models and control for electrical power line temperatur

Daniel Bienstock

Jose Blanchet

Juan Li

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Stochastic models and control for electrical power line temperature
Daniel Bienstock1;2, Jose Blanchet1 and Juan Li1
Abstract—In this paper we consider the evolution of the
temperature of a power line under stochastic exogenous factors
such as ambient temperature. We present a solution to the
resulting stochastic heat equation and we propose a number
of control algorithms designed to maximize delivered power
under chance constraints used to limit the probability that a
line exceeds its critical temperature.
I. INTRODUCTION
When a power line overheats it becomes exposed to a
number of risk factors. If the overheating is severe the phys-
ical/mechanical attributes of the line may be compromised,
rendering it unusable. Under less severe overheating the line
may sag, thus bringing it into proximity with other objects,
and thereby potentially causing a contact or arc which will
trip the line. If overheating is determined, the line will be
protectively tripped (be taken out of service). In each of
these cases we see a situation where a line is no longer
available; as a result, power previously ﬂowing on that line
will be transferred to other lines, in a complex fashion (i.e.
following laws of physics) possibly congesting those lines
and causing them to overheat, as well. In a failure scenario of
a transmission system, this sequence of events may result in
a cascade resulting in a large-scale blackout. The Northeast
U.S.-Canada blackout of 2013 produced precisely this type
of event, see [10].
The temperature of a power line primarily depends on
the amount of (active, or real) power ﬂowing on that line
(we refer the reader to [2], [3] or [9] for background on
power engineering). However, high-voltage power lines are
uninsulated and exposed to numerous exogenous factors,
such as in particular wind and ambient temperature, among
many. All of these factors can and do inﬂuence power line
temperature. IEEE Standard 738 [7] amounts to a determinis-
tic codiﬁcation of the impact of a very large number of such
factors, starting from a base model that relies on the classical
heat equation. Even though this is a very thorough approach,
an examination of the standard highlights the potential for
mis-estimation due to erroneous, missing or variable data.
The previously mentioned report [10] describes instances
during the 2013 cascade where incorrect calibration of a
power line lead to unexpected tripping which contributed
to system instability. A somewhat more nuanced analysis of
*This work was partially supported by DTRA grant HDTRA1-13-1-0021.
1Department of Industrial Engineering and Operations
Research, Columbia University, New York, NY 102272, USA,
dano@columbia.edu, jl3035@columbia.edu,
jose.blanchet@columbia.edu
2Department of Applied Physics and Applied Mathematics, Columbia
University, New York, NY 102272, USA.
power line temperature based on the heat equation is given
in [1].
In this paper we model the aggregated impact of un-
certainty in exogenous factors using a generic stochastic
model. In a previous work [6] we focused on time-dependent
stochasticity of exogenous factors. In this paper instead we
assume that randomness is primarily of a spatial nature
and ignore the time component. This is a reasonable model
given the scope of current control practice (with various
levels of control applied every ﬁfteen minutes and even more
frequently) and the length of typical lines in a transmission
system (short lines may measure 50 miles, and long lines
much more). A comprehensive model that accounts for short-
term exogenous variability over a large spatial domain may
prove challenging; though in future work we may address
this point. We compute a general solution to the resulting
stochastic heat equation, and suggest several control mecha-
nisms relying on so-called “chance constraints” to maximize
delivered power while maintaining an acceptable level of
risk.
II. FORMULATION
We now focus on a particular power line on the time
domain [0;]. The line is modeled as a one-dimensional
object parameterized by x, 0  x  L. Let
 I = I(t) denote the current of that line at time t,
with the dependence on t highlighted so as to allow
for control actions.
 T(x;t) the temperature at x at time t.
The heat equation states:
@T(x;t)
@t
= 
@2T(x;t)
@x2 +I2(t) (T(x;t) Text(x;t));
(1)
where   0,   0 and   0 are (line dependent)
constants, and Text(x;t) is the ambient temperature at (x;t).
In order to account for stochasticity, we model Text(x;t) =
G(h(x)) where h(x) denotes a random variable at x, with
distribution that is either known or can be estimated (in what
follows boldface will be used to denote uncertain quantities)
and G() > 0. We thus obtain
@T(x;t)
@t
= 
@2T(x;t)
@x2 + I2(t)   (T(x;t)   G(h(x)):
(2)
We will further assume  = 0. This is consistent with the
use of the heat equation in [7], [1]; it is justiﬁed by noting
that propagation in the time domain is much faster than in
the spatial domain. [We will acount for the randomness of
exogenous conditions in the spatial domain in an average,or aggregated manner made precise below]. We therefore
obtain:
@T(x;t)
@t
= I2   (T(x;t)   G(h(x)): (3)
Integrating on both sides with respect to x, and dividing by
L, we have
1
L
Z L
0
@T(x;t)
@t
dx = I2(t)  

L
Z L
0
T(x;t)dx
+

L
Z L
0
G(h(x))dx: (4)
Denoting by H(t) average internal temperature along the
line at time t, by R the average ambient temperature along
the line, i.e.,
H(t) ,
1
L
Z L
0
T(x;t)dx; R ,
1
L
Z L
0
G(h(x))dx;
we therefore have
dH(t)
dt
=
d
dt
1
L
Z L
0
T(x;t)dx =
1
L
Z L
0
@T(x;t)
@t
dx:
Then (4) becomes:
dH(t)
dt
= I2(t)   H(t) + R; (5)
with solution
H(t) = (6)
Z t
0
e (t s)(I2(s) + R)ds + Ce t
=
Z t
0
e (t s)I2(s)ds + R(1   e t) + Ce t;
where
C = H(0) =
1
L
Z L
0
T(x;0)dx:
We note that the quantity R is not observed – however
we can aasume that its distribution can be estimated). We
are interested in control schemes that vary I(t) in response
to observed conditions. As criterion for stability, we will
enforce the chance-constraint [5], [8]
P

max
t2[0;]
H(t) > k

 ; (7)
where k > 0 is a given limit and  > 0 is small.
III. CONSTANT I(t), t 2 [0;]
The case where I(t) is constant in the time window of
interest is of special interest because of its simplicity. We
are interested in computing those values  I such that setting
I(t) =  I for 0  t   satisﬁes (7). From the closed-form
solution above we obtain
H(t) = (= I2 + R)(1   e t) + Ce t (8)
from which it follows that H0(t) > 0 for  I large enough (and
negative or zero otherwise). Assuming, thus, that H(0) < k,
it follows therefore that (7) is equivalent to
P (H() > k)  : (9)
Using (8) this implies
 I2 


k   Ce    r(1   e )
1   e  ; (10)
where r is the -quantile of R. As discussed above we
assume that the distribution of R is known, and consequently
the bound in (10) is computable.
For future use, we denote by L(;k) the right-hand side
of expression (10).
Now consider an entire grid where several lines are
considered to be thermally stressed. We can then compute
the upper bound on the current for each such line that is
implied by computation in (10). These values can then be
used in e.g. generator dispatching at time t = 0.
IV. ADAPTIVE CONTROL
As an enhancement to the analysis above, we consider
a setup where additional information becomes available at
some point in the time window [0;]; this information can
then be used to improve the line ratings obtained as in e.g.
(10). Deﬁne the random variable
W , R(1   e );
and to simplify we assume that W is discretely distributed,
P(W = wi) = pi;i = 1;2;:::;n with a known distribution
(i.e., the distribution of R is known). We are interested in a
control scheme with the following characteristics
1 At time  = 0, we compute values I1, and I2;i for
i = 1;2;:::;n. These values are used as follows
2 In the time window [0;=2] current is set to the constant
value I1.
3 At time =2, we observe the value of R and thus of W.
Assuming W = wi then current is set, in the interval
[=2;], to the constant value I2;i.
The values I1, and I2;i, 1  i  m are computed according
to the following criteria
(a) P(H() > k) < :
(b) I1  L(=2;k).
(c) Where F : R2 ! R+ is coordinate-wise monotonely
increasing, we want to maximize, in expectation
n X
i=1
F(I1;I2;i)pi (11)
We now discuss these modeling features. Note that as per
the analysis in the previous section, for any choice of the
values I1, I2;i we will always have that H(t) is monotonely
increasing or decreasing for t 2 [0;=2] and also for
t 2 [=2;]. Thus, our control scheme may possibly result
in a realization where e.g. H(=2) > k. However, (a)
guarantees that even if this were to be the case, the line
temperature will have reached a safe value by time . And
of course (b) helps ensure that the probabilty of this event
is small.
Regarding item (c), many examples are reasonable. For
example, we could set F(I1;I2) = 1I1 + 2I2 where
1;2  0. Or, F(I1;I2) = 1I2
1 +2I2
2. Below we discuss
several cases.We will now cast the choice of the values I1, I2;i as an
optimization problem. To do so, suppose that W = wi. Then,
using (6),
H() = v1I2
1 + v2I2
2(i) + wi + Ce ; (12)
where
v1 ,
Z =2
0
e ( s) ds (13)
and
v2 ,
Z 
=2
e ( s) ds: (14)
Deﬁne
z1 , v1I2
1; z2(i) , v2I2
2(i); 1  i  n; (15)
and, for 1  i  n
 ki , k   Ce    wi:
Using this notation we have that, when W = wi
H() > k is equivalent to: z1 + z2(i) >  ki:
Moreover, let us deﬁne
~ F(z1;z2) , F(
p
z1=v1;
p
z2(i)=v2); (16)
which is simply recasting function F in terms of the z
variables. It follows that we can write our optimal control
problem as
P1 : max
n X
i=1
~ F(z1;z2(i))pi;
s.t.
n X
i=1
Ifz1 + z2(i) >  kigpi   (17)
z1  v2
1L2
1(=2) (18)
z1 + z2(i)  ui; 8i; (19)
z1  0; z2(i)  08i: (20)
In (17), I is the indicator function; the square roots in the
objective and (18) arise from our deﬁnition of the z variables.
Constraint (19) models a reasonable requirement: that the
line temperature at time  not exceed an absolute maximum
limit, with probability 1. Of course, this constraint may
render the problem above infeasible – however assuming that
H(0) is “safe” the problem will be feasible (if necessary by
setting z1 = z2(i) = 0 for all i) assuming realistic R. We
assume that ui >  ki.
Lemma 1: Let z
1, z
2(i) (1  i  n) be an optimal
solution to problem P1. Then, for each 1  i  n
z
1 + z
2(i) =  ki or ui:
Proof. Suppose that for some i, z
1 + z
2(i) < ki. Then
increasing z
2(i) maintains feasibility, and the monotonicity
asumption on F implies that the objective improve. The
same reasoning applies if ui < z
1 + z
2(i) < ki.
Using this observation we can simplify the optimization
problem. Deﬁne, for 1  i  n a binary variable yi such
that
yi =

0 when z1 + z2(i) =  ki
1 when z1 + z2(i) = ui
(21)
Then the above optimization problem can be recast as:
P2 :
max
n X
i=1
~ F(z1; ki   zi)pi(1   yi) + ~ F(z1;ui   zi)piyi
s.t.
n X
i=1
uipiyi  
0  z1  minfv2
1L2
1(=2); min
i
fuigg (22)
yi = 0 or 1; all i:
Problem P2 is a nonlinear, binary optimization problem. We
are interested in methodologies and special cases that render
it practicable. We will describe a general approximation
method that should prove effective when the parameter n
is small. We also describe a provably good approximation
algorithm for the case where n is large, which however only
applies in a special case of the function F.
As an aside, the issue of the magnitude of n concerns
several practical questions, primarily with regards with how
accurate a representation of the random variable R can be
constructed in “real time”. Adequate sensorization of power
lines should help in this regard, however there is a larger
issue of how data uncertainty can arise in this context (e.g.,
the spatial distribution of exogenous temperatures over a
small time window).
A. General approach for small n
Suppose that in problem P2 we were to ﬁx the variable z1
to a value ^ z1 satisfying (22). The remaining part of problem
P2 has the following general structure:
P2(^ z1) :
max
n X
i=1
~ fi(^ z1)yi
s.t.
n X
i=1
uipiyi  ;
yi = 0 or 1; all i:
Problem P2(^ z1) is a (binary) knapsack problem. Knapsack
problems are NP-hard – however in this case we are dealing
with small n. In fact, it is fair to say that knapsack problems
are the easiest of the NP-hard problems (and, commercial
mixed-integer program solvers dispatch them with ease even
for largish n).
These observations suggest the following (grid-like) ap-
proach:
(1) Enumerate equally spaced values of ^ z1 between the two
bounds in 22.
(2) For each enumerated value, solve P2(^ z1).While suffering from an enumerative component, this ap-
proach does have the attribute of handling any objective
function F in the deﬁnition of our problem.
A separate issue regarding the small n case concerns the
robustness of the computed answers with respect to e.g.
the (necessarily estimated) parameters pi and wi. Using a
small n has the effect of accumulating more probability mass
into fewer values, with a resulting increase in numerical
sensitivity (to the choices for the pi and wi).
B. Large n
Suppose now that n is large. As stated above we expect
that even in this case a good mixed-integer programming
solver should be able to solve the problems P2(^ z1). Nev-
ertheless, we would like to discuss a case where a solution
with sound theoretical foundation can be found.
Recall the formula (12) for H() as well as (15), and
note that z1+z2(i) appears in H() and thus, in the chance
constraint (17). Consider the special case where
F(I1;I2) = v2
1I2
1 + v2
2I2
2 (23)
for all I1; I2. Below we will discuss the implications of this
assumption. Using (23), it will follow that (using (16),
~ F(z1;z2) = z1 + z2; for all z1;z2: (24)
Thus, problem P2 can be equivalently restated as:
P3 : max
n X
i=1
~ F( ki)pi(1   yi) + ~ F(ui)piyi
s.t.
n X
i=1
uipiyi  
0  z1  minfv2
1L2
1(=2); min
i
fuigg (25)
yi = 0 or 1; all i:
We can see that constraint (25) is redundant. In short, P3
can be rewritten in the form:
P0
3 : max
n X
i=1
fi yi
s.t.
n X
i=1
qiyi  
yi = 0 or 1; all i;
for appropriate quantities fi > 0 and qi > 0; this constitutes
a standard 0   1 knapsack problem.
In comparing this approach to that used for the small n
case, we can see that see that we have simpliﬁed the problem
(no enumeration over the ^ z1 values). Of course, we do have
to solve the possibly large knapsack problem P3. As we
discussed before, this should prove routine (and very fast)
even for n in the hundreds, if not more.
However, from a theoretical perspective, more can be said.
The following result can be read from those in [4].
Theorem 2: Consider a 0   1 knapsack problem on N
variables
K : max
N X
j=1
pjxj
s.t.
n X
j=1
ajxj  b;
xj = 0 or 1; all j:
For each ﬁxed tolerance 0 <  < 1 there is a linear program
LP with the following properties
 The number of variables and constraints in LP is
O(N2).
 The xj are among the variables of LP.
 The solution of LP, together with a simple rounding for
the xj variables yields a (binary) solution for K that
is guaranteed to have value within tolerance  of the
optimum for K.
Now we comment on (23). We would argue that this is a
“reasonable” functional form for F(I1;I2) in that it amounts
to a weighted sum. Of course the weights are not ﬂexibly
chosen. Nevertheless, note (see (13), (14)) that v2 > v1.
Thus, (23) places more emphasis on what happens in the time
interval [=2;]. We would argue that this is a reasonable
approach, in the sense that we focus in the later time interval,
where, coincidentally, we are able to make more precise
decisions since randomness has been resolved.
C. Robustness of the solutions
REFERENCES
[1] M. Anghel, K.A. Werley, and A.E. Motter. Stochastic model for power
grid dynamics. Proceedings of the 40th Hawaii Int. Conf. on System
Sciences, 2007.
[2] G. Andersson. Modelling and Analysis of Electric Power Systems.
Power Systems Laboratory, ETH Z¨ urich, 2004.
[3] A.R. Bergen and V. Vittal. Power Systems Analysis. Prentice-Hall,
1999.
[4] D. Bienstock and B. McClosky. Tightening simple mixed-integer sets
with guaranteed bounds. Math. Programming, 133:337–363, 2012.
[5] A. Charnes, W. Cooper, and G. Symonds. Cost horizons and certainty
equivalents: an approach to stochastic programming of heating oil.
Management Science, 4 (1958), pp. 235–263.
[6] J. Blanchet, D. Bienstock and J. Li. Power Line Control under
Uncertainty of Ambient Temperature. to appear, CDC ’13 (2013).
[7] IEEE Std. 738-2006. IEEE standard for calculating the current-
temperature of bare overhead conductors. pages 1 – 59, 2006.
[8] L. Miller and H. Wagner. Chance-constrained programming with joint
constraints. Operations Research, 13 (1965), pp. 930–945.
[9] J. D. GLover, M. S. Sarma, and T. J. Overbye. Power System Analysis
and Design. CENGAGE Learning, 2012.
[10] U.S.-Canada Power System Outage Task Force. Report on the
august 14, 2003 blackout in the united states and canada: Causes and
recommendations. https://reports.energy.gov, (2004).
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