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Conference Paper

NREL/CP-5D00-76401

July 2020

Bayesian Structural Time Series for 

Behind-the-Meter Photovoltaic 

Disaggregation

Preprint

Peter Shaffery, Rui Yang, and Yingchen Zhang

National Renewable Energy Laboratory

Presented at the 2020 IEEE Conference on Innovative Smart Grid 

Technologies (IEEE ISGT)

Washington, D.C.

February 17–20, 2020
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Conference Paper

NREL/CP-5D00-76401

July 2020

Bayesian Structural Time Series for 

Behind-the-Meter Photovoltaic 

Disaggregation

Preprint

Peter Shaffery, Rui Yang, and Yingchen Zhang

National Renewable Energy Laboratory 

Suggested Citation 

Shaffery, Peter, Rui Yang, and Yingchen Zhang. 2020. Bayesian Structural Time Series 

for Behind-the-Meter Photovoltaic Disaggregation: Preprint. Golden, CO: National 

Renewable Energy Laboratory. NREL/CP-5D00-76401. 

https://www.nrel.gov/docs/fy20osti/76401.pdf.
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Bayesian Structural Time Series for

Behind-the-Meter Photovoltaic Disaggregation

Peter Shaffery

National Renewable Energy Laboratory

Golden, Colorado 80401

Email: peter.shaffery@nrel.gov

Rui Yang

National Renewable Energy Laboratory

Golden, Colorado 80401

Email: rui.yang@nrel.gov

Yingchen Zhang

National Renewable Energy Laboratory

Golden, Colorado 80401

Email: yingchen.zhang@nrel.gov

Abstract—Distributed photovoltaic (PV) generation often occurs “behind the meter”: a grid operator can only observe the

net load, which is the sum of the gross load and distributed

PV generation. This lack of observability poses a challenge

to system operation at both bulk level and distribution level.

The lack of real-time or near-future disaggregated estimates of

gross load and PV generation will lead to over scheduling of

energy production and regulation reserves, reliability constraints

violations, wear and tear of controller devices, and potentially

cascading failures of a system. In this paper we propose the use

of a Bayesian Structural Time Series (BSTS) model with local

solar irradiance measurements to disaggregate the summed PV

generation and gross load signals at a downstream measurement

site. BSTSs are a highly expressive model class that blends

classic time series models with the powerful Bayesian state space

estimation framework. Disaggregation is done probabilistically,

which automatically quantifies the uncertainties of the estimated

PV generation and gross load consumption. Depending on the

data availability in real-time, it can be used to disaggragate PV

and gross load at customer site, or can be used at the feeder

level. In this paper, we focus on solving the problem at feeder

level. We compare the performance of a BSTS model as well

as a handful of state-of-the-art methods on a Pecan Street AMI

dataset, using the National Solar Radiation Database (NSRDB)

to estimate local irradiance.

I. INTRODUCTION

In the last decade distributed photovoltaic (PV) generation

has become a significant source of renewable energy, and

its adoption is projected to continue growing for the next

several decades [1]. Increased penetration of distributed PV

poses new challenges to the operation and control of distribution power grids, which are frequently compounded by

the fact that these resources are typically unobserved [2]–

[4]. Most of distributed PV systems are located behind-themeter on customer premises, where utilities do not have direct

access to their output but only net load measurements. Realtime estimation of the behind-the-meter PV generation is

This work was authored by the National Renewable Energy Laboratory,

managed and operated by Alliance for Sustainable Energy, LLC, for the U.S.

Department of Energy (DOE) under Contract No. DE-AC36-08GO28308.

Funding provided by the National Renewable Energy Laboratory’s Laboratory

Directed Research & Development Program. The views expressed in the

article do not necessarily represent the views of the DOE or the U.S.

Government. The U.S. Government retains and the publisher, by accepting

the article for publication, acknowledges that the U.S. Government retains a

nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce

the published form of this work, or allow others to do so, for U.S. Government

purposes.

crucial to improving the situational awareness and reducing

the uncertainties of distributed PV generation in distribution

systems. By disaggregating PV generation from net-metered

measurements, distribution system operators can improve the

forecasting accuracy of net load, reduce the reserve deployment, and optimize the voltage profile.

To this end, a number of methods have been proposed to

disaggregate net load measurements into PV generation and

gross load components at several scales. These methods can be

divided into supervised disaggregation and unsupervised approached. Supervised methods use labelled, pre-disaggregated

data to perform disaggregation [5]–[7], while unsupervised

methods use only the aggregate signal, structural knowledge

of the system (eg. panel configurations), and environmental

measurements (eg. solar irradiance or temperature) [8]–[10].

More recently, [11] proposed using a “synthetic” PV generator

model combined with a Hidden Markov regression model for

gross load to perform disaggregation at the household-level.

Similarly, [12] also used a synthetic PV generator model as

a component of a probabilistic, load forecasting pipeline for

disaggregation at the regional level. Notably, while the forecast

is probabilistic, the disaggregation itself is not.

In this paper, we introduce a probabilistic model whose

structure mimics well-understood time series models, and

for which high-performing fitting algorithms already exist:

Bayesian Structural Time Series (BSTS) [13]. In this approach we construct a synthetic state space model, which is

designed so that the model has a structural similarity to classic

time series models (eg. the AR/I/MA class of models). This

methodology fundamentally differs from the aforementioned

unsupervised approaches by providing probabilistic estimation

of PV generation and gross load consumption. Instead of producing a deterministic or “point” estimate of the disaggregated

signals, the BSTS model defines a joint probability density

function (PDF) over the gross load and PV generation at each

time point, and the dispersion and covariance structure of the

joint PDF automatically quantifies the uncertainties associated

with the disaggregated PV and load signals.

In addition to quantifying uncertainty, the BSTS model has

the advantage of being robust to noise in the input data.

The underlying methodology of BSTS disaggregation is to

consider all possible disaggregated signal shapes, and weight

them according to their plausibility with the model covariates

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 1
and net load data. The BSTS approach will therefore only

make significant changes to its estimates of PV and gross load

when the input data exhibit a change large enough to alter the

plausibility of a substantial range of potential disaggregations,

reflecting real changes in the system.

These features of the BSTS approach can improve the

efficiency of distribution system operations. The joint PDF

produced by the BSTS model provides real-time bounds for

gross load and PV generation, and also accounts for correlation

in the errors between the two quantities, allowing the operator

to more precisely determine necessary reserve deployment.

Furthermore, robustness in the disaggregation estimate helps

prevent the operator from taking unnecessary control actions

due to noise in the data. These unique advantages contribute

to the novelty of our proposed method.

II. DATA

AMI data from Pecan Street Inc. Dataport was used to

assess the performance of our disaggregation model. This

data includes household-level power usage and PV generation

data at a 1-minute time resolution for 123 houses in the

Mueller neighborhood of Austin, TX. We test our proposed

method on 7 days in January and August, 2017. As our model

relies on local measurements of solar irradiance and ambient

temperature to disaggregate PV generation (quantities highly

correlated with PV and gross load), we supplemented the

Pecan Street data with data from the National Solar Radiation

Database (NSRDB). The NSRDB provides irradiance and

climate data (temperature, humidity, etc.) at a 30-minute time

resolution and a 4 km2spatial resolution. To create a synthetic

distribution feeder, we summed the household-level Pecan

Street time series data, and then downsampled the sum to

30-minute resolution, matching the NSRDB data. We did not

include line loss or other grid effects in this feeder model.

III. DISAGGREGATION MODEL

A. Structural Time Series Model

Let Pt denote the observed, feeder-level net power consumption at time t, St denote the downstream, distributed

PV generation, and Lt denote the gross load consumption.

In addition, we denote the solar irradiance metric as φt, and

the 1 × m gross load covariate vector as Xt (see III-B below

for an explanation of the temperature covariates). We model

these quantities as evolving as follows:

St = βtφt + 

(s)

t

βt = βt−1 + 

(β)

t

Lt = Xtγ + Lt−1 + 

(l)

t

Pt = St + Lt

(1)

where the random terms 

(·)

t ∼ N(0, σ2

·

), which in the case of

the terms 

(s)

t

and 

(l)

t

represents evolution in the gross load

and PV generation not explained by their respective covariates.

Since the right-hand side of Eq. (1) is linear in the state

variables, we can define a “synthetic” state variable yt =

[St, βt, Lt, 1]Tand then rewrite (1) into a state space model:

yt = Zt(γ)yt−1 + ωt

Pt = Ayt

(2)

where:

A

T = [1, 0, 1, 0]

Zt(γ) =









0, φt, 0, 0

0, 1, 0, 0

0, 0, 1, Xtγ

0, 0, 0, 1









(3)

ωt = [

(s)

t

, 

(β)

t

, 

(l)

t

, 0]T

In addition to specifying the model structure as in (1) or (2),

the Bayesian approach also requires setting prior distributions

over all unknown parameters. We use the following priors:

S1 ∼ N(μS, Σ

2

s

)

β1 ∼ N(0, Σ

2

β

)

L1 ∼ N(μL, Σl)

γ ∼ N(0, Σγ)

(4)

There are several options to handle the unknown Σ

2

·

in

(4), as well as the σ

2

·

and from (1). The package used here,

bsts for the R statistical computing language, assigns them

empirically-derived inverse-gamma priors and then estimates

them simultaneously with the other unknown parameters [14].

However, for simplicity of presentation we will treat them as

simply known.

B. Load Model

We model the relationship between gross load and temperature using a lagged temperature model, which reflects

that load commitments are often not due to instantaneous

ambient temperature. Furthermore, model performance using

lagged temperature demonstrated improvements over other

temperature models used in previous model iterations, which

for brevity are not discussed here [15]. To determine the

appropriate temperature lags we randomly designated one

day in both August and January as training data, and the

remaining six days as testing data. All performance metrics

were averaged over all such partitions of training and test data.

Given training temperature data T ̃

t and gross load L ̃t,

temperature lags were determined using the cross-correlation

function:

CC(k) = F

−1

h

F[T ̃ ]

∗

(ω)F[L ̃ ](ω)

i

(k) (5)

Where F is the Fourier transform. All timepoints k1, ..., km

corresponding to local maxima of |CC(k)| were selected as

lags for our model. Define the k-lag operator by:

Lk(Tt) = (

0, t < k

Tt−k, t ≥ k

Then the covariate vector Xt was defined as:

Xt = [Lk1

(Tt), ..., Lkm(Tt)]

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 2
C. Model Fitting

Let y = [y1, ..., yt]

T

and P = [P1, ..., Pt]

T

(with similar

vectors defined for all other times series). Given the priors in

(4), we can now write the joint posterior distribution over the

state space:

Pr[y, γ|P ] ∝ L(P ; y, γ)π[y, γ]

Pr[S, L, β, δ, γ|P ] ∝ N(μS, Σ

2

s

)N(0, Σ

2

β

)N(μL, Σ

2

l

)×

N(0, Σ

2

γ

)× (6)

Yn

t=1

[N(βtφt, σ2

S

)N(βt−1, σ2

β

)×

N(XT

t γ + Lt−1, σ2L)]

To estimate the disaggregated S and L we marginalize over

the state variables β and γ:

[S, L|P ] = Z[S, L, β, γ|P ]dβdδdγ

A point estimate of the disaggregated PV generation and true

load can then be obtained by taking the expectation of the

joint posterior over S and L:

(Sˆ, Lˆ ) = E[S,L|P ][S, L]

Uncertainty in this point estimate is represented by the dispersion of [S, L|P ] about the expectation. While there are a

number of measures of dispersion, the most straightforward

choice for this purpose (and the one we will use) is the p%

probability region centered at the mode of [S, L, β, δ, γ|P ]

(ie. the p% quantile region).

In practice, the functional form of Eq. (6) is too complex

to handle analytically. We therefore use Markov Chain Monte

Carlo (MCMC) to sample from it, and use the empirical statistics of the samples (mean, quantiles, etc.) as approximations

of the true quantities. Given the model (2), we can perform

MCMC using a generalization of the Kalman smoother [13],

[16].

IV. RESULTS

A. Model Comparisons

We compared the performance of our model to five other

state-of-the-art methods for disaggregation which use optimization approaches. We tested the Contextually Supervised

Generation Estimation (CSGE) model [8], as well as the four

models proposed in [9], which we will refer to as Models

A-D (reflecting their labels in the original source). Multiple

versions of CSGE have been proposed, however due to data

availability constraints we implemented an older iteration

found in [17]. Full descriptions of these models can be found

in their corresponding citations.

To fit the BSTS model we iterated the sampling algorithm

for N = 105iterations, using the bsts package for the

R statistical computing language. Burn-in length was estimated automatically by bsts. Fitting the comparison models

was performed with the CVXOPT and the SciPy Optimize

packages for Python. All fitting was performed on a laptop

computer (Intel Core i5-4300U with 16GB of RAM). Posterior

sampling for the BSTS model took approximately 6 minutes

and 38 seconds, while fitting the CSGE model took .7 seconds,

and fitting Models A-D took .08 to 82.1 seconds.

We compare the performance of the six models under

consideration using the relative Root Mean Square Error

(rRMSE). For an estimate xˆt being compared to true values

xt, this is defined:

rRMSE =

s

1

N

P

N

t=1

(ˆxt − xt)

2

maxt(|xt|)

(7)

Each model’s performance estimating PV generation and gross

load is listed in Table I. For our proposed method, rRMSE

has been averaged over all training/testing partitions. We see

that the BSTS method achieves significantly lower rRMSE

than other state-of-the-art methods, and that therefore on

average it estimates PV and gross load more closely than

the optimization methods. While in January the BSTS model

performs only slightly better than CSGE at estimating true PV,

it performs substantially better at estimating gross load than

any other method.

In addition to these metrics, which are valid for all considered models, we also assess the performance of the BSTS

estimates using metrics which are only defined for probabilistic estimates. Specifically, we look at the Prediction Interval

Coverage Probability (PICP) [18], which is defined as:

PICP =

1

N

X

T

t=1

1t (8)

where 1t is an indicator variable which is 1 when the target

variable is within the bounds of the prediction interval (in

this case the 95% quantile of the posterior density), and 0

otherwise. Since our model assumes that PV generation is

strongly determined by GHI, when GHI=0 the prediction interval collapses and so PICP for these times will not accurately

reflect model performance. We therefore calculated PICP using

all time-points, as well as using only time-points where GHI

TABLE I

DISAGGREGATION PERFORMANCE FOR JANUARY AND AUGUST DATASETS

Model St rRMSE (Jan) St rRMSE (Aug)

BSTS 5.37% 9.74%

CSGE 6.75% 22.36%

Model A 11.54% 15.93%

Model B 10.41% 15.80%

Model C 10.52% 18.54%

Model D 12.07% 15.70%

Lt rRMSE (Jan) Lt rRMSE (Aug)

BSTS 11.77% 9.40%

CSGE 18.43% 21.53%

Model A 24.88% 15.25%

Model B 24.48% 15.17%

Model C 24.01% 30.07%

Model D 26.02% 15.04%

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TABLE II

BSTS CALIBRATION AND PRECISION BY TARGET VARIABLE

Metric St (Jan) Lt (Jan) St (Aug) Lt (Aug)

PICP 24.41% 78.08% 44.96% 85.50%

PICP, daytime 53.41% 52.84% 77.30% 76.40%

PINAW 5.89% 21.15% 15.29% 20.74%

> 0 (ie. the daytime), which gives a better picture of model

accuracy.

While the PICP is determined by prediction accuracy, it

also depends on the choice of prediction interval width. By

choosing an especially wide interval, PICP can be increased,

without a concurrent increase in model performance. We therefore pair it with the Prediction Interval Normalized Average

Width (PINAW), which measures the interval width of our

estimates. It is defined:

PINAW =

1

NW

X

T

t=1

(Ut − Lt) (9)

where W is the range of the target variable, and Ut and Lt

are the prediction upper and lower bounds at time t. When

the PICP is near 1 (high accuracy) and the PINAW is near 0

(high precision) it implies that the probabilistic algorithm is

performing very well.

The performance of the BSTS model in terms of these

probabilistic metrics is given in Table II. The daytime PICP

indicates that our 95% prediction intervals contain the true

value only about 50-75% of the time, despite being wide

relative to the scale of the data. This suggests that our model

is overconfident, which is a common problem with Kalman

Filters when the underlying model doesn’t fully capture the

true dynamics of the data. Indeed we see that this problem

is worse for the January data, where the load model seems

structurally unable to capture an unusual mid-day reduction

in gross load (as shown in Fig. 1). We expect that further

developing our model to more faithfully track load dynamics,

for example by leveraging reactive power measurements, could

yield improvement in this area.

V. DISCUSSION

The BSTS model generally performed quite well relative

to the benchmark comparisons (Table I), however there were

some cases where no model performed well. Figs. 1 and 2

show disaggregated signal estimates from the BSTS and CSGE

models for a day where both models struggled to accurately

estimate true load. This appears to be due to a sudden decrease

in load which occurred mid-day several times in the January

dataset that was not captured by the covariates used in either

model. This unexpected drop leads to different behaviors in

both the BSTS and CSGE models, which highlight some of

the core differences between the BSTS approach and the other

optimization approaches.

In Fig. 2 we see that the estimates produced by CSGE

exhibit large, unfounded swings during the middle of the day.

This is likely because the surface of the loss function used by

0

50

100

150

200

0 5 10 15 20

Time (hr)

PV Generation (kW)

Estimated PV

True PV

80

120

160

200

0 5 10 15 20

Time (hr)

Gross Load (kW)

Fig. 1. BSTS estimated PV generation and gross load for one day in the

January dataset. Gray ribbon indicates 95% credible interval.

0 5 10 15 20

Time (hr)

0

25

50

75

100

125

150

175

200

PV Generation (kW)

Estimated Load

Gross Load

0 5 10 15 20

Time (hr)

80

100

120

140

160

180

200

220

Gross Load

Fig. 2. CSGE estimated PV generation and gross load for one day in the

January datase

CSGE is relatively flat near its minimum and so its argmin is

not a good estimate, as there are many solutions near it which

perform almost identically; the CSGE model is “indifferent”

between a range of possible disaggregations. The flatness of

the loss function is likely exacerbated by the drop in gross load

being inconsistent with the model (although similar problems

occurred on days where the drop was not present). On the

other hand, the BSTS estimate does not exhibit this erratic

behavior. Instead, the model represents its indifference with

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 4
increasing uncertainty (visible in the gray confidence band in

Fig. 1), and since the posterior expectaion averages over this

uncertainty, the resulting Lˆ in the BSTS model is better-fitting

and less volatile.

Another feature of the BSTS model which contributes to its

performance is that it allows the βt term to evolve in time in a

systematic way. This is a unique feature of the BSTS approach,

and it allows our model to overcome deficits in the GHI data,

which does not always correlate well with PV generation due

to the discrepancy in spatiotemporal resolution between the

NSRDB and the Pecan Street dataset, as well as the absence

of PV generation site configuration data (the PV cell zenith and

azimuth at each generating site). While we employed a fairly

simple model of the coefficient’s evolution, more sophisticated

models are possible accounting for seasonalities in the solar

site generating capacity. Similar models can be constructed

for other covariates, depending on the particular needs of the

dataset, for example allowing the power factor to have timedependence in a model which includes reactive power.

While our developed BSTS method performs well, relative

to other methods, it also faces some challenges. In Table II

we see that our model is somewhat overconfident, suggesting

the our model is unable to capture some dynamics in the PV

generation and gross load. Furthermore, the MCMC algorithm

used to fit the BSTS model had a higher computational cost

than any of the models based on optimization. Although

our model was fit within the time resolution of the data,

shorter timescales may prove challenging. Fitting time can

be reduced by running multiple sampling chains over several

nodes. Furthermore, the comparison in fitting time is not

perfect as the BSTS model also produces uncertainty estimates

“for free”. Estimating uncertainty in an optimization approach

requires, such as bootstrapping, which adds computational cost

not reflected in their implementation here.

VI. CONCLUSION

In this paper we have analyzed the efficacy of Bayesian

Structural Time Series for probabilistically disaggregating

consumer load and PV generation at the feeder-level. Using

Pecan Street Inc. AMI data we demonstrated that our proposed

method performs better than other state-of-the art PV disaggregation models, particularly in the August dataset. In addition

to accurately estimating the component signals, the novelty

in our approach is that it also quantifies the uncertainty in

the estimates. We also show that it has more favorable failure

modes, behaving less erratically when the problem is weaklyconstrained. The work presented here suggests a number of

ways to develop the BSTS model further. Estimation accuracy

of disaggregated PV generation and gross load can be achieved

by employing more sophisticated load models, and fitting time

can be reduced through a more sophisticated implementation

of the sampling algorithms. This framework can also be

extended to estimate generation and load at the household

level, although this would likely require higher resolution

irradiance and environmental data. Finally, given forecasts of

the relevant covariates, the BSTS approach can be extended

to provide probabilistic forecasts, both of the net load as well

as of the disaggregated, behind-the-meter PV generation.

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This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 5