Global Sensitivity Analysis in Load Modeling via Low-Rank Tensor

Abstract

Growing model complexities in load modeling have created high dimensionality in parameter estimations, and thereby substantially increasing associated computational costs. In this paper, a tensor-based method is proposed for identifying composite load modeling (CLM) parameters and for conducting a global sensitivity analysis. Tensor format and Fokker-Planck equations are used to estimate the power output response of CLM in the context of simultaneously varying parameters under their full parameter distribution ranges. The proposed tensor structure is shown as effective for tackling high-dimensional parameter estimation and for improving computational performances in load modeling through global sensitivity analysis.

Full text

IEEE TRANSACTIONS ON SMART GRID 1
Global Sensitivity Analysis in Load Modeling via
Low-rank Tensor
You Lin, Student Member, IEEE, Yishen Wang, Member, IEEE, Jianhui Wang, Senior Member, IEEE,
Siqi Wang, Member, IEEE, and Di Shi, Senior Member, IEEE
Abstract—Growing model complexities in load modeling have
created high dimensionality in parameter estimations, and
thereby substantially increasing associated computational costs.
In this paper, a tensor-based method is proposed for identifying
composite load modeling (CLM) parameters and for conducting
a global sensitivity analysis. Tensor format and Fokker-Planck
equations are used to estimate the power output response of CLM
in the context of simultaneously varying parameters under their
full parameter distribution ranges. The proposed tensor structure
is shown as effective for tackling high-dimensional parameter
estimation and for improving computational performances in
load modeling through global sensitivity analysis.
Index Terms—Dimensionality reduction, load modeling, pa-
rameter estimation, sensitivity analysis, tensor.
I. INTRODUCTION
LOAD modeling has been a critical component in power
system stability analysis for decades. Accurate load
modeling leads to more precise system operation limits and
can improve system operation and economics [1]. Inaccurate
load modeling can result in system operating states that are
vulnerable to contingencies, thus emphasizing the need for
informative load modeling in power system operations.
Advances in power system measurement equipment, specif-
ically PMUs, allows for the collection of key electrical infor-
mation of specific buses, such as voltage magnitude V, voltage
angle θ, real power P, and reactive power Q. The objective
of load modeling, thus, is to construct a mathematical model
fto replicate the behaviors of Pand Qwith input Vand θ,
as shown in (1).
P=f1(V, θ|π)and Q=f2(V, θ|π)(1)
By estimating the load modeling parameters π,π=
[π1,...,πM]T, we can calculate the power output response
given the input Vand θ. Here, Mis the total number of
parameters, which is usually 4 in ZIP and 11 in composite load
modeling (CLM) represented by ZIP and 3rd-order induction
motor (IM).
The more complex and detailed the load model structure that
is to be mimicked for different types of dynamic responses,
the more parameters there are to be estimated [2]. The trade-
off between computation and model structure, therefore, drives
This work is funded by SGCC Science and Technology Program under
contract no. SGSDYT00FCJS1700676.
Y. Lin, Y. Wang, S. Wang, and D. Shi are with GEIRI North America, San
Jose, CA 95134, USA.
Y. Lin and J. Wang are also with the Department of Electrical and Computer
Engineering, Southern Methodist University, Dallas, TX 75205, USA.
the non-trivial nature of the problem for solving and analysis.
Additionally, the correlation among parameters increases the
difficulty in identifying some parameters. Existing parameter
estimation methods in load modeling are highly dependent on
the initialization performance.
The existence of multiple locally optimal solutions is a
common problem in the current load modeling research field
[1]. One common method to obtain a global optimal solution
is by calculating the response curve fitting accuracy based on
enumerations of all possible parameter sets. However, it is
impractical to attain an optimal solution using this method
when the number of parameters is large. Moreover, there
are several issues still to be addressed in load modeling
parameter estimation, including: How sensitive the real and
reactive power are to the variation of each parameter? How to
efficiently obtain global optimal parameters in their full range?
In this paper, we use tensor structure methods to address
the high dimensional parameter estimation problem. We also
evaluate the influence of jointly varying parameters to provide
global sensitivity analysis. Tensors allow for the mapping of
high-dimensional features to low-dimensional representations
by explicitly building the tensor in a low-rank form [3].
With such a structure, we can directly control the effective
dimensionality of a set of parameters, thereby improving the
overall computation performance and solution quality. The
contributions of this paper are as follows:
•Efficient sampling of a large number of parameters in
high dimensional parameter space via tensor;
•Computing varied distributions of all parameters within
their entire feasible range;
•The ability to provide a global optimal reference for all
model parameters.
II. OPTIMIZATION PROBL EM I N LOA D MODELING
The composite load model in this paper includes the ZIP
and IM. The static load component is represented by the ZIP
model, whose real power PZIP,t and reactive power QZIP,t
at time t are calculated from (2) and (3) given the parameters
ap,bp,cp,aq,bq, and cq. Here, cp= 1 −ap−bp, and cq=
1−aq−bq.
PZI P,t =PZI P,0 apVt
V02
+bpVt
V0+cp!(2)
QZI P,t =QZI P,0 aqVt
V02
+bqVt
V0+cq!(3)

IEEE TRANSACTIONS ON SMART GRID 2
The dynamic load component is represented by IM. With the
measured bus voltage Vt∠θtand power output ˜
St=Pt+jQt,
we can derive the d/q transformation of the voltage, Ud,t and
Uq,t, from the reference axis of the overall system to d/q
axis of motors. The reference axis for the measurements is
the global reference axis for the system, which needs to be
converted to the d/q axis.
δt= tan−1XsItcos ϕt−RsItsin ϕt
Vt−(RsItcos ϕt+XsItsin ϕt)(4)
where, ϕt=θt−αt, and ˙
It=˜
S
Vt∠θt∗
=It∠αt,Rsand
Xsare the stator resistance and reactance of IM, respectively.
We then can get Ud,t =−Vtsin(δt), and Uq,t =Vtcos(δt).
The third-order IM model can be described by the following
differential and algebraic equations (DAEs) [4].
dv0
d,t
dt =−Rr
Xr+Xmv0
d,t +X2
m
Xr+Xm
iq,t+sv0
q,t (5)
dv0
q,t
dt =−Rr
Xr+Xmv0
q,t −X2
m
Xr+Xm
id,t−sv0
d,t (6)
dst
dt =1
2HTm0(1 −st)2−v0
d,tid,t −v0
q,tiq ,t(7)
id,t =Rs(Ud,t −v0
d,t) + X0(Uq,t −v0
q,t)
R2
s+X02(8)
iq,t =Rs(Uq,t −v0
q,t)−X0(Ud,t −v0
d,t)
R2
s+X02(9)
X0=Xs+XmXr
Xm+Xr
(10)
where Rr, Xr, Xm, H, s, v0
d,t, v0
q,t, id,t , iq,t are the rotor resis-
tance and reactance, the excitation reactance, IM inertia and
slip, the transient voltages and currents of IM in d/q axis,
respectively.
The real and reactive power output, PIM,t and QIM ,t, of
the IM can be calculated from the following equations:
PIM ,t =Ud,tid,t +Uq,t iq,t (11)
QIM ,t =Ud,tiq,t −Uq ,tid,t (12)
The estimated real and reactive power output, ˆ
Pand ˆ
Qof
the composite load model can be calculated from (13) given
the static load proportion ωrepresented by ZIP.
(ˆ
Pt=ω·PZI P,t + (1 −ω)·PI M,t
ˆ
Qt=ω·QZI P,t + (1 −ω)·QIM,t
(13)
The load modeling formulation in (2)∼(13) can be repre-
sented with the DAEs as:
(dxt=µ(xt,π)dt
M(xt,π)xt=b(14)
Given the voltage measurements as the inputs, the optimiza-
tion problem in load modeling parameter identification is to
find the optimal parameter πand state variable xto achieve
the minimized fitting errors between the P/Q measurements
and estimated ˆ
P / ˆ
Qresponses. Here, xt={v0
d,t, v0
q,t, st}and
π={ap, bp, aq, bq, Rs, Xs, Rr, Xr, Xm, H, ω}. This fitting
error is measured by the root mean square error (RMSE)
between the estimated and measured values of real or reactive
power.
The load modeling parameter identification is a highly non-
linear and nonconvex optimization problem, which commonly
leads to local optimal solutions. These local optimal solutions
are also highly dependent on the selection of the initial values.
Considering these issues and disregarding the computation
limitations, a simple and straightforward method would be to
obtain the global optimal solution through enumeration. By
calculating the model output accuracy corresponding to all
the possible combinations of all dependent parameters in their
varying ranges, we can arrive at a global optimal solution.
III. STOCHASTIC SOLUTIONS OF DAE EQUATI ON S
Before analyzing the sensitivity of all parameters, we must
first find the solution of the DAEs in (14) and then present
reasonable distributions of the parameters, which can be
achieved simultaneously using the Fokker-Planck operator.
The Fokker-Planck operator is a partial differential equation
that describes the time evolution of the joint probability density
function of its state variables and parameters [5]. With the
DAE equations in (14), the Fokker-Planck equation for the
probability distribution p(x|π)of the state vector xgiven the
parameter values πis:
∂
∂t p(x|π) = −
N
X
i=1
∂
∂xi
[µi(x, π)p(x|π)] (15)
where Nis the number of state variables. With the parametric
Fokker-Planck operator A(x, π), (15) can be expressed as,
A(x, π)p(x|π) = −
N
X
i=1
∂
∂xi
[µi(x, π)p(x|π)] (16)
We then use the tensor structures to compute p(x|π)simul-
taneously for all parameters πwithin their feasible varying
space as shown in the following section.
IV. PARAMETER ESTIMATION VIA TENSOR
In this paper, a tensor structure is used to tackle the high
dimension computation burden. Define the N-dimensional
varying space Ωx=I1× · · · × INof state variables xt
and M-dimensional varying space Ωπ=J1× · · · × JMof
model parameters π. Here, the feasible range of state variable
xi,t(i= 1,· · · , N )is Ii= [lx
i, ux
i], and the feasible range
of parameter πj(j= 1,· · · , M )is Jj= [lπ
j, uπ
j]. Here, we
discretize the space of Ωxinto (nd+1)Nnodes with step sizes
hx
i= (ux
i−lx
i)/nd. Similarly, discretizing the space of Ωπinto
(md+ 1)Mpoints with step sizes hπ
i= (uπ
i−lπ
i)/md. The
values of p(xt|π)are discretized into (N+M)−dimension
tensor ˜
p∈Ri1×···×iN×j1×···×jNwith entries
˜
pi1×···×iN×j1×···×jN
=p(xi1,t,· · · , xiN,t |πj1,t,· · · , πjM,t)(17)
If we process (17) in conventional vector and matrix format,
the searching space we face is (nd+ 1)N(md+ 1)M. The

IEEE TRANSACTIONS ON SMART GRID 3
storage cost of tensor ˜
pwill increase exponentially with Nand
M, which is computationally intractable. Thus, we propose to
use the tensor decomposition to approximate the probability
distribution p(xt|π)[6], shown as:
p(xt|π) =
R
X
r=1
pr
1(x1,t)· · · pr
N(xN,t )pr
1(π1)· · · pr
N(πN)(18)
where, Rrepresents the tensor rank. Based on the low-rank
representation in (17), the mathematical operations on the
probability distribution p(xt|π)in N+Mdimensions can be
performed using combinations of one-dimensional operations,
and the storage cost is reduced to (N+M)R.
If choosing a sufficiently large computation domain,
i.e., ndand mdare large enough, we can approximate
the distribution p(xt|π)by eigenvector ˜
pof A(xt,π)
corresponding to the minimum eigenvalue λmin clos-
est to zero. Thus, given the initial value of all state
variables x0={v0
d0, v0
q0, s0}and parameters π0=
{ap0, bp0, aq0, bq0, Rs0, Xs0, Rr0, Xr0, Xm0, H0, ω0},˜
pcan
be solved from (19) using the alternating minimum energy
method (AMEN) method [7] as the linear solver in tensor
format.
A(xt,π)˜
p=λmin ˜
p(19)
Since the solution ˜
pis presented in tensor format shown in
(18), we extract the univariate probability density function of
each state variable and parameter, represented by pi(xi,t)and
pj(πj), respectively.
By examining the probability densities of each parameter,
the best parameter estimation π∗at its largest probability is
obtained. Simultaneously, several local optimal estimations at
its lower probability positions are also provided.
Given the optimal parameter estimation π∗, the real and
reactive power output of the CLM can be obtained using
(2)∼(13). Meanwhile, the model fitting error can be calculated.
V. FRAMEWORK OF THE PROP OS ED MO DEL
The pseudo code of the tensor-based parameter estimation
method is presented in Algorithm 1.
Algorithm 1 tensor-based parameter estimation algorithm
1: Input: Measured bus voltage magnitude Vt, voltage angle θt, measured
power P/Q.
2: Initialize the state variables x0={v0
d0, v0
q0, s0}and model parameters
π0={ap0, bp0, aq0, bq0, Rs0, Xs0, Rr0, Xr0, Xm0, H0, ω0}.
3: Define the varying range of xtand π.
4: Formulate the DAEs shown in (14).
5: Calculate the Fokker–Planck operator A(xt,π)in tensor format.
6: while A(xt,π)p(xt|π)6= 0 do
7: Adjust the AMEN algorithm.
8: i=i+ 1.
9: end while
10: Output: Individual distributions of xtand π, represented by pi(xi,t)and
pj(πj); global optimal solution for model parameters π∗; and response
of power output ˜
Por ˜
Qcorresponding to π∗.
VI. NUMERICAL RE SU LTS
To achieve a more realistic simulation, we use the simulated
PMU data based on the WECC composite load model as
historical PMU measurements. This WECC load model is
connected to the IEEE 39-bus system. Using the measurement
data, we calculate the distributions of all parameters in the
formulated ZIP + IM load model based on our proposed
method. In this section, we will first analyze the effectiveness
of load modeling parameter solutions and the benefits of the
proposed method in computational performances. Then, we
provide the comparisons with benchmark methods, including
Monte Carlo simulation and the quasi-Newton algorithm.
A. Parameter solutions and power output
The proposed model can estimate the time-varying param-
eters of load models. For each time step t, we can obtain the
joint load parameter distributions and a set of corresponding
optimal load parameters at this snapshot. As an example,
the estimated joint distributions of static load proportion ω
and motor stator reactance Xsin different time snapshots are
shown in Fig. 1(a) and (b). The estimated distributions of ω
and Xsconcentrate in values around 0.5 and 0.1, respectively.
From the joint distributions presented in Fig. 1, multiple
local optimal values of these two parameters are also found.
Similarly, the estimated joint distributions of motor inertia H
and the rotor reactance Xrare presented in Fig. 1(c) and (d).
Note that the estimated joint distribution actually characterizes
all 11 load parameters, and the 4 parameters are randomly
selected for the data visualization. The corresponding power
responses (P/Q)resulting from the voltage disturbance are
further obtained by combining all joint distributions of load
parameters and state variables. The power responses (P/Q)
calculated from the optimal parameters are presented in Fig.
2. From this figure, we can observe that the proposed method
based on ZIP+IM model can identify load parameters and fit
the real and reactive power responses reasonably well.
B. Benefits of the proposed method
There are 11 parameters and 3 state variables in the ZIP+IM
model. In the simulation case, we divide the entire feasible
ranges of each parameter and each state variable into 210
segments. Therefore, the total variables in Section IV are set
as nd=md= 210. The grid size, also called the dimension
size, of all parameters’ space is (210 + 1)11. The grid size
of all state variables is (210 + 1)3. The overall dimension we
are processing is (210 + 1)11 ·(210 + 1)3= 1.41 ×1042. For
such high-dimensional solution space, the tensor structure in
our proposed method makes it possible to process within only
25s on average for each time step. On the contrary, to the
best of our knowledge, there is no load modeling parameter
estimation and sensitivity analysis methods can process such
large solution space.
One common practice for processing high dimensions pa-
rameter space is using Monte Carlo simulation. It is usually
executed by sampling one percent or one thousandth of the
original high-dimensional space. However, it is very difficult

IEEE TRANSACTIONS ON SMART GRID 4
(a) Joint distributions at time t. (b) Joint distributions at time
t+1
120 s.
(c) Joint distributions at time t. (d) Joint distributions at time
t+1
120 s.
Fig. 1: Time-varying distributions of different load model
parameters.
0 1 2 3 4 5
Time(s)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
P/Q(p.u.)
Estimated P (Proposed method)
Estimated P (Quasi-Newton)
Measured P
Estimated Q (Proposed method)
Estimated Q (Quasi-Newton)
Measured Q
Fig. 2: Simulated real and reactive power
to conduct Monte Carlo simulation to iterate over a huge
number of scenarios. We compare the computation time of
the proposed method and the conventional Monte Carlo sim-
ulation as shown in Table I. From the table, the computation
time of Monte Carlo for 100 iterations is 29.84s. The total
Monte Carlo computation time for the 1037 iterations will be
8.29 ×1032h, which is entirely computationally intractable.
We also compare the power response curves calculated
from the parameter estimation solutions of the quasi-Newton
algorithm as shown in Fig. 2. As seen in the figure, the quasi-
Newton identified local solutions cannot accurately reproduce
the measured real and reactive power with RMSE value of
0.0878. As a contrast, the RMSE of our proposed method
is only 0.0169. In order to produce good results, the proce-
dure of quasi-Newton requires initialization of the parameters
close enough to their true values or local optimal values [8].
Otherwise, the algorithm will diverge. As the load model es-
TABLE I: Computation Time of the Proposed Method and
Monte Carlo Simulation
Method Proposed Monte Carlo Monte Carlo
method 100 Scenarios 1037 scenarios
Average computation 25s 29.84s 8.29 ×1032h
for each time step
sentially represents an aggregate model for all the downstream
connected loads, the true value of the load parameters will be
unknown in the real system. Thus, the initialization and local
optimization issues limit the applications of the quasi-Newton
algorithm. However, our proposed method overcomes the two
above difficulties as well as provides the full parameter time-
varying joint distributions.
VII. CONCLUSION
A global sensitivity analysis approach is proposed to per-
form load modeling parameter estimation via tensor. Simulta-
neously, varying distributions of all load model parameters in
their feasible range can be efficiently estimated based on the
Fokker–Planck equations in tensor format. The low-rank tensor
representation of the formulated load modeling can tackle high
dimensional problems with efficient computation. Global opti-
mal references of parameters are obtained from the estimated
distribution of all load model parameters. Furthermore, the
sensitivities of all parameters are intuitively presented in the
estimated parameter distributions.
REFERENCES
[1] A. Arif, Z. Wang, J. Wang, B. Mather, H. Bashualdo, and D. Zhao, “Load
modeling—a review,” IEEE Transactions on Smart Grid, vol. 9, no. 6,
pp. 5986–5999, 2017.
[2] X. Wang, Y. Wang, D. Shi, J. Wang, and Z. Wang, “Two-stage wecc
composite load modeling: A double deep q-learning networks approach,”
arXiv preprint arXiv:1911.04894, 2019.
[3] T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,”
SIAM review, vol. 51, no. 3, pp. 455–500, 2009.
[4] P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability and
control. McGraw-hill New York, 1994, vol. 7.
[5] H. Risken, “Fokker-planck equation,” in The Fokker-Planck Equation.
Springer, 1996, pp. 63–95.
[6] S. Liao, T. Vejchodsk`
y, and R. Erban, “Tensor methods for parameter
estimation and bifurcation analysis of stochastic reaction networks,”
Journal of The Royal Society Interface, vol. 12, no. 108, p. 20150233,
2015.
[7] S. V. Dolgov and D. V. Savostyanov, “Alternating minimal energy
methods for linear systems in higher dimensions,” SIAM Journal on
Scientific Computing, vol. 36, no. 5, pp. A2248–A2271, 2014.
[8] Y. Li, H.-D. Chiang, B.-K. Choi, Y.-T. Chen, D.-H. Huang, and M. G.
Lauby, “Load models for modeling dynamic behaviors of reactive loads:
Evaluation and comparison,” International Journal of Electrical Power &
Energy Systems, vol. 30, no. 9, pp. 497–503, 2008.