Modeling Uncertainty in Support Vector Surrogates of Distributed

Energy Resources

Enabling Robust Smart Grid Scheduling

org Bremer

and Sebastian Lehnhoff

Department of Computing Science, University of Oldenburg, Uhlhornsweg, Oldenburg, Germany

R&D Division Energy, OFFIS – Institute for Information Technology, Escherweg, Oldenburg, Germany

Keywords:

Uncertainty, SVDD, Smart Grid, Distributed Generation.

Abstract:

Robust proactive planning of day-ahead real power provision must incorporate uncertainty in feasibility when

trading off different schedules against each other during the predictive planning phase. Imponderabilities like

weather, user interaction, projected heat demand, and many more have a major impact on feasibility – in the

sense of being technically operable by a speciﬁc energy unit. Deviations from the predicted initial operational

state of an energy unit may easily foil a planned schedule commitment and provoke the need for ancillary

services. In order to minimize control power and cost arising from deviations from agreed energy product

delivery, it is advantageous to a priori know about individual uncertainty. We extend an existing surrogate

model that has been successfully used in energy management for checking feasibility during constraint-based

optimization. The surrogate is extended to incorporate conﬁdence scores based on expected feasibility under

changed operational conditions. We demonstrate the superiority of the new surrogate model by results from

several simulation studies.

1 INTRODUCTION

The current upheaval regarding control in the chang-

ing electricity grid leads to growing complexity and

a need for new control schemes (Nieße et al., 2012).

A steadily growing number of renewable energy re-

sources like photovoltaics (PV), wind energy con-

version (WEC) or co-generation of heat and power

(CHP) has to be integrated into the electricity grid.

This fact leads to a growing share of hardly pre-

dictable feed-in. The behavior of such units often

depends on uncertain prediction of projected weather

conditions, user interaction (e. g. hot water usage), or

similar. An algorithm for robust control would coor-

dinate distributed energy resources with a proactive

planning that already takes into account such uncer-

tainty issues for scheduling in order to minimize the

need for ancillary services in case of deviation from

planned electricity delivery. Without loss of gener-

ality, we will focus on algorithms for virtual power

plants (VPP) as an established control concept for

renewables’ integration (Nikonowicz and Milewski,

2012) for the rest of the paper. All concepts are nev-

ertheless applicable for different organizational struc-

tures, too.

Many balancing algorithms for a bunch of differ-

ent control schemes have already been proposed as a

solution to the problem of assigning a suitable, feasi-

ble schedule to each energy unit such that the sum of

all schedules resembles a desired load proﬁle while

concurrently other objectives like minimal cost are

met, too. Among such solutions are centralized algo-

rithms as well as decentralized approaches for VPP as

organizational entity. A VPP can be seen as a cluster

of distributed energy resources (generators as well as

controllable consumers) that are connected by com-

munication means for control. From the outside, the

VPP cluster behaves like a larger, single power plant.

A VPP may offer services for real power provision

as well as for ancillary services (Nieße et al., 2012;

Lukovic et al., 2010; Braubach et al., 2009).

Traditionally, energy management is implemented

as centralized control. However, given the increasing

share of DER as well as ﬂexible loads in the distribu-

tion grid today, the evolution of the classical, rather

static (from an architectural point of view) power sys-

tem to a dynamic, continuously reconﬁguring system

of individual decision makers (e. g. as described in

(Ili

c, 2007; Nieße et al., 2014)), it is unlikely for such

centralized control schemes to be able to cope with

the rapidly growing problem size. Thus, the seminal

work of (Wu et al., 2005) identiﬁed the need for de-

Bremer, J. and Lehnhoff, S.

Modeling Uncertainty in Suppor t Vector Surrogates of Distributed Energy Resources - Enabling Robust Smart Grid Scheduling.

DOI: 10.5220/0005691600420050

In Proceedings of the 8th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2016) - Volume 2, pages 42-50

ISBN: 978-989-758-172-4

centralized control. Examples for VPP are given in

(Coll-Mayor et al., 2004; Nikonowicz and Milewski,

2012). An overview on existing control schemes and

a research agenda can e. g. be found in (McArthur

et al., 2007; Ramchurn et al., 2012).

In order to additionally address the integration of

the current market situation as well as volatile grid

states, (Nieße et al., 2014) introduces the concept of

a dynamic virtual power plant (DVPP) for an on de-

mand formation and situational composition of en-

ergy resources to a jointly operating VPP. In this ap-

proach VPPs gather dynamically together with re-

spect to concrete electricity products at an energy

market and will diverge right after delivery. Such dy-

namic organization even more relies on assumptions

and predictions about individual ﬂexibilities of each

(possibly so far unknown) energy unit when going

into load planning.

Anyway, a general problem for all algorithms is

the presence of individual local constraints that re-

strict possible operations of all distributed energy re-

source (DER) within a virtual power plant. Each DER

ﬁrst and foremost has to serve the purpose it has been

built for. But, usually this purpose may be achieved in

different alternative ways. For example, it is the (in-

tended) purpose of CHP to deliver enough heat for the

varying heat demand in a household at every moment

in time. Nevertheless, if heat usage can be decou-

pled from heat production by use of a thermal buffer

store, different production proﬁles may be used for

generating the heat. This leads, in turn, to different

respective electric load proﬁles that may be offered as

alternatives to a VPP controller. The set of all sched-

ules that a DER may operate without violating any

technical constraint (or soft constraint like comfort) is

the sub-search-space with respect to this speciﬁc DER

from which a scheduling algorithm may choose solu-

tion candidates. Geometrically seen, this set forms a

sub-space F ⊆ R

in the space of all possible sched-

ules.

In (Bremer et al., 2011) a model has been pro-

posed to derive a description for this sub-space of

feasible solutions that abstracts from any DER model

and its speciﬁc constraint formulations. These surro-

gate models for the search spaces of different DER

may be automatically combined to a dynamic opti-

mization model by serving as a means that guides an

arbitrary algorithm where to look for feasible solu-

tions. Due to the abstract formulations all DER may

be treated the same by the algorithm and thus the con-

trol mechanism can be developed independently of

any knowledge on the energy units that are controlled

afterwards.

Up to now, this approach takes into account

merely a hard margin that isolates feasible and infea-

sible schedules. A schedule is either feasible or not.

But, in real life problems this feasibility depends on

predictions about the initial operational state of the

unit from which the schedule is operated. If this initial

state deviates from the predicted, the schedule might

or might not still be operable. The uncertainty in pre-

dicting the initial state of the unit is reﬂected by an

uncertainty about the feasibility of any schedule. This

fact results in a need for a fuzzy deﬁnition of the fea-

sible region that contains all feasible schedules of a

unit.

In this contribution, we extend the model given

in (Bremer et al., 2011) to unsupervised fuzzy deci-

sion boundaries after (Liu et al., 2013) and demon-

strate the superior quality when deciding on feasibil-

ity of schedules under uncertain conditions. We start

with a discussion on related work and brieﬂy recap the

used model technique before we deﬁne the extension

and propose a measure for the conﬁdence of arbitrary

schedules. We conclude with several simulation re-

sults that support the extended approach.

2 RELATED WORK

2.1 Uncertainty in the Smart Grid

Several works scrutinize the problem of uncertainty

within the smart grid in general; mainly by using pre-

deﬁned stochastic models (Alharbi and Raahemifar,

2015). Uncertainty in long term development exami-

nations like (Zio and Aven, 2011) are not in the scope

of this work. Lots of work has been done in the ﬁeld

of wind (or photovoltaics) forecasting, e. g. (Zhang

et al., 2013; Sri et al., 2007), or on integration into

stochastic unit commitment approaches (Wang et al.,

2011), respectively. But, so far surprisingly low ef-

fort has been spent on integration into energy resource

modeling for the case of operability. In (Wildt, 2014)

uncertainty about demand response is integrated di-

rectly into a multi agent decision making process, but

in an unit speciﬁc and not in an abstract way. Integrat-

ing models of correlation in unit behaviour may be

handled by using factory approaches for the scenario

as has been demonstrated for the energy sector e. g. in

(Bremer et al., 2008). An example for modeling reli-

ability and assessment differentiated for different unit

types is given in (Blank and Lehnhoff, 2014).

On the other hand, a need for an abstract unit-

independent surrogate model of individual feasible re-

gions in distributed generation scenarios can for ex-

ample be derived from (Nieße et al., 2012; Hinrichs

et al., 2013a).

Modeling Uncertainty in Support Vector Surrogates of Distributed Energy Resources - Enabling Robust Smart Grid Scheduling

2.2 Surrogate Models for DER

Abstract surrogate models in the energy management

sector are usually built on a set of feasible schedules

that serve as a training set for deriving the surrogate

model. The procedure starts with initializing a unit

behavior model with a parametrization from the

physical unit or – in case of simulation – from its

simulation model. These parameters may be directly

read from the unit reﬂecting its current operation

state or may be further projected onto a future state

using the current operation schedule and predictions

on future operation conditions.

Whereas in the ﬁrst case exact parameters are de-

rived, the latter case usually suffers from uncertainty

from different forecast sources. The initialization

deﬁnes the initial state of the unit at the starting

point from whence alternative schedules are to be

determined by sampling the behavior model which

simulates the future ﬂexibilities of the energy unit. If

the initial operational state of the unit at the starting

point of the time frame over which the energy load

is balanced or optimized is ﬁxed, a surrogate model

can be derived that abstracts from the speciﬁc unit at

hand and allows for an ad hoc integration at runtime

into the scheduling algorithm.

Such models based on support vector approaches

have been presented e. g. in (Bremer et al., 2011).

We will brieﬂy recap this technique before extending

the ideas to uncertainty integration. The model is

based on support vector data description (SVDD) as

introduced by (Tax and Duin, 2004). The goal of

building such a model is to learn the feasible region

of the schedules of a DER by harnessing SVDD to

learn the enclosing boundary around the whole set of

operable schedules.

We will brieﬂy introduce SVDD approach as for

instance described in (Ben-Hur et al., 2001; Bremer

et al., 2011). This task is achieved by determining a

mapping function Φ : X ⊂ R

→ H , with x 7→ Φ(x)

such that all data points from a given region X is

mapped to a minimal hypersphere in some high-

or indeﬁnite-dimensional space H (actually, the

images go onto a manifold whose dimension is at

maximum the cardinality of the training set). The

minimal sphere with radius R and center a in H that

encloses {Φ(x

)}

can be derived from minimizing

kΦ(x

) − ak

≤ R

+ ξ

with k · k denoting the

Euclidean norm and with slack variables ξ

≥ 0 that

introduce soft constraints for sphere determination.

Introducing β and µ as the Lagrangian multipliers,

the minimization problem for ﬁnding the smallest

sphere becomes

L(ξ, µ, β) = R

−

∑

+ ξ

− kΦ(x

) − a

)kβ

−

∑

(1)

∑

is a penalty term and determines size and accu-

racy of the resulting sphere by determining the num-

ber of rejected outliers. Usually C reﬂects an a priori

ﬁxed rejection rate.

After introducing Lagrangian multipliers and fur-

ther relaxing to the Wolfe dual form, the well known

Mercer’s theorem may be harnessed for calculating

dot products in H by means of a Mercer kernel in data

space: Φ(x

) · Φ(x

) = k(x

, x

); cf. (Sch

olkopf et al.,

1999). In order to gain a more smooth adaption, it is

known (Ben-Hur et al., 2001) to be advantageous to

use a Gaussian kernel:

, x

) = e

−

2σ

−x

. (2)

Putting it all together, the equation that has to be max-

imized in order to determine the desired sphere is:

W (β) =

∑

k(x

, x

)β

−

∑

i, j

k(x

, x

). (3)

With k = k

we get two main results: the center a =

∑

Φ(x

) of the sphere in terms of an expansion into

H and a function R : R

→ R that allows to determine

the distance of the image of an arbitrary point from

a ∈ H , calculated in R

by:

(x) = 1 −2

∑

, x) +

∑

i, j

, x

). (4)

Because all support vectors are mapped right onto the

surface of the sphere, the radius R

of the sphere S can

be easily determined by the distance of an arbitrary

support vector. Thus the feasible region can now be

modeled as

F = {x ∈ R

|R(x) ≤ R

} ≈ X . (5)

Initially, such models have for example been used for

handwritten digit or face recognition, pattern denois-

ing, or anomaly detection (Chang et al., 2013; Park

et al., 2007; Rapp and Bremer, 2012). A relatively

new application is that of modeling feasible regions

and constraint abstraction for distributed optimization

problems especially in the ﬁeld of energy manage-

ment, e. g. as used in (Hinrichs et al., 2013a).

Using SVDD as surrogate model within a VPP

control algorithm starts with generating a training set

of feasible schedules for a speciﬁc energy unit with

the help of a simulation model of this unit. Thus, a

schedule is a vector x ∈ R

consisting of d values

for mean active power to be operated during the re-

spective time inerval. In a ﬁrst step, the simulation

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

−

classiﬁcation

schedule distance

k k k

comparison

support vectors

questionable

schedule

sgn( )

Figure 1: Scheme for using SVDD as surrogate model for

checking feasibility of operation schedules in distributed

energy management after (Sch

olkopf, 1997).

model is parametrized with the estimated initial oper-

ation state of the unit (e. g. the temperature of a ther-

mal buffer store attached to a co-generation plant) at

the future point in time that marks the start of the time

frame for which a cluster schedule for the VPP is to be

found. A cluster schedule as result of an (distributed)

optimization process assigns a schedule to each en-

ergy unit within the VPP such that the sum of all

individual schedules resembles a given target sched-

ule (often an energy product to be sold at market) as

close as possible. This simple case is an instance of

the multiple choice constraint optimization problem

(Hinrichs et al., 2013b); for each unit a schedule has

to be chosen from the feasible region of that unit. Of-

ten, further objectives like cost are concurrently opti-

mized. Because the units are not necessarily known

at compile-time and in order to be able to implement

the control strategy independently, surrogates with a

well-deﬁned interface are used for checking feasibil-

ity during optimization. In this way, a simulation

model for each unit is parametrized with a predicted

initial operation state and generates a training set of

feasible schedules for training the SVDD classiﬁer

that in turn is used by the problem solver for check-

ing feasibility (Bremer et al., 2011). The process of

checking feasibility for the VPP case is depicted in

ﬁgure 1. A so far unaddressed problem is that of inte-

grating uncertainty issues in such models. Integrating

uncertainty into support vector data description has so

far led to only a few approaches. For instance, (Zheng

et al., 2006) introduced a fuzzy approach for the data

clustering use case. With the help of a fuzzy deﬁ-

nition of membership that determines for each point

whether it belongs to the training set or not, they con-

trol the rate of hyper volume and outlier acceptance.

Another approach with fuzzy constraint treatment is

(k)

∗

Figure 2: Basic idea of an decoder for constraint-handling

based on SVDD (Bremer and Sonnenschein, 2014).

given by (GhasemiGol et al., 2010). A different ap-

proach is taken in (Liu et al., 2013). An individual

weighting is introduced allowing for a differentiated

consideration of accepted errors. Thus, data points

with a higher conﬁdence have a larger impact on the

decision boundary. Equation (6) shows the respective

extension to (1) in the last term.

L(ξ, µ, β) = R

−

∑

+ ξ

− kΦ(x

) − ak

)β

−

∑

(κ[x

]ξ

(6)

A deﬁnition of a problem speciﬁc differentiated con-

ﬁdence value for weighting has so far not been intro-

duced. Liu et al. used the SVDD distance of a ﬁrst

training run as weighting for a second run. Here,

we may later harness some a priori information for

a more speciﬁc weighting.

In equation (6) the last term determines the trade-

off between accepted error and hypersphere volume

like the last term in equation (1). In contrast to the

standard version equation (1), each point x

is individ-

ually weighted according to its individual conﬁdence

of membership to the positive class by κ[x

]. κ gives a

measure for the reasonability of x

We will later use this approach for modelling un-

certainty in the use case of energy management. Be-

forehand we brieﬂy discuss a specialized application

for the SVDD model of feasible region: the ability to

be used as decoder.

Modeling Uncertainty in Support Vector Surrogates of Distributed Energy Resources - Enabling Robust Smart Grid Scheduling

2.3 Decoders for Scheduling

In order to be able to systematically generate feasi-

ble solutions directly from the search space model, a

decoder approach had been developed on top of the

support vector model instead of merely telling feasi-

ble and infeasible schedules apart. In (Bremer and

Sonnenschein, 2013) a so called support vector de-

coder has been introduced. Basically, a decoder is a

constraint handling technique that gives an algorithm

hints on where to look for feasible solutions. It im-

poses a relationship between a decoder solution and a

feasible solution and gives instructions on how to con-

struct a feasible solution (Coello Coello, 2002). For

example, (Koziel and Michalewicz, 1999) proposed

a homomorphous mapping between an n-dimensional

hyper cube and the feasible region in order to trans-

form the problem into a topological equivalent one

that is easier to handle. In order to be able to derive

such a decoder mapping automatically from any given

energy unit model, (Bremer and Sonnenschein, 2013)

developed an approach based on the mentioned sup-

port vector model (Bremer et al., 2011).

Provided the feasible region of an energy unit has

been encoded by SVDD, a decoder can be derived as

follows. The set of alternatively feasible schedules

after encoding by SVDD is represented as pre-image

of a high-dimensional sphere S. Figure 2 shows the

situation. This representation has some advantageous

properties. Although the pre-image might be some

arbitrary shaped non-continuous blob in R

, the high-

dimensional representation is a ball and geometrically

easier to handle with the following relations: If a

schedule is feasible, i.e. can be operated by the unit

without violating any technical constraint, it lies in-

side the feasible region (grey area on the left hand

side in ﬁgure 2). Thus, the schedule is inside the pre-

image (that represents the feasible region) of the ball

and thus its image in the high-dimensional represen-

tation lies inside the sphere. An infeasible schedule

(e. g. x in Fig. 2) lies outside the feasible region and

thus its image

lies outside the ball. But, some im-

portant relations are known: the center of the ball, the

distance of the image from the center and the radius

of the ball. One can now move the image of an infea-

sible schedule along the difference vector towards the

center until it touches the ball. Finally, the pre-image

of the moved image

is calculated to get a sched-

ule at the boundary of the feasible region: a repaired

schedule x

∗

that is now feasible. No mathematical

description of the original feasible region or of the

constraints are needed to do this. More sophisticated

variants of transformation are e. g. given in (Bremer

and Sonnenschein, 2013).

svdd

classiﬁer

usw-svdd

classiﬁer

train train

foreach

schedule

check feasibility

conﬁdence score

model

vary

predict

model

training set X

generate

parametrize

& instantiate

parametrize

& instantiate

model class

scenario

compare

Figure 3: Integration of conﬁdence scores and evaluation

scheme for comparing both classiﬁers.

3 MODELING CONFIDENCE

In order to model the uncertainty in a schedule’s op-

erability we deﬁne the conﬁdence of a schedule as the

share of variations of the initial state that still allows

operating the schedule without any modiﬁcation. Let

X be a set of d-dimensional schedules x

that is going

to serve as training set for building the SVDD model.

X has been generated by assuming operation of a unit

U starting from an initial operation state z

∈ Z

at a

certain future point in time with the set Z

of all pos-

sible operation states. This set is unit speciﬁc. To give

an example, Z

in the case of a co-generation plant

might be in the simplest version the set of assignments

for the state of charge (SOC) of an associated thermal

buffer store. Let Ω(z

) bet a set of variations of z

and

F [Ω(z

)] the set of schedules x

∈ X that are opera-

ble from any state in Ω(z

) without modiﬁcation. We

now deﬁne the conﬁdence of a schedule p ∈ X as the

ratio

κ[p] = P(p ∈ F [Ω(z

)]|p ∈ F [z

])

|{x|x ∈ F [z]∀z ∈ Ω(z

)}|

|Ω(z

(7)

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

Table 1: Improved classiﬁcation for a boiler with different water drawing proﬁles and different variations in usage prediction.

draught w

σ / kJ SVDD csw-SVDD SVDD csw-SVDD SVDD csw-SVDD ∆ / %

135 0.313 ± 0.341 0.513 ± 0.381 0.281 ± 0.324 0.501 ± 0.372 0.308 ± 0.339 0.505 ± 0.384 74,37

90 0.527 ± 0.341 0.674 ± 0.319 0.532 ± 0.340 0.699 ± 0.298 0.527 ± 0.341 0.674 ± 0.319 30.24

45 0.841 ± 0.190 0.904 ± 0.149 0.860 ± 0.187 0.918 ± 0.120 0.841 ± 0.190 0.904 ± 0.149 6.99

27 0.943 ± 0.085 0.965 ± 0.072 0.949 ± 0.081 0.969 ± 0.066 0.943 ± 0.085 0.965 ± 0.072 2.15

18 0.969 ± 0.053 0.981 ± 0.048 0.973 ± 0.047 0.984 ± 0.038 0.970 ± 0.053 0.981 ± 0.048 0.91

In this way, the conﬁdence is the probability of still

being operable if a given variation is applied to the

initial operation state that had been taken as assump-

tion for generating the training set of feasible sched-

ules.

What remains open is the deﬁnition of variation

in initial states. The actual design of such variation

highly depends on the unit type at hand and on its

embedding into the actual operation site. For this rea-

son, this question cannot be answered in general here.

In this paper we deﬁne variation for our simulations

in a scenario speciﬁc way.

By using equation (6) instead of equation (1) in

the SVDD part of the surrogate model for the feasi-

ble regions of energy units (and for the derived de-

coder) and by using the expectation value of the fea-

sibility of a schedule under changed conditions for

the units operations as deﬁned in (7) as a score for

the conﬁdence of the schedule, we deﬁne the con-

ﬁdence score weighted extension to the surrogate

model (csw-SVDD) used in (Bremer et al., 2011).

4 RESULTS

We tested the approach with a simulation study. For

this purpose we used appliances with a characteris-

tics that allows for a well deﬁned simulated varia-

tion in initial operation state. We have chosen an

under-counter water boiler, a co-generation plant and

a fridge as example units for electricity generation as

well as demand. All models had already been used

in several studies and projects for evaluation (Bremer

et al., 2010; Bremer and Sonnenschein, 2013; Neuge-

bauer et al., 2015; Hinrichs et al., 2013a; Nieße and

Sonnenschein, 2013).

Fridge: A fridge allows for modelling different vari-

ations. We tested two variants: variations in

changing the thermal mass and variation of the ex-

pected start temperature.

Co-generation: For co-generation plants (CHP) we

modeled errors in expected weather conditions re-

sulting in differences for the usage of the con-

currently produced heat. Hence, we co-simulated

Table 2: Comparison of decoding errors as portion of cor-

rectly constructed schedules for different forecast devia-

tions using the example of a boiler with predicted hot water

demand.

σ / kJ SVDD csw-SVDD

9 0.074 ± 0.243 0.003 ± 0.056

18 0.350 ± 0.445 0.025 ± 0.152

27 0.524 ± 0.469 0.098 ± 0293

45 0.738 ± 0.411 0.594 ± 0.468

67.5 0.842 ± 0.355 0.804 ± 0.387

CHP together with the heat losses of a house

based on weather forecasts.

Water boiler: By keeping a water reservoir within a

certain temperature range by an electrical heating

device, electricity consumption can be scheduled

with rather few constraints. Assuming the tech-

nical insulation setting as ﬁxed, losses are merely

dependent on the ambient temperature difference.

On the other hand, possible variations in schedul-

ing load depend on the predicted usage proﬁle for

water drawing. Setting the ambient temperature

ﬁxed, the initial state for scheduling is determined

by the temperature of the water in the tank and

the proﬁle for predicted water drawing during the

scheduling horizon. For variations, we modeled

different prediction errors for the usage proﬁle.

In order to evaluate the improvement of the mod-

iﬁed model we trained two models with basically the

same training set of feasible schedules generated from

the simulation model of the unit which is also used for

evaluation of both surrogates. Figure 3 shows the set-

ting of the basic evaluation scenario.

Each scenario comprises a speciﬁc model class for

an energy unit and a prediction for an initial state

which serves as parametrization for instantiating a

model of the energy unit. From this model a training

set X of feasible schedules is generated. Each sched-

ule consists of a ﬁxed number d of values for consec-

utive mean real power at which the unit can be op-

erated without violating any constraint. This training

set serves for training a classic SVDD classiﬁer surro-

gate model for testing feasibility of a given schedule

without a need for the actual energy unit model. At

the same time each scenario contains a unit speciﬁc

Modeling Uncertainty in Support Vector Surrogates of Distributed Energy Resources - Enabling Robust Smart Grid Scheduling

deﬁnition of variation σ for the initial state. This vari-

ation is used to generate a set of models, each with

a random variation. Each schedule in the training set

is then checked for feasibility with each of these var-

ied models. The expectation value of feasibility un-

der a certain variety of initial operation states (given

the schedule x was feasible under the ﬁxed, predicted

initial operation state) serves as conﬁdence score κ[x]

for training a csw-SVDD. Finally, both classiﬁers can

be compared by using classical classiﬁer evaluation

methods (Powers, 2008; Witten et al., 2011). To eval-

uate the classiﬁer performance, we calculated the con-

fusion matrix by comparing classiﬁer and the origi-

nal model that had been used for generating the train-

ing set and derived standard indicators for comparison

(Powers, 2011). Feasibility of a randomly (equally

distributed) generated schedule is tested for feasibil-

ity once with help of the classiﬁer and once with the

help of the unit model. In each scenario, 10000 varia-

tions have been used to ﬁnd the expectation value κ.

Table 1 shows some results for a water boiler.

In this scenario we estimated a given water proﬁle

for hot water drawing as predicted usage. Hot water

usage strongly determines feasibility of a given

electrical proﬁle. As variations we generated random

deviations from the given water proﬁle of a given

size by adding normally distributed values with

given standard deviation σ (negative drawings were

corrected to zero for plausibility) ranging from 18

to 135 kJ per 15 minute time interval. We tested

scenarios with a duration of one hour with a 15

minute resolution and the following artiﬁcial draw-

ing proﬁles: w

= (180 kJ, 0 kJ, 0 kJ, 720 kJ),

= (0 kJ, 1440 kJ, 180 kJ, 540 kJ) and

= (180 kJ, 90 kJ, 90 kJ, 180 kJ).

The absolute performance (depicted is the recall

value) degrade fast with growing uncertainty in both

classiﬁers. This is as expected because of the growing

deviation from the expected initial state. Neverthe-

less, the csw-SVDD performs better in all cases and

the mean relative improvement and thus the advan-

tage grows with growing error in prediction. Table

2 shows the results (error rate of not correctly gener-

ated schedules) for a decoder built from the respective

classiﬁer for proﬁle 3 only. The results for the decoder

part are not as good as for the classiﬁer model part but

nevertheless signiﬁcant.

For evaluating the classiﬁers we primarily use the

Table 3: Comparison of classiﬁer recall indicators (higher

values are better) for further scenarios.

scenario SVDD csw-SVDD

fridge 1 0.927 ± 0.045 0.994 ± 0.025

fridge 2 0.747 ± 0.028 0.829 ± 0.085

chp 0.838 ± 0.260 0.884 ± 0.226

Table 4: Comparison of classiﬁer accuracy for a simulated

boiler with 24-dimensional schedules and deviations (σ) in

predicted water usage of different size in a different number

of time intervals.

σ, n SVDD csw-SVDD

60, 3 0.8431 ± 0.0939 0.9371 ± 0.0966

120, 1 0.8644 ± 0.1323 0.9507 ± 0.1051

120, 2 0.6802 ± 0.2196 0.9372 ± 0.1062

120, 3 0.5175 ± 0.2658 0.9027 ± 0.1291

recall indicator. The precision degrades in both cases

signiﬁcantly. This is immediately apparent. The pre-

cision reﬂects the likelihood of a found schedule be-

ing feasible (Powers, 2011). Because feasibility here

is checked under changed preconditions and feasi-

bility is a property of the schedules, precision de-

grades in both classiﬁers at approximately the same

level. The new csw-SVDD classiﬁer for energy re-

sources surrogate modeling shows but a higher recall

behavior, because the recall reﬂects the likelihood of

a schedule being feasible even under changed condi-

tions. But this is exactly what is needed for the use

case of checking feasibility of a schedule during en-

ergy management operations.

Table 3 shows some further results for a 2 hour

time frame. For fridge 1 an unpredictable user in-

teraction was simulated by adding a random thermal

mass (30 ± 5 kJ, equating to about 500 g of food with

room temperature) to the reefer cargo in the fridge.

For fridge 2 a variation of the predicted starting tem-

perature was introduced. In the CHP scenario the

thermal demand was varied to simulate a deviation

from the weather forecast.

Finally, table 4 shows some results for longer time

periods with 24-dimensional schedules. Again, these

are boilers with variations in a limited number of n

time periods. Due to a lack of real world data, a nor-

mal distribution of the variations has been assumed in

all simulations according to (Stadler, 2005). This as-

sumption is likely to become invalid in practice. An

advantage of the chosen approach for the csw-SVDD

surrogate is the ability to derive the decision boundary

unsupervised from the conﬁdence scores of the indi-

vidual schedules in the training set regardless of the

underlying distributions. In this way, the approach

can be used unchanged for individual variations of

newly implemented and integrated energy resource

models; even if they are introduced later at run time.

5 CONCLUSION

Predictive energy management for balancing or plan-

ning electricity demand and production according to

operation schedules needs predictions of future opera-

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

tion alternatives and thus information about ﬂexibili-

ties of all devices for the scheduler to choose from.

Such predictions on ﬂexibility found meta-models

as representations of individually restricted search

spaces. Whether such a predicted operation schedule

is actually still operable when it comes to ﬁnally oper-

ating the assigned (optimal) ones depends on several

certain predictions that where made while construct-

ing the training set of probably feasible schedules.

A robust planning algorithm should take into ac-

count this uncertainty of operability already during

the phase planning. For the use case scrutinized in

this contribution, robustness of a schedule is deﬁned

by the operability even under changed circumstances

and preconditions. This ability of a schedule is con-

densed into a conﬁdence value that allows individ-

ual weighting during the training phase of the search

space meta-model.

To achieve this goal, we adapted an approach for

conﬁdence integration in classiﬁcation, added a con-

ﬁdence model speciﬁc for electric ﬂexibilities for op-

eration schedules, and demonstrated its applicability

with several use cases. The results are already promis-

ing for the model part and call for further extension

especially regarding concrete deﬁnitions of variety

and conﬁdence for speciﬁc unit types.

Future work will have to target a better integra-

tion of uncertainty into decoders as well. If this

is achieved, a more robust scheduling within virtual

power plants will lead to a better support for the

integration of ﬂuctuating renewable resources. For

the case of surrogate modelling this was already im-

proved with the approach proposed here.

balance

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