Prosumer Response Estimation Using SINDyc in Conjunction with Markov-Chain Monte-Carlo Sampling
Abstract
:1. Introduction
2. Methodology
2.1. A Sparse Nonlinear System Identification Algorithm
- Structure selection is computationally demanding, especially for high dimensional problems.
- The extrapolation capabilities of the power series are sub-optimal.
- Polynomial models suffer heavily from the curse of dimensionality.
- The capability to approximate a broad group of target problems;
- Low sensitivity to noise;
- Global explanatory capabilities.
Algorithm 1: Sweep over the set of regularization coefficients and identification of SINDyc models. It returns when the sparsity level satisfies a chosen criterion, and at a minimum, an evaluation function examines the stability properties of the model. |
|
2.2. Probabilistic Model
2.3. Probabilistic Model Inference
Algorithm 2: Probabilistic model inference: using a candidate model derived using SINDyc in Algorithm 1, MCMC is performed on the system observations . |
|
2.4. Excitation Model
3. Results
3.1. Polynomial Prediction Model
3.2. Model Coefficient Distribution Inference Using MCMC
3.2.1. An Exemplary Prior PDF
3.2.2. Probabilistic Model
4. Discussion
5. Materials and Methods
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Algorithmic Symbols | |
count_nonzeros | Count nonzeros in |
count_values | Count values in |
evaluation_function | Model evaluation function |
MCMC_function | Function performing MCMC |
MCMC_params | Parameters to MCMC_function |
nnz | Number of non-zeros in |
nval | Number of values in |
select_Xistar_function | Function selecting from the set of candidate models XI |
SINDyc_params | Arguments passed to the SINDyc algorithm |
sparsity_threshold | Permissible fraction of nonzero elements in |
XI | List of candidate models |
Mathematical Symbols | |
Regularization coefficient | |
Free system dynamics | |
Set of regularization coefficients | |
Forced system dynamics | |
Prior of residual error | |
Prior of sparse model coefficients | |
Sparse candidate model coefficients | |
Posterior predictive check of the system observations z | |
Prior mean of residual error | |
Prior dispersion of the residual error | |
Prior dispersion of candidate model coefficients | |
Model structure | |
Prior of the residual error | |
Posterior distribution of sparse model coefficients | |
MCMC seed | |
Sparse model coefficients | |
model coefficient | |
b | Sweep bounds |
Cluster 0, Cluster 1 | |
Model 0, Model 1 | |
f | Nonlinear function |
m | Probabilistic model prior |
N | Number of observations |
n | Node n |
Number of candidate regularization coefficients | |
Burn-in iterations (warm-up) | |
Numbers of chains | |
Iterations per chain | |
Cluster 0, Cluster 1 | |
P | Prosumer price response |
p | Price offer |
Excitation signal | |
Sampling rate | |
v | Measurement noise |
X | Sys-ID data (system observations) |
x | System states |
y | Model output |
z | System observations |
Abbreviations
DR | demand response |
EV | electric vehicle |
ICo | indirect control |
LASSO | least absolute shrinkage and selection operator |
MCMC | Markov-chain Monte-Carlo |
MLE | maximum likelihood estimation |
NUTS | no-u-turn sampler |
ODE | ordinary differential equation |
probability density function | |
PPC | posterior predictive check |
PR | prosumer response |
RES | renewable energy source |
SINDy | sparse system identification of nonlinear dynamics |
SINDyc | sparse system identification of nonlinear dynamics with control |
SMPC | stochastic model predictive controller |
Sys-ID | system identification |
V2G | vehicle-to-grid |
Appendix A
Symbol | Relation | Distribution |
---|---|---|
∼ | ||
∼ | ||
∼ | ||
v | ∼ | |
∼ | ||
∼ | ||
Symbol | Relation | Value |
= | 1s | |
= | 50 | |
= | 5 | |
= | 100 | |
b | = | |
= | 1000 | |
= | 4 | |
= | 500 | |
= | 101 |
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Banis, F.; Madsen, H.; Poulsen, N.K.; Guericke, D. Prosumer Response Estimation Using SINDyc in Conjunction with Markov-Chain Monte-Carlo Sampling. Energies 2020, 13, 3183. https://doi.org/10.3390/en13123183
Banis F, Madsen H, Poulsen NK, Guericke D. Prosumer Response Estimation Using SINDyc in Conjunction with Markov-Chain Monte-Carlo Sampling. Energies. 2020; 13(12):3183. https://doi.org/10.3390/en13123183
Chicago/Turabian StyleBanis, Frederik, Henrik Madsen, Niels K. Poulsen, and Daniela Guericke. 2020. "Prosumer Response Estimation Using SINDyc in Conjunction with Markov-Chain Monte-Carlo Sampling" Energies 13, no. 12: 3183. https://doi.org/10.3390/en13123183