Security-Constrained Power System Transfer Capability Assessment S. H. Goh*
[email protected]
Z. Y. Dong
[email protected]
T. K. Saha
[email protected]
School of Information Technology and Electrical Engineering The University of Queensland Brisbane Qld 4072 Australia Abstract Till now, many transfer capability techniques are performed in a simplified manner, with limited consideration of security constraints. This is unacceptable as the risks for not evaluating security impacts are high. There have been doubts on whether current techniques could provide secure transfer capability assessment and thus introducing a new challenge. This paper presents an optimal power flow based (OPF-based) approach for transfer capability assessment incorporating a security index which determines the system security level utilizing various techniques. Line stability indices are also implemented to quantify how ‘close’ the system is to the point of static voltage collapse and to identify the critical lines connecting weak buses in a stressed system. A case study of a 9-bus system is presented to demonstrate the proposed technique. Keywords:
Total transfer capability, available transfer capability, power system security assessment, optimal power flow, power flow feasibility limits, stability indices.
1. Introduction Over the past decade, electric power industries in many countries including Australia, has undergone market restructuring and privatization in a bid to promote open access. This eventually leads to networks being much more heavily-loaded, and an emerging trend of greater network interconnections for increasing electricity trades, thus transfer capability becomes a key issue in power system operation and planning [1, 2]. There are a number of interconnector projects undergoing in Australia over the last few years, namely Murraylink (VIC and SA interconnection); SNI (SA and NSW interconnection); and SNOVIC (Snowy region of NSW – VIC interconnection). As shown in [3], a proposed combination of the SNI and SNOVIC interconnections is aimed at maximizing the net economic benefits out of increased transfer capability. Such projects involve millions of dollars, and important decisions of these projects that are to be made will be based mainly on transfer capability assessment. Any inaccuracy of this assessment can impose a drastic negative effect on the expected net benefits of such projects. Many currently available techniques are performed in a very simplified way, with limited consideration of security constraints. In many cases, only thermal limits and voltage limits are considered. There has been a lack of a systematic approach to properly include security constraints for transfer capability *Corresponding author: S. H. Goh Email:
[email protected] Tel: +61-7-33658309
assessment. For example, transfer capability of the interconnection in many occasions is assessed only by repeated power flow analyses with various contingencies under different forecasted load conditions. This may introduce risks of not evaluating the security impacts on transfer capability assessment, and consequences can be very serious in a competitive market situation. To summarize, there are uncertainties on whether current techniques could provide secure transfer capability assessment, hence introducing a new challenge for further investigation. 2.
State-of-Art of Transfer Capability
Transfer capability can be represented in several ways, among which the available transfer capability (ATC) and the total transfer capability (TTC) are the two most widely used ones. ATC of a power system as defined by the North American Electric Reliability Council (NERC) is the measure of transfer capability remaining in the physical transmission network for further commercial activity over and above already committed uses [4]. To determine TTC, the objective is to maximize the power transfer between two areas without any violation of thermal, voltage and stability limits. A standard TTC problem formulation can be written as follows [1]: Maximize
Pr =
∑
Pkm m∈R ,k∉R
(1)
Subjected to Pi − ViV jYij cos θ ij + δ i − δ j = 0
(
∑ j∈N
Qi −
)
(2)
+ δi − δ j = 0
)
(3)
Pg ,min ≤ Pg ≤ Pg ,max
(4)
Q g ,min ≤ Qg ≤ Qg ,max
(5)
S ij ≤ S ij ,max
(6)
Vi,min ≤ Vi ≤ Vi,max
(7)
∑V V Y sin (θ i
j∈N
j ij
ij
where Eqn. (1) is the interchange real power between the sending and receiving areas; k denotes buses not in receiving area and m denotes buses in receiving area. R is the set of buses in receiving area and N is the set of all the buses. Eqn. (2-3) are the power flow constraints. Eqn. (4-5) represent real and reactive power generation limits, Eqn. (6) stands for the thermal constraints, Eqn. (7) is the voltage level constraint. See [1] for more details. It should be noted that other constraints may also be included in this approach depending on the specific transfer capability assessment requirement ATC assessment requires serious consideration of network topologies, generation dispatches, customer demand levels, system contingencies, and other simultaneous transactions on the network [5, 6]. Moreover the trend of greater interconnections, either for economic or reliability reasons, introduces additional complexity to the quantification of ATC [7]. In recent years, there has been a keen rapidlygrowing interest for power engineers to formulate and solve this complex transfer capability problem. As a result, many techniques and methods have been introduced since then. Although many methods and techniques have been developed, very few methods have proved to be capable of comprehensively assess security constraints during transfer capability assessment. The most commonly-used methods can be classified into the following three categories: continuation power flow (CPF) method; repeated power flow (RPF) method; and optimal power flow (OPF) method [8]. CPF is first introduced for determining the maximum loadability, however it is adaptable for other applications including ATC computation without changing its principles. The advantage of CPF is that it will not encounter numerical difficulties of illconditioned power flow equations, thus CPF yields solutions even at voltage collapse points. However a
major disadvantage is that it involves complicated implementation of its parameterization, predictor, corrector and step-size control elements [9,10]. OPF and security-constrained OPF (SCOPF) are powerful tools that have been under very active development for the past 30 over years [8]. They can be implemented by using different optimization approaches, for example, using neural networks [1], sequential quadratic programming (SQP) [2], bi-level optimization [11], and linear programming (LP) [12,13]. OPF can be used to maximize the power transfer between two areas assuming that all OPFoptimized parameters can be centrally dispatched. Ejebe et al. [14] presents a linear method utilizing distribution factors which has been proven to be very efficient in computing ATC values. Another good example is the linear step ATC solver of PowerWorld Simulator [15]. These methods are extremely computationally efficient and give very good ATC approximations, thus making them very suitable for online dispatches whereby computation time is the major concern. However, they do introduce certain degree of error when compared to more precise nonlinear methods [8]. 3. State-of-Art of Security Assessment It is well-known that power systems are highly nonlinear and complex especially under competitive electricity market situations. A power system can be modeled by differential and algebraic equations as follows [16]: •
x s = f (x s , xa , p ) 0 = g (x s , x a , p ) where xs is a vector of dynamic state variables, xa is a vector of algebraic variables, and p is a vector of power system parameters.
3.1 Security Assessment Techniques Computational approaches for locating the aperiodic and oscillatory stability conditions include continuation and direct methods. The equilibrium conditions of the nonlinear system given above are substituted with the simplified expression of F (x, p ) = 0 . Aperiodic instability points can be located by solving the following set of equations,
F (x, p ) = 0 Fx v = 0 or wT Fx = 0 v = 1 or w = 1
where v, w ∈ R N is the left and right eigenvector of the power flow Jacobian Fx at an equilibrium
defined by F (x, p ) = 0 . The last equation ensures the nontrivial condition. The problem can be solved The with Newton-Raphson-Seydel method. neighboring equilibrium points close to the saddle node bifurcation point can be calculated by solving the following equations, F (x, p ) = 0
(Fx − εI )v = 0
where I is the identity matrix of the same order as the Jacobian Fx , ε is a small real number. It is evident that the bifurcation point is obtained with ε = 0 [17].
Oscillatory instability is associated with Hopf bifurcation which is featured by a pair of conjugate pure imaginary eigenvalues. It can be computed by solving the equations below. Here the system Jacobian is the reduced form, J s = f x − f p g −p1 g x . F ( x, p ) = 0
J sT J sT
(x, p )v'+wv' ' = 0 (x, p )v' '−wv' = 0 v =1
where 0 ± jw are the eigenvalues corresponding to the Hopf bifurcation, and v = v'± jv ' ' are the corresponding left eigenvalues. The last equation is the nontrivial condition. Again it can be solved by Newton like optimization method. The above two approaches belong to the direct method. Makarov et al. [16] developed a general method, which is capable of revealing most of the small disturbance stability conditions in one optimization approach in the parameter space. Given the nonlinear system Jacobian as J , the general method is based on solving the following optimization problem: Subjected to
α 2 ⇒ min/ max f (x, p0 + τ∆p ) = 0
J T (x, p0 + τ∆p )l '−al '+ wl ' ' = 0
J T (x, p0 + τ∆p )l ' '−al ' '− wl ' = 0 li '−1 = 0 li ' ' = 0 where ω and w are the real and imaginary parts of an eigenvalue of J and l '+ jl ' ' is the corresponding eigenvector. The last two equations are to make sure
the conditions are nontrivial. Examples of utilizing this technique are given in [17].
3.2 Power Flow Feasibility Limit Index Security constraint check is a complex task for transfer capability assessment. However it is very important in today's electricity industry under a market situation whereby power systems have been stressed toward its security limits. There are many methods of assessing security constraints and on how to incorporate them into the system planning problem. In this paper, we proposed the framework of security-constrained transfer capability assessment. For simplicity, we use the critical eigenvalue of power flow Jacobian (or feasibility limit index) to indicate the security constraints during transfer capability assessment at this stage. As shown in [18], the conventional power flow Jacobian matrix can be written as follows:
F J (V , θ ) = θ Gθ
FV GV where Fθ, FV, Gθ and GV are four sub-matrices: containing the partial derivatives of real power (F) and reactive power (G) with respect to voltage magnitudes (V) and angles (θ). By applying singular value decomposition (svd) to the power flow Jacobian (J), the matrix decomposition will become as follows [18]:
J = U Σ VT =
n
∑σ
i
ui v i T
i =1
where U and V are n x n orthonormal matrices, the singular vectors ui and vi are the columns of the matrices U and V respectively, and Σ is a diagonal matrix where σi ≥ 0 for all i. Σ(J) = diag { σi (J) } i = 1, 2, …, n The diagonal elements in the matrix ∑ are usually ordered such that σ1 ≥ σ2 ≥ … ≥ σn ≥ 0. A fast and efficient power flow feasibility limit index can be easily found from the minimum singular value. If this minimum singular value is equal to zero, then the studied power flow Jacobian matrix J is singular (determinant of J is equal to zero), thus no power flow solution can be obtained − see [18] for more details. Hence, from [18], we introduced this power flow feasibility index termed as λ (proximity index to singularity of J), λ = min ( svd ( J ) ) ≥ 0 where λ must be greater than zero to have a feasible load flow solution (determinant of J is not zero).
3.3 Line Stability Index Moghavvemi et al. [19] and Musirin et al. [20] both proposed line stability indices for voltage collapse prediction and critical line identification (leading to weak buses). This paper also incorporates Moghavvemi’s formulation of line stability index, which is as follows [19] – see Fig. 1– :
Lmn =
4 XQr
[Vs sin (θ − δ )]2
≤1
where θ is the impedance angle and δ = δ 1 − δ 2 is the voltage angles difference. R + jX
s
r
VS ∠δ1
Vr ∠δ 2
S s = Ps + jQs
S r = Pr + jQr
∑
Lmn is the stability index of the line connecting the two buses, and will be computed for all transmission lines in a power system. The system is said to be stable, in the sense of transmission lines, as long as Lmn remains much less than 1; and approaches 1 towards the point of bifurcation. The most critical line connecting the weak buses in the system can be easily identified from the value of Lmn closest to 1 − see [19] for more details. 4.
Security-Constrained Transfer Capability Assessment 4.1 Problem Formulation The proposed OPF-based technique enable increment of the complex load with uniform power factor at every PQ bus in the sink area, and increment of the injected real power at PV buses in the source area as well. The mathematical formulation can be expressed as follows [1]:
∑P
w1
Gi
+ w2 λ + w3 Lmn
(8)
i∈source
Subjected to all bus
PGi − PDi −
∑ V V (G i
j
all bus
QGi − QDi −
∑V V (G i
j =1
j
)
(9)
)
(10)
cos δ ij + Bij sin δ ij = 0
ij
j =1
ij
sin δ ij + Bij cos δ ij = 0
PGi ,min ≤ PGi ≤ PGi ,max (11)
QGi ,min ≤ QGi ≤ QGi,max (12) S ij ≤ S ij ,max
λ (P, Q, V , δ ) ≥ 0 (15) Lmn (Q, V , X , θ , δ ) ≤ 1 (16) where w1, w2 and w3 are weighing factors to denote that the emphasis is not only placed solely on source real power injection, but also the relevant security or stability indices. Note that w1 + w2 + w3 = 1 , and the actual weighing factors are problem dependent. Eqn. (9-10) are power flow constraints, Eqn. (11-12) represents real and reactive power generation limits, Eqn. (13) stands for the thermal constraints, Eqn. (14) is the voltage level constraint, and Eqn. (15) is the power flow feasibility limit index [18], and Eqn. (16) is the line stability index [19]. ATC can then be expressed as follows: ATC = PGimax − PGi0
Figure 1 – Typical oneline of transmission line
Maximize
Vi,min ≤ Vi ≤ Vi,max (14)
(13)
where
∑P
i∈source
Gimax
∑
i∈source
i∈source
is the total maximum source real
power injection, and
∑P
i∈source
Gi0
is the total base case
source injection. 4.2 Computation Procedures This section summarizes the steps required to compute ATC for a selected source/sink transfer case. 1. Select a transfer case (normal or contingency). 2. Establish and solve base case OPF. 3. Increment complex sink load with uniform power factor by a step increase ∆. 4. Solve power flow problem of step 3 by OPF. 5. Check solution of step 4 for violations. If no violation, go back to step 3. 6. If there is any violations, decrease complex sink load with minimum possible amount to eliminate them. This is the ATC number for the selected case. 7. Return to step 1 to select another transfer case. Go to step 8 if there is no more case. 8. Compute ATC for source/sink transfer case, which is the minimum value of all ATC values from every case considered. 5. Case Study This section presents a case study to illustrate the basic notion of transfer capability. The proposed technique is firstly applied to a 3-generator 9-bus system [21] – see Fig. 2– for transfer capability assessment. This 9-bus system is divided into two areas, and ATC is computed for area 1 to 2 and vice
versa. Detailed parameter values of this system can be found in [21]. Note that, for simplicity, only generators and tie-lines contingencies are considered here, and the final ATC value is determined from the most limiting contingency case.
aimed at developing multi-objective algorithms to maximize power transfer without compromising system security, furthermore it will be able to freely manipulate source generation and sink load, all in one optimization approach. Table 1 - ATC calculations for area 1 to area 2 transfer
2
3 7
2
3 6
8 9
5
Area 2
4 Area 1 1
λ
Lmn
Critical line
104.65
0.4322
0.3266
5-6
57.43
0.4668
0.1934
1-4
0
1.1247
0.3246
8-9
63.38
0.9266
0.2513
5-6
88.39
0.2958
0.1999
5-6
Transfer Case
ATC (MW)
Normal Line 5-6 outage Line 4-9 outage Gen 2 outage Gen 3 outage
Final ATC (MW)
0
1 Table 2 - ATC calculations for area 2 to area 1 transfer Figure 2 - The 3-generator 9-bus system
ATC values obtained give a very good illustration of the remaining system’s strength or an area’s strength to facilitate additional power transfers either for economic or reliability purposes. From Table 1, the final ATC value is found to be zero when one of the tie-lines (line 4-9) experiences an outage. When this happens, area 1 does not have the capability for any additional power transfer to area 2. From Table 2, it can be seen that area 2 is ‘stronger’ than area 1, having a final ATC value of 29.52 MW. The critical lines connecting the weak buses in the system are also identified in all transfer cases. Increased power transfers may be achieved by having appropriate reactive support at these weak buses. In addition, the power flow feasibility limit index (λ) provides a proximity to singularity of the power flow Jacobian matrix J that indicates the actual system security levels corresponding to each transfer cases. This proposed technique can also be applied to large practical power system. By applying this technique, both system transfer capability and the corresponding security levels can be obtained. Depending on the different security levels the system manager can then choose specific plans with associated risks as revealed by the security level indices in grid expansion planning or planning for new market participants. One of the limitations of this proposed OPF-based technique is that it lacks the flexibility to manipulate source generation and sink load. Future work is
Transfer Case
ATC (MW)
λ
Lmn
Critical line
Normal
145.20
0.2758
0.2852
8-9
29.52
0.4246
0.2397
8-9
103.91
0.4461
0.3322
8-9
75.96
0.3928
0.2324
8-9
95.69
0.0889
0.3141
8-9
53.10
0.2863
0.2130
8-9
Line 5-6 outage Line 4-9 outage Gen 1 outage Gen 2 outage Gen 3 outage
Final ATC (MW)
29.52
6. Conclusions It has been widely known that ATC values are indispensable to power system operators and planners today. System operators utilize these ATC values to determine how ‘strong’ a system or an area is, meaning their ability to facilitate additional power transfers either for economic or reliability purposes. These values are also vital to system planners for grid expansion or planning for market entry for new participants etc.
However many current transfer capability techniques neglect power system security considerations, which may introduce high risks in a market situation today. Many past experiences and incidents have highlighted the importance of power system security, and this should never be neglected as the cost of its outcome can be catastrophic. This paper presents a security-constrained transfer capability method to reduce such risks in power system planning based on secure ATC information.
This paper firstly reviews the state-of-art of transfer capability and security assessment in power systems in brief. Then, it proposes an OPF-based technique used to compute ATC values providing relevant security level of each transfer case, and moreover critical lines (connecting weak buses) can also be identified. Utilizing this proposed technique, system operators and planners can then safely use the secure ATC information revealed for whatever operation or planning purposes. 7.
References
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