(1)Jairo Quirós Tortós
Eddie Araya Padilla
Abstract
This paper presents a steady state approach to prevent voltage collapse in a power system. The power system operation close to stability boundaries is an important factor which may lead to voltage instability conditions. There are different ways to prevent voltage collapse in the power system; one of this is the correct coordination of the protection devices to avoid undesirable equipment disconnections.This paper proposes a new method to coordinate the back-up zone in the distance protection scheme. The methodology uses both Optimal Power Flow (OPF) and Continuation Power Flow (CPF) solutions to determine the impedance seen by the relay. These impedances are later compared to conclude which is the best option to set the parameter for the back-up zone in the relay. The methodology also considers the distance to Maximum Loading Point (MLP) and the corresponding Voltage Collapse Point (VCP). Therefore, undesirable disconnections can be avoided when the power system operates close to the stability boundaries. In order to demonstrate and validate the methodology, simulations are carried out on the IEEE 14-bus test system using MATLAB.
Keywords: Distance protection, optimal power flow, voltage collapse.
Resumen
Este artículo presenta un enfoque en régimen permanente para prevenir el colapso de tensión en un sistema de potencia. La operación del sistema de potencia cercana a los límites de estabilidad es un importante factor que puede conllevar a condiciones de inestabilidad de tensión. Existen diferentes formas de prevenir el colapso de tensión en el sistema de potencia, la correcta coordinación de los dispositivos de protección de modo tal que se evite la indeseada operación de los equipos es una alternativa.
Este artículo propone un nuevo método para coordinar la zona de respaldo en el esquema de protección de distancia. La metodología utiliza dos tipos de soluciones, la solución del flujo optimo de potencia (OPF, por sus siglas en inglés) y la solución del flujo de potencia de continuación (CPF, por sus siglas en inglés) para determinar la impedancia vista por el relé. Estas impedancias son luego comparadas para concluir cuál es la mejor opción para ajustar la zona de respaldo del relé de protección. La metodología también considera la distancia al máximo punto de cargabilidad (MLP, por sus siglas en inglés) y el correspondiente punto de colapso de tensión (VCP, por sus siglas en inglés). Para demostrar y validar la metodología, simulaciones son llevadas a cabo en el sistema de prueba IEEE-14 barras utilizando MATLAB.
Palabras clave: Protección de distancia, flujo óptimo de potencia, colapso de tensión.
The continuous changes in the power system topology to meet the expansion plans and the interconnections with other countries have caused an increase in the number of power system blackouts. The impact of these blackouts in the economy has encouraged more researchers to determine different methods to avoid these undesirable events. By properly setting the protection devices, undesirable cascading outages can be avoided.
Voltage stability can be described as the capability of the system to maintain their bus voltage magnitudes within a permissible operating threshold under normal operating conditions and after the disturbances arise (Kundur, 1994). Massive system interconnections, increased demands, insufficient generation, transmission expansion, economical and environmental factors have led power systems to operate very close to their limits.
In real power systems, the experiences of voltage collapse are massive. In (Jonsson, 2003; Ohno & Imai, 2006; Ruiz, 2006; Sancho & Vega,
2001) the authors describe what happened in the power system causing the voltage collapse. Common characteristics have been identified when a voltage collapse happens. Some of these common features are:
• Slow dynamic behaviour: Operation of On-Load Tap Changers (OLTC) and over excitation current limiters in generators.
• The decrease in the voltage magnitude at buses more than expected.
• The disconnection of the faulted line by a conventional relay: distance relay or differential protection scheme.
• Undesirable trip out of important transmission lines and sources of reactive power.
Voltage stability studies have been typically carried out using static approaches such as power flow analysis (Gan, Thomas, & Zimmerman,
2000; Momoh & Shu, 1999; Vaahedi, Tamby, Manssur, Li, & Sun, 1999), P-V curves (Johansson & Sjögren, 1995; Jonsson, 2003; Kundur, 1994; Nagao, Tanaka, & Takenaka, 1997; Rajagopalan, Lesieutre, Sauer, & Pai, 1992; Repo, 2001; Taylor, 1994), V-Q curves (Chowdhury & Taylor, 2000; Johansson & Sjögren, 1995; Jonsson, 2003; Kundur, 1994; Overbye, Dobson, & DeMarco, 1994; Taylor, 1994), modal analysis (Gao, Morison, & Kundur, 1992; Morison, Gao, & Kundur, 1993), and others techniques including continuation power flow techniques and the correspondent λ-V curves (Alinezhad & Kamarposhti, 2009; Milano, 2005; Venkataramana, 2006; Venkataramana & Christy, 1992). More complex methods including detailed and realistic models of power system elements and their controllers are considered on dynamic simulations which can be effectively simplified using the so-called Quasi Steady State Simulation (Van Cutsem & Vournas, 1998).
This paper presents a methodology to determine the settings for the back-up zone in the distance protection scheme to avoid voltage collapse in a power system. By using the λ-V curve, obtained from the CPF algorithm (Milano, 2005; Venkataramana, 2006; Venkataramana & Christy, 1992), the distance to the VCP can be easily determined as the distance in megawatts between the actual operating point and the bifurcation point (nozzle of the curve). At this point, the corresponding electrical variables are determined to compute the settings for the back- up zone (generally third zone) in the distance protection scheme.
When a system is subject to a severe disturbance, depending on the loading conditions of the system, it can move the operating point of the system into a condition where voltages in parts of the system are below the permissible limits. In this sense, part of the system will face an unstable condition which might cause undesirable operation of protective devices. By this, unexpected disconnection of reactive power sources accelerates the voltage collapse (Van Cutsem & Vournas, 1998). To avoid system blackout caused by mal-operation of relays, it is necessary to make a correct setting of the protection devices, which is the main goal of this paper.
This paper is organised as follow. The continuation power flow algorithm and the optimal power flow algorithm are presented in Section 2 and Section 3, respectively. The proposed methodology to coordinate the distance protection scheme is then presented in Section 4. Simulations are carried out in Section 5 to demonstrate and validate the proposed method. Final remarks are given in Section 6.
The continuation power flow is an important tool to determine the power flow in a power system starting from a base case and leading to the steady state voltage stability limit of the system (Venkataramana, 2006; Venkataramana & Christy, 1992). Therefore, this algorithm is used worldwide to determine the power flow solution when the system operates close to the VCP. The employed algorithm avoid the singularity of the Jacobian by reformulating the power flow equations and applying locally parameterized continuation technique (Venkataramana, 2006; Venkataramana & Christy, 1992).
As it can be noticed in Figure 1, the CPF determines the power flow solution for a base case first. Then, a tangent predictor is used to estimate the next solution corresponding to a different value of the load parameter. This estimate is then corrected using the same Newton-Raphson algorithm employed by a conventional power flow. The local parameterization previously mentioned provides a means of identifying each point along the solution path and plays an integral part in avoiding singularity in the Jacobian (Venkataramana, 2006; Venkataramana & Christy, 1992).
Based on the original nonlinear power system equations, F(x) = 0, where x = [v ,θ ], the CPF formulation introduces a load parameter λ to apply a locally parameterised continuation technique to the power flow problem. Therefore, the new set of nonlinear power system equations is represented by:
(1)
This parameter λ, which varies from 0 up to a critical value i.e. 0 < λ < λ critical, is used to reformulate the power flow equations in a power system with Nb buses. Therefore, if F is used to denote the whole set of equations, the problem can be expressed as follow:
(2)
Where v and q represent the vector of bus voltage magnitudes and phases. Here, λ = 0 represents the base load and λ = λcritical corresponds to the critical load. As it was previously mentioned, the base case solution ( v0, θ0, λ0) is known via a conventional power flow and the solution path is being sought over a range of λ.
These set of equations can be expressed in terms of the power load, power generation and power injection at each bus i as follows:
(3)
Figure 1. Predictor-corrector scheme used in the CPF.
Source: Venkataramana & Christy, 1992.
(4)
Where, (Equations 5 and 6)
and Pd,i (λ) and Qd,i (λ) are the active and reactive power load at the bus i, respectively, Pg,i (λ) and Qg,i are the active and reactive power generation at the bus i, respectively. In (5) and (6), Gij is the ij-th element of the bus conductance matrix G, Bij is the ij-th element of the bus susceptance matrix B, vi and θi are the bus voltage magnitude and phase at bus i.
(5)
(6)
To simulate the load change, Pd,i (λ) and Qd,i terms are modified as follows:
(7) 
(8)
Where Pd,i,0 and Qd,i,0 are the original active and reactive power load at bus i, respectively, and Kd,i is a multiplier to designate the rate of load change at bus i as λ changes.
Similar to the power load, power generation is also modified as follow:
(9)
Where Pg,i,0 is the active power generation at bus i in the base case and Kg,i is a constant used to specify the rate of generation change at bus i as λ changes.
When the base case solution is obtained, the prediction of the next solution is made by taking an appropriated size step in a direction tangent to the solution path. Thus, the tangent vector is obtained by taking the derivative of both sides of the power flow equations.
Factorizing,
(10)
(11)
As it can be noticed in (11), the matrix of partial derivative [Fv Fθ Fλ] represent the conventional power flow Jacobian augmented by one column (Fλ), while the vector of differentials [dv dθ dλ]T is the vector being sought.
At this point, it is important to visualise that as one additional unknown was added when λ was inserted into the power flow equations, but the number of equations remained unchanged, and, therefore, one more equation is now needed. This problem is solved by choosing a non-zero magnitude for one of the components of the tangent vector.
(12)
Where r represents the tangent vector with a corresponding dimension c = 2n1 + n2 + 1 (n1 and n2 are the number of load and generation buses, respectively). This results in equation 13. Where ek is a row vector with all elements equal to zero except the k-th, which is equal to one. When the tangent vector is obtained by solving (13), the prediction can be made as follows:
(13)
(14)
Where l+1 represents the predicted solution for a subsequent value of λ, and σ is a scalar that designated the step size. This step size should be selected so that the predicted solution is within the radius of convergence of the corrector. When the prediction is made, the corrector is developed. Local parameterization is used in this paper, thus,
the reformulated power flow equations (2) are expressed as follow:
(15)
Where η is an appropriate value for the k-th
element of x.
The continuation parameter is selected at each step based on the state variable with the largest tangent vector component. The load parameter λ is a good guess for power systems. Then, equation is used to calculate xk, the continuation parameter.
(16)
Where k corresponds to the component of the
tangent vector that is maximal. As the VCP represented the point at which dλ = 0, by computing dλ and by monitoring when this value changes from positive to negative the VCP is determined.
The optimal power flow algorithm uses optimization techniques to find the optimal state of a power system while satisfying a set of physical and operational constraints (Momoh & Shu, 1999). Most OPF problems can be represented as follows:
(17)
Where x is the vector of explicit decision variables, f(x) is a scalar function representing the power system’s planning or operation optimization goal, g(x) is a nonlinear vector containing the equality constraints on the system, h(x) is a nonlinear vector of functional variables with lower bound hmin and upper bound hmax representing specific operational limits on the system. Îx is a vector with components of x that have finite bounds (lower bound xmin and upper bound xmax) corresponding to physical and specified operational limits on the system, Î is an incidence matrix to obtain Îx from x. It is assumed that f(x), g(x) and h(x) are continuous and differentiable at least twice for x. In this paper, n number of primal variables xi, m number of equality constraints, p number of nonlinear functional bound constraints, and q number of simple bound constraints are considered.
Similar to the CPF, and as the equality constraints, the OPF algorithm considered a double set of Nb, nonlinear active and reactive power balance equations at each bus i. These are formulated similarly as the CPF algorithm, but considering λ = 0, this is:
(18)
(19)
Where the active and reactive power injections at bus i are computed by using (5) and (6), respectively. Pd,i and Qd,i are the active and reactive power load at bus i, respectively. Pg,i and Qg,i are the active and reactive power generation at bus i, respectively.
The inequality constraints are formed of two sets of Nl, where Nl represents the number of branches in the power system, branch flow limits as nonlinear functions of the bus voltage magnitudes and phases, one for the sending end i and one for the receiving end j of each branch.
(20)
(21)
Where the power flow in branches is computed by
considering the tap setting representation and the
π-model of the transmission line as follows:
(22)
(23)
In (22) and (23), tij, gij, bij, and b sh are the tap setting, the series conductance, the series
susceptance and the shunt susceptance of the branch (i, j), respectively. In these equations, tij = 1 if the branch (i, j) is a transmission line and bijsh = 0 if branch (i, j) is a transformer.
In the algorithm, the vector of explicit decision variables x is represented by the vector of voltage magnitudes v, the vector of voltage phases θ (the reference bus is defined as θ1 = 0°), the vector transformer tap settings t, and the vectors of active (Pg) and reactive (Qg) power generations as it can be noticed in (24).
(24)
Each bus, tap setting, and generators would
include a double set of finite bounds which is
represented as follows:
(25)
Figure 2. Flowchart for correct coordination of the distance protection to avoid voltage collapse.
Figure 3. Single line diagram of the IEEE 14- bus test system.
Source: University of Washington, 2010.
Figure 4. λ-V curve for base case on bus 4 and bus 5.
As voltage stability problems are under studied, the system loss for a given load and generation represents the objective function in the proposed OPF formulation. This is a function of the reactiv1 e power flow, which is mainly dictated
1
by the system node voltage magnitudes, as the
variation in the node phases calculated in the
initial load flow is small.
Where vi is the phasor of positive sequence voltage at the relay location and Iij is the phasor of positive sequence current flowing from bus i to bus j. Equation (27) can be also expressed in terms of the transferred complex power as follow.
(26)
Or:
(28)
High voltage transmission lines are commonly protected by distance protection schemes (Jonsson, 2003; Jonsson & Daalder, 2003; Yang, Shi, & X.,
2006). The main objective of the protection devi- ces is to clear the fault and isolate the faulted trans- mission line as quickly as possible considering the critical clearing time previously determined by grid codes. On the other hand, voltage collap- se sometimes occurs after undesirable trip of the transmission lines (Johansson & Sjögren, 1995; Jonsson, 2001, 2003; Ohno & Imai, 2006; Sancho & Vega, 2001). Thus, the correct coordination of the distance protections to reduce the number of undesirable trips is vital in current power systems.
The third zone in the distance protection scheme has been used as a back-up zone (Abba-Aliyu, 2009; Horowits & Phadke, 2006; Jonsson & Daalder, 2003). Therefore, the incorrect setting in this zone might produce an undesirable disconnection of the transmission line during voltage instability, accelerating the process of blackout.
As the voltage instability is a symmetric phe- nomenon, i.e. only positive sequence components are involved (Jonsson, 2003), the impedance seen by the relay can be modelled by as follow.
(27)
(29)
According to (29), if the voltage at the relay location and power in the transmission line, which is protected by the relay, are known, it is possible to determine the impedance to set the protection device in order to avoid line disconnection under voltage instability conditions.
The methodology to determine the smallest impedance is shown in Figure 2. This method is iteratively carried out for the each considered contingency. In this figure, the λ-V curve is obtained for those critical buses to determine the critical point as previously shown in Figure 1. When the MLP is known, the CPF and the OPF algorithms are run for a load condition around this MLP. The electrical variables, voltage, current, and power flow, are then saved to compute the impedance according to each method. When these impedances to set the back-up zone are known, a direct comparison of both methods is carried out to select the smallest impedance.
In order to demonstrate and validate the proposed methodology, simulations are carried out on the IEEE
14-bus test system University of Washington, 2010. The topology of this test system is shown in Figure
3. The IEEE 14-bus has 5 generating units with bus
1 chosen as the slack bus. The system includes
16 transmission lines, 4 transformers and 11 load buses. The system has 7 transmission lines in the high voltage side. Because of this, the system has
14 different distance relays, each one in both sides of the high voltage transmission lines.
Table l. Results from CPF simulations to coordinate distance protection to avoid voltage collapse
Figure 5. A-V curve for buses 4 and 5 under the outege of line 1-5.
≥
Figure 6. λ-V curve for buses 4 and 5 under the outege of line 4-5.
Using the CPF algorithm, the λ-V curves are built for critical buses. Figure 4 shows the λ-V curve at bus 4 and bus 5 for the case when no contingency is considered. As it can be calculated from Figure 4, the critical point is reach when
λcritical = 4.0303 p.u.
Different contingencies in the high voltage level, where distance protections are placed, are considered to analyse the system response. For every contingency, the voltage, active, and reactive power in the transmission lines are computed. Then, by using the proposed methodology, the smallest impedance for each side of the transmission line can be determined. These calculated impedances are shown in Table 1. This impedance (in per unit) will be then used to adjust the impedance setting for the back-up zone to avoid undesirable transmission line disconnections.
Analysing all contingencies, two critical scenarios are identified. The first one happens when the transmission line 1-5 is disconnected and the second one occurs with the outage of transmission line 4-5. For these two cases, important reduction of the voltage levels and high power flows through the transmission lines are obtained. Figure 5 and 6 show the λ-V curves at bus 4 and bus but considering transmission lines
1-5 and 4-5 disconnected, respectively. As it can be noticed, the maximum loading point for these cases are obtained when λcritical = 3.6539 p.u. and when λcritical = 3.9348 p.u., respectively.
Similar simulations are carried out for all ou- tages considered in Table 1 but using the OPF al- gorithm presented in Section 3. By minimising the power losses, the reactive sources are improved and, therefore, the voltage stability margins are enhanced.
The simulations using the proposed OPF are carried out for the same operating point as it was obtained from the CPF, i.e. in the MLP. These si- mulation results are summarized in Table 2. It is necessary to say that when using the OPF, the si- mulations are not simulated considering exactly the same MLP found by the λ-V curves due to OPF convergence problems when approaching the VCP, therefore, the closest point in this curve is used.
As it is noticed in Table 2, voltage magnitudes are clearly improved i.e. these are closer to 1 p.u. Also, in comparison with the CPF solution, the im- pedances are larger. By using the OPF algorithm, it is noticed that voltage stability problem is not an issue. In fact, as the constraints included in the formulation consider voltage limits as well as loa- dability of the branches, the obtained impedances are expected to be larger, as it is noticed in Table 2.
When the impedance of each transmission lines is computed by both algorithms, a direct comparison is carried out. As it is noticed in Table 3, the calculated impedance from the OPF algorithm to set the back-up zone in the protection devices is considerably bigger than the same setting using the CPF method.
This result is due to the considered constraints in the OPF formulation. As it can be noticed in Table 2, the voltage magnitude at each bus is closer to the nominal voltage when using the OPF method. In addition, the OPF method reduces the reactive power losses in the transmission lines. As a consequence, this minimization produce that power flows through the transmission lines are smaller than the power flows obtained from the CPF method.
As it can be noticed in Table 3, the correct settings to avoid an undesirable trip of the distance protections are obtained by the CPF. Table 4 shows the final settings in distance protection to avoid voltage collapse in IEEE 14-bus test system.
A new methodology to coordinate the back- up zone in the distance protection devices using the two algorithms was proposed in this paper. Different contingencies were considered and the parameters to set the protection were calculated for both algorithms. A direct comparison was then carried out and the smallest impedance computed was selected. Simulations on the IEEE 14-bus test system were done to demonstrate and validate the proposed methodology. According to the results, the CPF method should be used to compute the parameters to set the back-up zone in the distance protection. When using the OPF algorithm, the voltage magnitude at all buses is close to the nominal value and the power flow in transmission lines is low, in comparison with the CPF. This result is obtained because of the constraints introduced in the OPF formulation. By setting the back-up zone with the calculated parameters, undesirable disconnection of transmission lines can be avoided and, therefore, the power system would be operable under stressful conditions.
Bij ij-th element of the bus susceptance matrix B
bij series susceptance of the branch
(i, j)
b sh shunt susceptance of the branch
Pg,i active power generation at bus i
Pg,i,0 active power generation at bus i in
the base case
Pi active power injected at bus i
Pij active power from the sending end
in the transmission line (i, j)
Pji active power from the receiving end in the transmission line (i, j)
(i, j)
c dimension of the tangent vector
CPF Continuation Power Flow
scalar nonlinear objective function
Qd,i reactive power load at bus i
Qd,i,0 reactive power load at bus i in the
base case
Qg vector of reactive power
f(·)
in the OPF formulation with n
unknowns
generation
Qg,i reactive power generation at bus i
g(·) m-vector of equality constraints in
the OPF formulation
Gij ij-th element of the bus
conductance matrix G
Gij ij-th element of the bus
conductance matrix G
gij series conductance of the branch
(i, j)
h(·) p-vector of functional variables in the OPF formulation
hmax upper bound of h(·)
hmin lower bound of h(·)
IEEE Institute of Electrical and
Electronics Engineers
incidence matrix to obtain Îx,
Î which has q number of simple bound constraints
Qi reactive power injected at bus i
Qij reactive power from the sending
end in the transmission line (i, j)
Qji reactive power from the receiving end in the transmission line (i, j)
r tangent vector
t vector of transformer tap settings tij tap setting of the branch (i, j) VCP Voltage Collapse Point
v vector of voltage magnitudes
vi bus voltage magnitude at bus i
vi phasor of positive sequence
voltage at the relay location
VSM Voltage Stability Margin
x vector of explicit decision variables
I -1 phasor of positive sequence
Zrelay
impedance seen by the relay
current flowing from bus i to bus j
Kd,i multiplier to designate the rate of load change at bus i
Kg,i multiplier to designate the rate of generation change at bus i
MLP Maximum Loading Point
n1 number of load buses
n2 number of generation buses
Nb number of buses in the power
systems
Nl number of transmission lines in the power systems
OLTC On-Load Tap Changers OPF Optimal Power Flow p.u Per unit
Pd,i active power load at bus i
Pd,i,0 active power load at bus i in the
base case
Pg vector of active power generation
θi bus voltage phase at bus i
λ loading parameter
λcritical maximum loading parameter
* represent the conjugated of the complex number
σ scalar that designated the step size
θ vector of voltage phases
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Lecturer in the Department of Electric Machines and Power Systems, School of Electrical Engi- neering, University of Costa Rica and Ph.D. stu- dent at the University of Manchester. His main research interests are application of intelligent methods to power system restoration, and con- trolled islanding, power system monitoring, pro- tection and control, voltage stability assessment, and power system dynamics. At the end of his Ph.D. studies, he will return to the University of Costa Rica as a Professor and Researcher.
E-mail: jairoquirostortos@ieee.org
Cathedratic Professor in the Department of Electric Machines and Power Systems, School of Electrical Engineering, University of Costa Rica. His main research interests are nonlinear dynamics, FACTS devices and wind generation.
E-mail: earaya@eie.ucr.ac.cr