Abstract-Electric power engineers and researchers need ap propriate randomly generated grid network topologies for Monte. Carlo experiments to test and ...
A Novel Measure to Characterize Bus Type Assignments of Realistic Power Grids Zhifang Wang, Seyyed Hamid Elyas
Robert J. Thomas
Electrical and Computer Engineering Virginia Commonwealth University Richmond, VA, USA {zfwang,elyassh}@vcu.edu
Electrical and Computer Engineering Cornell University Ithaca, NY, USA rjtl @cornell.edu
Abstract-Electric power engineers and researchers need ap propriate randomly generated grid network topologies for Monte Carlo experiments to test and demonstrate new concepts and methods. Our previous work proposed a random topology power grid model, called RT-nested-smallworld, based on a compre hensive study of the real-world grid topologies and electrical properties. The proposed model can be used to produce a sufficiently large number of power grid test cases with scalable network size featuring the same kind of small-world topology and electrical characteristics found in realistic grids. However, the proposed RT-power grid model has a shortcoming that is its random assignment of bus types. And our recent study has shown that the bus type assignment of a realistic power grid is not random but a correlated one. Generally speaking,the buses in a power grid can be grouped into three categories: generation buses (G), load buses
(L), and connection buses (C). When studying
the dynamics of a grid we need to take into account not only its "electrical" topology but also the generation and load settings including their locations, which are equivalent to the bus type assignments in our model. In this paper we define a novel measure to characterize typical bus type assignments of realistic power grids. The proposed measure will enable the recognition of the specific set of bus type assignments, consistent with that of a realistic grid, from those generated from random permutation. This will prove useful for designing an optimal algorithm to improve our random topology power grid modeling. The proposed measure, called the Bus Type Entropy, incorpo rates both bus type ratios and the link type ratios. Therefore it provides a quantitative means to identify the presence of correlation among the bus type assignments of a realistic grid. We then experiment with this entropy measure on a NYISO system and the IEEE 300-bus system. The numerical results from both test cases verify the effectiveness of the proposed measure to characterize the bus type assignment of a real-world power grid.
Index Terms-Power grid modeling, random topology, vulner ability, graph network.
I.
INTRODUCT ION
Electric power grid is one of the most important critical in frastructures. It is widely distributed and contains vast dynamic complexities resulting from the large number of interconnected grid components [1]. During the past several decades under risk operating of the US and other electric power grid has received considerable attention, because outages of such a system may lead to severe impacts on the essential functioning of a society and its economy.
In order to demonstrate and test new concepts and methods for grid vulnerability analysis and controls, electric power engineers and researchers may find appropriate randomly generated grid network topologies useful for Monte Carlo ex periments. If the random networks are truly representative, the new concepts or methods, if tested well in this environment, will test well on any instance of such a network, such as the IEEE-30, 57, 118, or 300-bus systems [2] or other existing realistic or synthetic grid models [3]-[6]. Many researchers have also realized the need to generate random-topology power grid test cases with scalable network size. A number of models have been proposed in the literature based on observed statistical characteristics of the grids. For example, reference [3] proposed a Tree-topology power grid model to study power grid robustness and to detect critical points and transitions in transmission flows to cause cascading failure blackouts. Reference [5] used Ring-structured power grid topologies to study the pattern and speed of contingency or disturbance propagation. Reference [6] first proposed sta tistically modeling a power grid as a small-world network in their work on random graphs. Reference [7] provided a statistical model for power networks in an effort to grasp what class of communication network topologies need to match an underlying power network. Reference [8] used a small-world graph model to study the intrinsic spreading mechanism of the chain failure in a large-scale grid. All these models provide useful perspectives of power grid characteristics. However, the topology of the generated power grids, such as the ring or tree-like structures and the small-world graph networks, fails to accurately or fully represent a realistic power system, especially its distinct sparse connectivity and scaling property versus the grid size [9]. We note that power grid networks are much more than a graph topology. In order to facilitate numerical simulations for grid controls and operations, one also needs to include realistic electrical parameter settings such as line impedances, the generation and load settings. Our previous work [9] proposed a random topology power grid model, called RT-nested-smallworld, based on a com prehensive study of the topology and electrical properties of realistic grids. The proposed model can be used to produce a large number of power grid test cases with scalable network size featuring the same kind of small-world topology and
TABLE I RATIO OF Bus T YPES IN REAL-WORLD POWER NETWORKS
(n,m)
rG/L/c(%)
IEEE-30
(30,41)
20/60/20
IEEE-57
(57,78)
12/62/26
(l18,179)
46/46/08
IEEE-1l8 IEEE-300 NYISO
(300, 409)
23/55/22
(2935,6567)
33/44/23
electrical characteristics found in realistic power grids. This model has been adopted by several researchers to study grid vulnerabilities, PMU placement,smart grid communication ar chitecture, etc. However, the proposed RT-power grid model has a shortcoming that is the random assignments of bus types. Generally speaking, the buses in a grid can be grouped into three categories as follows with minor overlaps: I G L C
the generation buses which connect generators, the load buses which support custom demands, the connection buses which form the transmission network.
In a typical power grid, 10",40% are generation buses, 40",60% are load buses, and 10",20% are connection buses, as seen in Table I. When studying the dynamics of a grid we not only need to consider its electrical topology but also the generation and load settings including their locations, due to the immediate dependence of the grid dynamics on the latter. In our recent study [10] we found that there exists non-trivial correlation between the bus types (GIL/C) and other network topology metrics such as the node degree distribution and the clustering coefficient; and that random permutation of bus type assign ments in a grid, although retaining the same ratios of GILIC buses, may significantly change the grid dynamics such as its vulnerability to cascading failures. Therefore, in order to accurately formulate a random topology power grid we need to assign bus types in a way consistent with that of a realistic grid. In this paper we seek an effective way to characterize the bus type assignment of realistic power grids, by defining a novel measure, called the Bus Type Entropy. The proposed measure will allow us to recognize the specific set of bus type assignments, either directly extracted from a realistic grid or formulated consistent with the former, from those randomly generated from permutation. Therefore it will provide very useful guidance for an optimal algorithm to improve our random topology power grid modeling. The proposed measure incorporates both bus type ratios and the link type ratios. Therefore it provides a quantitative means to identify the presence of correlation among the bus type assignments of a realistic grid. We then experiment this entropy measure on the NYISO and the IEEE 300-bus systems, I It is possible that a smaU number of buses in the grid belong to more than one categories.
where the former represents an interconnect grid of US and the latter a synthetic grid representing part of the New England System. The numerical results from both test cases verify the effectiveness of the proposed measure to characterize the bus type assignment of a real-world power grid. The rest of the paper is organized as follows: Section II presents the system model for our analysis of a power grid net work; Section III gives the definition of our proposed measure to characterize the realistic grid's bus type assignment; Section IV examines the effectiveness of the proposed measure with the experiments on a NYISO reduced system and the IEEE 300 bus system, and Section V concludes the paper. II.
S Y ST E M MODEL
The dynamics of a power system can be dominated mainly by the generationlload settings and its network admittance matrix Y. The generation and load settings, given the bus type assignment, may assume independent statistical models, as have been discussed in [7]. The network admittance matrix can be formulated as: (1) where Zl is the line impedance, A (zll) represents a diagonal matrix with entries of {zl-l}. and A the line-node incidence matrix. For a grid network with n nodes and m lines, its line node incidence matrix A := (Al,k)mxn , arbitrarily oriented, is defined as: Al,i = 1; Al,j = -1, if the lth link is from node i to node j and Al,k = 0, for k i- i, j. The Laplacian matrix L can be obtained as L = AT A which has
£(i, j)
�
{
- 1,
k, 0,
if there exists a line (i - j), with k = - L#i L(i,j), otherewise,
for j i- i for j = i
(2) with i, j = 1, 2, ··· ,n. As indicated in [9], the Laplacian of a grid network fully defines its topology therefore all the topological metrics can be derived from it; while the admittance matrix Y can be viewed as a complex weighted Laplacian and fully describes the "electrical" topology of a grid. Reference [11] introduced a stochastic cascading failure model which incorporates the statistics of the generation and loads in a grid therefore to derive the statistics of the line flow process. For the tractability of the problem, the DC power flow approximation was utilized to characterize a power grid network, which is a standard approach widely used in optimizing flow dispatch and for assessing line overloads [12]. Consider a power grid transmission network with n buses interconnected by m transmission lines, the time-varying network flow equation can be written as follows: P(t) = B'(t)B(t), F(t) = A (Yl(t)) AB(t)
(3)
where P(t) represents the vector of injected real power, B(t) the phase angles, and F(t) the flows on the lines. The matrix B'(t) is defined as B'(t) = AT A (Yl(t)) A, where Yl(t) = Sl(t) / Xl with Xl the line reactance and Sl(t) the line state;
Sl(t) = 0 if line l is tripped (off), and Sl(t) = 1 if the line is closed (on); A (Yl(t)) represents a diagonal matrix with entries of {Yl(t), 1= 1, 2, ·· . ,m}. The vector of line states, s(t) = [Sl(t), S2(t),··· , sm(t)]T with Sl(t) E {O, 1}, is defined as the network state. III. T HE
PROPOSED ME ASURES
With the system model presented Section II, we can easily see that the dynamics of a power grid not only depend on the "electrical" topology but also the generation and load settings including the locations. The location setting of generation and loads is equivalent to the bus type assignment in our grid modeling. A. The Bus Type Entropy
']['i =
1, if bus i is a (G)eneration bus, 2, if bus i is a (L)oad bus, 3, if bus i is a (C)onnection bus,
bus type
(4)
where i = 1, 2, ··· , n. If the grid is taken as an undirected graph, each transmission line will assume one of the following six link types, i.e.
{GG, GL, GC, LL, LC, CC},
(5)
which is fully determined by the bus types of its terminal buses if given the grid topology represented by its Laplacian matrix L . Therefore the link type vector of a grid lL(']['IL) = [lLzlmx1 can be defined as a function of '][' with: lLj
E
{1, 2, 3, 4, 5, 6}, j = 1, 2, ··· ,m,
(6)
where the link type value of line j is numbered according to the set (5) in the same order. The results of [9] indicated that the bus type assignment of realistic power grid is not random but a correlated one. There exists non-trivial correlation between the bus types (GIL/C) and other network topology metrics such as the node degree distribution and the clustering coefficient, as shown in Table II and III, where (k) is the average node degree, C the clustering coefficient, and p(x, y) = represents
EJX-X)(Y-Y)} var(x)var(y)
the Pearson coefficient between two vectors x and y. However, the correlation index used in [9], when evaluated on a number of realistic power grids, is found to have both positive and negative values. The inconsistency of the correlation index value lead to difficulties when we try to characterize an appropriate bus type assignment. In this paper we seek a practical way to characterize the bus type assignment '][' of a realistic power grids, by defining a novel measure. By using this measure we will be able to recognize the specific set of bus type assignments {']['* }, extracted from the realistic grids, among all the generic ones generated from random permutation P(,][,). The proposed
pet, kt)
(k)
(k)c / (k)L / (k)c
IEEE-30
2.73
2.00 / 2.61 / 3.83
0.4147
IEEE-57
2.74
3.86 / 2.54 / 2.67
-0.2343
IEEE-ll8
3.03
3.56 / 2.44 / 3.40
-0.2087
IEEE-300
2.73
l.96 / 2.88 / 3.15
0.2621
NYISO
4.47
4.57 / 5.01 / 3.33
-0.l030
TABLE III CORRELATION BETWEEN Bus T YPE t AND CLUSTERING COEFFICIENTS C OF REAL-WORLD POWER NETWORKS
p(t, Cd
Cc / CL / Cc
Call
W(,][,)
For a n-bus, m-Iine power grid, we define the vector '][' = [']['i]nxl with:
{
TABLE II CORRELATION BETWEEN Bus T YPE t AND NODE DEGREE (k) IN REAL-WORLD POWER NETWORKS
IEEE-30
0.2348
0.1944 / 0.2537/ 0.2183
0.0210
IEEE-57
0.l222
0.1524 / 0.l352 / 0.0778
-0.1064
IEEE-ll8
0.1651
0.1607 / 0.1969 / 0.0167
-0.0538
IEEE-300
0.0856
0.1227 / 0.0895 / 0.0364
-0.1428
NYISO-2935
0.2134
0.2693 / 0.2489 / 0.0688
-0.2382
measure W(,][,), called the follow:
Bus Type Entropy,
is defined as
W(,][,) = -�r=llog('TJ - ��llog(Rll)'
(7)
where 'T, = nTjn represent the bus type ratio of bus i and RILJ = mILJ/m the corresponding link type ratio of the jth line; and �r=lO(']['i - k), k = 1, 2, 3 �j=l o(lLj - k), k = 1, 2, · .. , 6;
(8) (9)
representing the total number of buses and lines in the grid that have some specified types respectively. The bus type entropy of (10) can also be written as: W1('][') = -��=llog('k)
x
nk - �%=llog(Rk)
x
mk, (10)
And we also give two variations of (10) as follows: (11) W2('][') =-��=llog('k) - �%=llog(Rk)' 1 1 3 W3('][') =-�k=llog('k) x - - �k6 =llog(Rk) X - (12) nk mk Obviously (11) can be viewed as a simplified version of (10) by omitting nk and mk in computing the entropy. This simplification brings the advantage that the resulting entropy will have a more stable range regardless of the grid size. And (12) uses link and 1/mk to replace nk and mk in (10) respectively. All the three measures have been defined based on both the bus type assignment '][' and the link types lL(']['IL). Therefore they have implicitly incorporated the correlated grid topology. W(,][,) We use empirical probability analysis to examine the per formance of the proposed measures of bus type entropy (10) (12) based on some realistic grid data. Given a real-world
B. The Empirical PDF of
power grid topology with n buses and m transmission lines, we can find out the original generation and load locations hence determine the realistic bus type vector, denoted as 'II' * , and the corresponding bus type entropy W * = W('II'*). Then we randomly permute 'II' * , which equals a random relocation of generation and loads in the original network topology. Obviously the random permutation of 'II' * will not change the total number of buses of each type , i.e., the bus type ratios rC/L/C will remain the same during the process. However, the link type vector and ratios, therefore the bus type entropy, will be changed accordingly. Theoretically there exist at most (13) different bus type entropies calculated from the all randomized bus vectors given a specific grid topology. Denote the set of randomized bus type vector with size kmax as T = P ('II'*), which is collected from kmax times Monte-Carlo experiments. If the sample size kmax » fi is sufficiently large, we can say with confidence that T and the following analysis of entropy based on it have sufficient statistical power. Based on the same connecting topology Lo of tl:!.e original grid, we can derive the resulting link type vector JL(TILcV' and then compute the corresponding bus type entropy W(TILo), using (10)-(12), of the randomized bus type assignments in T. And the empirical probability distribution function (PDF) of the random variable W can be generated through the normalized histogram analysis of W(TILo) as (14) where 06.(-) is a Dirac Delta function with a width of odx)
=
{ �'
