumar, S. N. Singh, and S. C. Srivastava, “Placement of FACTS
controllers using modal controllability indices to damp out power system
oscillations,” IET Generation, Transmission & Distribution, vol. 1, no. 2,
pp. 209–217, March 2007.
[16] M. H. Haque, “Improvement of first swing stability limit by utilizing
full benefit of shunt FACTS devices,” IEEE Trans. Power Syst., vol. 19,
no. 4, pp. 1894–1902, Nov 2004.
[17] Y. Tang and A. P. S. Meliopoulos, “Power system small signal stability
analysis with FACTS elements,” IEEE Trans. Power Del., vol. 12, no. 3,
pp. 1352–1361, Jul 1997.
[18] E. Ghahremani and I. Kamwa, “Optimal placement of multiple-type
FACTS devices to maximize power system loadability using a generic
graphical user interface,” IEEE Trans. Power Syst., vol. 28, no. 2, pp.
764–778, May 2013.
[19] R. S. Wibowo, N. Yorino, M. Eghbal, Y. Zoka, and Y. Sasaki, “FACTS
Devices Allocation with Control Coordination considering Congestion
Relief and Voltage Stability,” IEEE Trans. Power Syst., vol. 26, no. 4,
pp. 2302–2310, Nov 2011.
[20] A. Lashkar Ara, A. Kazemi, and S. A. Nabavi Niaki, “Multiobjective
Optimal Location of FACTS Shunt-Series Controllers for Power System
Operation Planning,” IEEE Trans. Power Del., vol. 27, no. 2, pp. 481–
490, Apr 2012.
[21] S. J. Galloway, I. M. Elders, G. M. Burt, and B. Sookananta, “Optimal
flexible alternative current transmission system device allocation under
system fluctuations due to demand and renewable generation,” IET
Generation, Transmission & Distribution, vol. 4, no. 6, pp. 725–735,
June 2010.
[22] S. Gerbex, R. Cherkaoui, and A. J. Germond, “Optimal location of multitype FACTS devices in a power system by means of genetic algorithms,”
IEEE Trans. Power Syst., vol. 16, no. 3, pp. 537–544, Aug 2001.
[23] M. H. Haque, “Optimal location of shunt FACTS devices in long
transmission lines,” IEE Proceedings - Generation, Transmission &
Distribution, vol. 147, no. 4, pp. 218–222, Jul 2000.
[24] S. Rahimzadeh, M. Tavakoli, and B. A. H. Viki, “Simultaneous application of multi-type FACTS devices to the restructured environment:
achieving both optimal number and location,” IET Generation, Transmission & Distribution, vol. 4, no. 3, pp. 349–362, March 2010.
[25] L. Ippolito and S. P., “Selection of optimal number and location of
thyristor-controlled phase shifters using genetic based algorithms,” IEE
Proceedings - Generation, Transmission & Distribution, vol. 151, no. 5,
pp. 630–637, Sept 2004.
[26] F. Bouffard and F. D. Galiana, “Stochastic security for operations
planning with significant wind power generation,” IEEE Trans. Power
Syst., vol. 23, no. 2, pp. 306–316, 2008.
[27] ——, “An electricity market with a probabilistic spinning reserve
criterion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 300–307, 2004.
[28] P. Guha Thakurta, R. Belmans, and D. Van Hertem, “Risk-based
management of overloads caused by power injection uncertainties using
power flow controlling devices,” IEEE Trans. Power Syst., vol. 30, no. 6,
pp. 3082–3092, 2015.
[29] F. Bouffard, F. D. Galiana, and A. J. Conejo, “Market-clearing with
stochastic security - Part I: Formulation,” IEEE Trans. Power Syst.,
vol. 20, no. 4, pp. 1818–1826, 2005.
[30] ——, “Market-clearing with stochastic security - Part II: Case studies,”
IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1827–1835, 2005.
[31] H. Yu and W. Rosenhart, “An optimal power flow algorithm to achieve
robust operation considering load and renewable generation uncertainties,” IEEE Trans. Power Syst., vol. 27, no. 4, pp. 1808–1817, 2012.
[32] D. Van Hertem, J. Rimez, and R. Belmans, “Power flow controlling
devices as a smart and independent grid investment for flexible grid
operations: Belgian case study,” IEEE Trans. Smart Grid, vol. 4, no. 3,
pp. 1656–1664, 2013.
[33] V. Frolov, S. Backhaus, and M. Chertkov, “Efficient algorithm for locating and sizing series compensation devices in large power transmission
grids: I. model implementation,” New Journal of Physics, vol. 16, no. 10,
2014.
[34] ——, “Efficient algorithm for locating and sizing series compensation
devices in large power transmission grids: Ii. solutions and applications,”
New Journal of Physics, vol. 16, no. 10, 2014.
[35] H. Lee Willis, Power Distribution Planning Reference Book. New York:
Marcel Dekker, Inc, 2004.
[36] “ILOG CPLEX optimization studio for MATLAB, http://www-03.ibm.
com/software/products/ru/ibmilogcpleoptistud.”
[37] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: Steady-State Operations, Planning and Analysis Tools for
Power Systems Research and Education,” IEEE Trans. Power Syst.,
vol. 26, no. 1, pp. 12–19, Feb 2011.