Privacy-Preserving Self-Organization in Distributed Energy Scheduling

Joerg 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

ﬁ

Keywords:

Privacy Preservation, Distributed Optimization, Multi-Agent Systems, Secret Sharing.

Abstract:

Negotiation among agents that are controlling and orchestrating a set of distributed processes often relies on

frequent data exchange to allow solution evaluation and thus convergence towards a joint solution. Solving

decentralized coordination problems with coalitions of agents that exchange messages and information to build

beliefs for problem solving, inevitably allows insight into other agents’ operational options. Keeping local

information private is thus of utmost importance for a wide user acceptance of such algorithms. We present an

extension to a distributed, self-organizing algorithm for energy scheduling in virtual power plants or energetic

neighborhoods that keeps all information about possible operations of participating energy resources private.

For calculations during optimization the algorithm relies on secret sharing and joint multi-party computations.

We evaluate the algorithm against the original non privacy-preserving standard version and present some

insights for future work.

1 INTRODUCTION

Multi-agent systems are widely seen as one of the

most promising solutions for optimization and asyn-

chronous coordination in future, autonomous cyber-

physical systems like the energy grid (Stark et al.,

2024). Digitalized energy systems can be character-

ized by a high degree of complexity in monitoring and

controlling a large number of distributed energy re-

sources. The infrastructure is dynamically optimized

with regard to generation, consumption, load ﬂow,

supply quality and safety, cost, comfort, and more.

Multi-agent systems are widely seen as the best

tool to integrate autonomy by self-organization prin-

ciples and to cope with the speciﬁc problem charac-

teristics in the smart grid (Ramchurn et al., 2012).

Whereas trust has been on the research agenda

in multi-agent systems for years (Ramchurn et al.,

2004), privacy has received less attention so far. Trust

is a major issue when humans have to interact with a

system. Many CEOs identify trust and reputation as

the key driver of their action. On the other hand, pri-

vacy is just as important for a broad acceptance of dis-

tributed, agent-based algorithms (Rapp and Bremer,

2023).

An example is given by the predictive scheduling

use case. In virtual power plants (VPP), a task that

has often to be solved is the scheduling problem that

assigns an optimal operation schedule to each energy

resource. Of course, the algorithm has to take into ac-

count a set of objectives like accurate resemblance of

the desired load proﬁle, robustness of the schedule,

costs, remaining ﬂexibility for subsequent planning

periods, and many more (Stark et al., 2024; Bremer

and Lehnhoff, 2020a). A schedule in this context is

a real-valued vector with each element denoting the

amount of energy (or mean active power) generated

or consumed during the respective time interval for

a given future discrete planning horizon. A simple

predictive scheduling algorithm just tries to assign a

schedule to each energy resource such that the sum of

all schedules resembles a desired aggregated schedule

as close as possible. Such aggregated schedule might

for example be the result of some market bidding and

the VPP wants to jointly operate this schedule.

As a distributed, self-organizing solution to pre-

dictive scheduling, (Hinrichs et al., 2013) proposed

the combinatorial optimization heuristic for dis-

tributed agents (COHDA). COHDA was proposed as

a solution to problems that can be decomposed on

an algorithmic level (Talbi, 2009) and can thus be

adapted to a wide range of different problems (Stark

et al., 2021; Narayanan et al., 2024; Bremer and

Lehnhoff, 2017b; Bremer and Lehnhoff, 2017a). The

general concept is closely related to Cooperative Co-

evolution (Potter and Jong, 2000). The key concept of

Bremer, J. and Lehnhoff, S.

Privacy-Preserving Self-Organization in Distributed Energy Scheduling.

DOI: 10.5220/0013094900003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 1, pages 253-262

ISBN: 978-989-758-737-5; ISSN: 2184-433X

253

COHDA is an asynchronous iterative approximative

best-response behavior that relies on collected infor-

mation from other agents for own decision making.

In the predictive scheduling use case in VPPs,

operable schedules are frequently exchanged (Hin-

richs and Sonnenschein, 2017) that contain informa-

tion about time and possible energy generation and/

or consumption schemes – often for a whole day with

15 minute (or smaller) time resolution. These sched-

ules are needed by the agents to calculate the dis-

tance of the joint schedule to the wanted target sched-

ule to evaluate solution candidates during local de-

cision making. Each schedule contains data of the

possible portion of energy that may be generated (or

consumed) during a given time period for a series of

multiple time intervals. With each negotiation round,

a new possible operable schedule is sent to several

other agents together with transient information on

other agents’ schedules. This is related to the widely

used gossiping principle that comes into play in many

distributed optimization algorithms (Poli, 1996). The

sent information can be collected, aggregated, and ex-

ploited by other agents.

Simulation examples for exploitation can be found

in (Dabrock, 2018), where detailed information on in-

ternal processes like heating proﬁles, and thus work-

ing hours, machinery load factors, or current capacity

utilization were derived. Moreover, in (Bremer and

Lehnhoff, 2022) individual time of use tariffs were

reconstructed from aggregated, collected schedules.

Parties that are potentially interested in participat-

ing in distributed energy management – e.g. in vir-

tual power plants (Naval and Yusta, 2021) or in en-

ergy neighborhoods (Wehkamp et al., 2020) – will

likely refrain from engaging in agent-based coordi-

nation if the risk of revealing data is larger than the

beneﬁt from participating. In order to achieve a broad

acceptance of such algorithms, it is essential to handle

the disclosure of data sparingly or to otherwise ensure

that it cannot be misused.

We extended the meanwhile widely used COHDA

protocol by integrating privacy preservation through

secret sharing. In this way, no schedules with pri-

vate information have to be sent around anymore and

all calculations for solution candidate evaluation are

done jointly in a way that no party can reveal any in-

formation of a single agent’s energy resource.

The rest of the paper is organized as follows. We

start with an overview on existing privacy concepts

in multi-agent systems and the smart grid. After de-

scribing the standard COHDA algorithm, we identify

the privacy gap and develop the integration of secret

sharing for keeping information private. Finally, we

evaluate privacy-preserving COHDA and its perfor-

mance with regard to solution quality, message vol-

ume and convergence.

2 RELATED WORK

Privacy is probably one of the oldest human concerns

(Schoeman, 1984; Schermer, 2007). Nevertheless,

with the advent of delegating more and more tasks

to autonomous systems that act on behalf of a user

ambiantly in the background, the technical perspec-

tive grows signiﬁcantly in importance. For the infor-

mation technological perspective several taxonomies

for classiﬁcation have been proposed, e.g. (Spiek-

ermann and Cranor, 2009; Bostwick, 1976; Kang,

1997). Thus, privacy as a unitary concept with a uni-

form value (Such et al., 2014).

According to (Solove, 2005), privacy can be

threatened by three main information-related activi-

ties: information collection, processing, and dissemi-

nation. The use case of self-organized, distributed en-

ergy scheduling covers all three and therefor requires

special attention and a holistic solution that covers

the complete algorithmic approach (Rapp and Bre-

mer, 2023). Due to the algorithmic level decomposi-

tion of the solved problem (Talbi, 2009), energy gen-

eration or consumption schedules are collected from

other agents, are processed several times to calculate

the objective value of different solution candidates

and are eventually disseminated to other agents for

further decision making. This general scheme occurs

(at least partly) also in other distributed solutions to

the problem, e.g. in (Bremer and Lehnhoff, 2016).

When it comes to privacy prevention measures in

multi-agent systems in general, several approaches

have already been developed. Some good overviews

can be found in (Such et al., 2014; Chandramohan

et al., 2015). For algorithmic applications, some ef-

fort has been put into consensus algorithms (Wang

et al., 2021; Fiore and Russo, 2019), but for dis-

tributed optimization only few examples using en-

cryption can be found. For example, (Huo and Liu,

2021) uses the Pailler cryptosystem (Wu et al., 2016)

with some inefﬁciency issues (Peng et al., 2004).

For the energy domain, several surveys explore so-

lutions for different use cases: e.g. for vehicle to grid

applications, communication, or metering (Han and

Xiao, 2016; Kumar et al., 2019; Finster and Baum-

gart, 2015). These works cover the collection per-

spective and are not concerned with any algorithmic

use of the data. As possible technologies for secur-

ing distributed optimization algorithms against pri-

vate data leakage, two techniques seem to be most

promising: (partial) homomorphic encryption and

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

254

multiparty computation based on secret sharing (Rapp

and Bremer, 2023).

The concept of homomorphic encryption was

originally introduced as privacy homomorphism by

(Rivest et al., 1978). The ﬁrst working fully ho-

momorphic encryption scheme has been proposed in

2009 by (Gentry, 2009) whereas partially homomor-

phic schemes have been known for longer. Since

then, great progress regarding performance and ap-

plicability has been made (Meftah et al., 2022; Moore

et al., 2014). But, this issue is still not fully solved.

In general, homomorphic encryption allows for arbi-

trary commutations to be conducted on the encrypted

data without a need to decrypt it ﬁrst. The result is

then a cypher text or number in the same cryptosys-

tem. Good overviews on fully homomorphic encryp-

tion schemes can be found in (Marcolla et al., 2022).

Partially homomorphic encryption schemes are

often more efﬁcient but support just a subset of op-

eration, e.g. multiplication or addition only. Ex-

amples can be found in (Wu et al., 2016; Guo

et al., 2022; Mikhail et al., 2014). Also sim-

ple order-preserving schemes exist (Agrawal et al.,

2004). Order-preserving encryption had previously

been used for predictive scheduling in energy man-

agement in the smart grid in (Bremer and Lehnhoff,

2020b). But, in order to make it work, the solved op-

timization problem had to be reformulated, what lim-

its the approach to a simpliﬁed problem version. The

results can thus not be generalized. This seems to be

the general case because no efﬁcient higher level op-

erations are supported that are usually needed in eval-

uating solution candidates in optimization.

Homomorphic encryption is still computationally

rather expensive and would slow down an optimiza-

tion process signiﬁcantly (Alaya et al., 2020; Moore

et al., 2014) or require extensive GPU or in-memory

calculations (Meftah et al., 2022; Reis et al., 2020;

Al Badawi et al., 2018) that cannot be guaranteed

in the ﬁeld in smart grid applications. Additionally,

homomorphic encryption with a public key scheme

(Rothblum, 2011) rather supports client-server or

peer-to-peer structures instead of agent coalitions as

in our use case. The ability to decrypt the result also

allows for decrypting the used data.

A different approach for keeping privacy relies on

the idea of secret sharing. In a secret-sharing scheme,

a dealer shares a secret with n parties P by distributing

shares to the parties such that: (Beimel, 2011):

1. given a set A of subsets of the n parties (the access

structure), a subset in A can reconstruct the secret

from the shares (all in the subset are needed),

2. no one outside A can reveal any (partial) informa-

tion of the secret.

perceive

decide

act

𝐾

for each agent in neighbours

inbox

Sum up most

recent schedules of

other agents

private search

space and

preferences

Figure 1: Flow chart for a single agent in standard COHDA.

This process chain is started each time a message arrives in

the in-box of the agent.

Often, A equals P; except for example in (Dawson

and Donovan, 1994). Hence, all members of P to-

gether are needed to reveal the secret (Gordon and

Katz, 2006). Several applications of secret sharing

for distributed computing can be found for exam-

ple for Byzantine agreement (Rabin, 1983), securing

multiparty computations (Ben-Or et al., 2019; Chaum

et al., 1988), threshold cryptography (Desmedt and

Frankel, 1991), access control (Naor and Wool,

1996), or generalized oblivious transfer (Tassa, 2011).

We used a scheme of secret sharing where each share

constitutes a random part of a horizontally strati-

ﬁed schedule – a vector of random shares of active

power for each time interval – and used multi-party-

computation (Catalano et al., 2005) to prepare objec-

tive evaluation in our distributed self-organizing algo-

rithm. In this way the computationally expensive part

has to be conducted only once for each agent decision.

3 ALGORITHM

3.1 COHDA

An asynchronous iterative approximate best-response

behavior is the core idea of COHDA. Each agent is re-

sponsible for one dimension of the algorithmic prob-

lem decomposition. The intermediate local solutions

of other agents (represented by published decisions)

are regarded as temporarily ﬁxed. Thus, each agent

only searches along a low-dimensional cross-section

of the search space and thus has to solve merely a

simpliﬁed sub-problem. Nevertheless, for evaluation

of the solution, the full objective function is used af-

ter aggregation of all agent’s contributions. In this

way, the approach achieves an asynchronous coordi-

Privacy-Preserving Self-Organization in Distributed Energy Scheduling

255

nate descent with the additional ability to escape local

minima by parallel searching different regions of the

search space; and because former decisions can be re-

vised if newer information becomes available.

Hence, all agents coordinate themselves by up-

dating knowledge and exchanging information about

each other that supports local decision-making. For

message exchange, the agents are logically drawn to-

gether by an artiﬁcial communication overlay net-

work. Most often, a small world topology (Watts and

Strogatz, 1998) is used for this purpose. Starting with

an arbitrarily chosen agent and by passing it a mes-

sage containing just the global objective, each agent

repeatedly goes through three stages: perception, de-

cide, and act.

perception stage. In this ﬁrst phase, the

agent prepares for local decision-making based

on incoming information. Every time an agent

receives a message from one of the neighboring

agents that precede in the directed communication

topology, it aggregates the data that is included

in the message(s) into the own knowledgebase K.

Each message contains two essential pieces of in-

formation: the result of the updated local decision

of the sending agent and the transient informa-

tion about decision updates of other agents that

led to this decision. The important privacy issue is

that this information contains operation schedules

of all other agents (or rather of their controlled

energy resources respectively) and probably also

cost annotations for these schedules.

decision stage. In this second phase, the

agent is making a local decision on a schedule for

the own controlled energy resource that puts the

coalition forward as best as possible. To do this,

a local optimization is solved. The agent deter-

mines the gap to the optimal solution by summing

up the schedules (that are the results of the most

recent decisions) of all other agents and by calcu-

lating the difference to the wanted target sched-

ule. To do this, the agent needs to know the deci-

sions of the others. During optimization, the agent

searches for the local schedule that ﬁlls the gap as

best as possible. As constrained-handling tech-

nique for the individual constraints of the energy

resource, often a decoder approach is used (Bre-

mer and Sonnenschein, 2013).

act stage. During this last phase, the agent com-

pares the best found solution with the previous so-

lution by using the global objective function. If

the new solution is better, the agent broadcasts

a message containing its new local solution con-

tribution together with everything it has learned

from preceding agents and their current local solu-

tion contributions (the decision base) to the imme-

diate neighbors in the communication topology.

Upon these messages, the receiving agents then

also go through these three stages, what in turn

may lead to revised local solution contributions

and thus to a further improved overall solution.

If no local solution can be found that improves the

overall solution, no message is sent and the process

ceases. After the system has generated a series of

intermediate solution candidates, the heuristic even-

tually terminates in a state where all agents know an

identical solution. This one is taken as the ﬁnal so-

lution of the negotiation. Properties like guaranteed

convergence and local optimality have been formally

proven in (Hinrichs and Sonnenschein, 2017).

3.2 Adding Privacy

In order to enable privacy preservation in COHDA,

we need to get rid of sending plain schedule infor-

mation to other agents. We achieve this goal by us-

ing a secret sharing approach. To do this, the local

process of summing up the schedules of all the other

agents was replaced by a joint summation that incor-

porates all agents; with each agent knowing only a

random part of the information. In privacy COHDA

schedules are stored locally so that only the agent that

controls a device knows the schedules that have been

selected as possible contributions to solution candi-

dates. Because each agent has to store several sched-

ules during the course of a negotiation and because

also older schedules need to be kept (for comparison

and because several paths are searched in parallel), we

chose a hash map for local storage. Only the identi-

fying hash code is sent to other agents instead of the

plain schedule. Any identiﬁer type would do.

In standard COHDA an agent a

(for the set of all

agents A) evaluates solution candidates (single objec-

tive case) during decision making by calculating

e = δ

|A|

∑

i=1

,ζ

, (1)

where δ represents a distance measure (often || · ||

)

that measures the similarity between the joint sched-

ule of all agents (that includes solution candidate x

of agent a

) and the target schedule ζ. This can be

converted to

e = δ





∑

∈A\a

,ζ





→

∈F

min. (2)

Thus, we need the sum of schedules of all other agents

to combine with our own solution candidates x

for

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

256

COHDA

msg?

perceive

yes

store

act: store locally

and send hash

sum up and

decide

initialize sum

sum up

split

request ?

partial

schedule?

partial

sum ?

all parts

received?

split and send

partial schedules

sleep until next asnyc. message comes in

store

all parts

received?

for each agent in MAS

send split

request

for each agent in MAS

for each agent in neighbors

inbox

yes

Send to

initiator

Figure 2: Flow chart for a single agent in privacy COHDA. This process chain is started each time a message arrives in the

in-box of the agent. The green framed part corresponds to the standard COHDA process with just perceive-decide-act phases.

evaluation. Only the elements of x

(from the local

feasible region F

) are mutable for optimization; the

sum of others is locally treated as ﬁxed. In privacy

COHDA, the agent does not know the other sched-

ules. Instead, this summation has to be done jointly

by a secret sharing approach. Let x

be the schedule

that is requested from agent a

to be summed up with

the other agents’ schedules. Schedules are scaled to

[0,1] (percent of rated power). Each agent splits up

its own schedule. Agent a

splits up x

into the set

= r

(1)

,...,r

(n)

of random pieces of the same di-

mension as the original schedule (stratiﬁcation):

= {r

(ℓ)

∑

ℓ=1

(ℓ)

= x

∧ r

(

ℓ) ∼ N (0, 1)

}. (3)

The elements r

(1)

,...,r

(n−1)

of set R

are dis-

tributed to the other n − 1 agents; one piece is kept.

This is done by each agent. Then, for each agent

holds: agent a

knows r

(k)

,...,r

(k)

and can calculate

the sum S

∑

(k)

that is sent to the requesting agent

(the one that wants to make a new decision). This

agent ﬁnally can calculate the sum of all schedules by

summing up the sums of random parts from all agents:

S = S

+ ···+ S

Neither the random pieces of schedules nor the

intermediate sums of random parts from different

agents contain any usable information. Yet, the sum

of all parts eventually yields the desired sum. The ran-

dom partition for schedule stratiﬁcation can be chosen

differently every time. Actually, it is even not neces-

sary to reveal the identity of a schedule part owner.

As a result, the decision stage was split up into two

parts and the summation process coordination was in-

tegrated by adding three new message types. Figure

2 shows the new overall process of a single agent.

When a standard COHDA message is received, the

standard perceive stage takes place with the only dif-

ference that the message contains only hash codes of

schedules. The actual schedules and are stored pri-

vately at the respective agents.

With the hash codes the summation of schedules

is initiated by sending a split request message to all

agents. This message contains for each agent: (1) the

individual hash code for the receiving agent (the one

that is part of the joint solution), (2) the agent ID of

the requesting agent (the one to which the results have

to be returned), and (3) a process ID for the new sum-

mation process. The process ID is necessary due to

the asynchronous nature of COHDA. While an agent

is waiting for the result, new COHDA messages might

come in from other agents that spawn additional sum-

mation processes.

If an agent receives a split request message, it

looks up the respective schedule using the received

hash code and splits it up into random parts for ev-

ery agent (including itself). After splitting, one dif-

ferent one of the random partial schedules is sent to

Privacy-Preserving Self-Organization in Distributed Energy Scheduling

257

each agent (including itself) together with the pro-

cess ID and the ID of the initiator. Because the agent

includes itself in the summation process, there is at

least on random part that is not revealed even if all

other agents work together against this agent. After

sending, the agent sleeps again until the next message

comes in.

If an agent receives a partial schedule, it stores the

part together with the process ID and waits for other

partial schedules with the same process ID from other

agents to come in. If the agent has received a par-

tial schedule from each agent with the same process

ID, it sums up these parts. From the received partial

schedules as well as from the sum, no information can

be derived. If all parts are received and the (partial)

sum is calculated, the sum is sent back to the initiator

together with the process ID.

If an agent receives a partial sum it stores it and

waits for the other partial sums. A partial sum is the

sum of random parts from each agent and thus does

also contain no information on a speciﬁc schedule. If

an agent receives a partial sum it is the initiator of

this summation process. As soon as all partial sums

are received, they are added up to the complete sum

of all schedules of all other agents that belong to a

speciﬁc solution candidate. Now, the agent can con-

tinue with the standard COHDA procedure by deter-

mining a schedule for the own controlled energy re-

source based on the sum and the gap to the goal. If

better than the previous solution, it stores the new so-

lution candidate and sends out the hash code for the

new schedule. In Fig. 2 the parts that correspond to

standard COHDA are marked by a green background

(cf. Fig. 1). The rest is an addition to the protocol to

achieve privacy by secret sharing.

4 RESULTS

For our evaluation experiments, we used the follow-

ing scenario: Groups of distributed co-generation

plants (CHP) are supposed to jointly generate a given

energy generation schedule (e. g. given from some

market). Each CHP is controlled by an agent and

all agents together form a coalition and negotiate the

individual operation planning together by using the

COHDA algorithm. As model for the CHP, we used a

well know simulation model that has been used and

evaluated e.g. in (Neugebauer et al., 2015). This

model comprises a micro co-generation plant with

4.7 kW of rated electrical power (12.6 kW thermal

power) and is bundled with a thermal buffer store.

Constraints restrict power band, buffer charging, gra-

dients, min. on and off times, and satisfaction of ther-

Table 1: Results for four CHP generator examples with

different numbers of agents, different time horizons, and

different problem difﬁculty averaged over 100 simulation

runs each. We compare the mean number of messages sent

during negotiation, the mean number of decisions that the

agents made to revise previous solution candidates and the

ﬁnal solution quality measured as symmetric mean absolute

percentage error.

standard COHDA privacy COHDA

50 agents, l = 8

sMAPE 0.46 ± 0.69 3.2 ± 0.81

decisions 4612.1 ± 2490.241 3597.8 ± 301.5

messages 109728 ± 59154.5 6992777.7 ± 501955.3

10 agents, l = 96

sMAPE 0.024 ± 0.095 0.78 ± 0.29

decisions 149.81 ± 45.9 338.43 ± 79.9

messages 620.07 ± 194.307 35971.74 ± 8773.9

100 agents, l = 96

sMAPE 0.12 ± 0.046 0.73 ± 0.39

decisions 13191.9 ± 3861.7 11383.45 ± 4415.5

messages 650751.8 ± 190802.5 93115416.5 ± 40164449.1

10 agents, l = 96, unsolvable

sMAPE 36 ± 1.9 37 ± 1.3

decisions 62.9 ± 7.58 74.55 ± 7.44

messages 259.9 ± 31.1 6906.3 ± 768.6

mal demand. Thermal demand is determined by sim-

ulating losses of a detached house (including hot wa-

ter drawing) according to given weather proﬁles.

100 200 300 400

time/ ticks

error/ ∥·∥

mean current solution

mean best solution

(a)

20 40

80 100 120

time/ ticks

error/ ∥·∥

mean current solution

mean best solution

(b)

Figure 3: Mean convergence behavior of privacy COHDA

(left) and standard COHDA (right).

First, we looked at the convergence of privacy

COHDA. Convergence of standard COHDA has

been formally proven in (Hinrichs and Sonnenschein,

2017). COHDA converges under the condition that

the global objective evaluation function is monotoni-

cally decreasing and no agent sends a message if no

improvement was made. We use the same evaluation

function as in the standard version. On the other hand,

in privacy COHDA messages can be sent also if no

improvement was made because an agent may parti-

cipate in some joint summation. But, such message

does not trigger any new COHDA action (act phase).

So, the conditions for termination still hold in privacy

COHDA. Nevertheless, we analyzed the convergence

behavior. From 100 optimization runs with 20 agents

each, we collected at each point in time the best so

far seen and the currently scrutinized solution quali-

ties from each agent. To do this in an asynchronously

acting system, we used an observer that queried all

agents at global time ticks. As all of these collected

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

258

Table 2: Results for a mixed and a pure consumption sce-

nario with 25 agents each. We compare the mean number

of messages sent during negotiation, the mean number of

decisions that the agents made to revise previous solution

candidates and the ﬁnal solution quality measured as sym-

metric mean absolute percentage error.

standard COHDA privacy COHDA

mixed scenario, 25 agents, l = 96

sMAPE 0.16 ± 0.12 3.7 ± 1.34

decisions 3424.2 ± 1107.52 5259.8 ± 1518.7

messages 41215.1 ± 13310.8 3142462.6 ± 935036.1

25 consumer agents, l = 96

sMAPE 0.36 ± 0.15 2.1 ± 1.45

decisions 2974.1 ± 445.9 4747.5 ± 1508.1

messages 35794.6 ± 5368.8 2557552.1 ± 921311.3

convergence series are of a different length, we aver-

aged the solution qualities for all agents for each time

tick for the ﬁrst 400 ticks.

A sample result is shown in Figure 3(a). Ob-

viously, a rather good solution appears on average

rather early, but needs some time to propagate to the

other agents. This propagation time is longer than in

standard COHDA due to the fact that the summation

takes some time until a solution candidate from other

agents can be considered. In (Hinrichs and Sonnen-

schein, 2014) it has been demonstrated that a certain

degree of additional disturbance in the system leads to

a better solution quality, because more solution can-

didates are considered in parallel. Such disturbance

is already incorporated in both COHDA version by

random message delays. An additional effect due to

the privacy related messages could not be observed al-

though the decision making delay obviously leads to a

larger exploration compared with standard COHDA.

For comparison: Fig. 3(b) shows the mean conver-

gence of standard COHDA for the same scenario.

Compared with this result, privacy COHDA still ex-

plores larger regions in the end instead of switching

to exploitation. Future work will show whether this is

a potential for improvement.

Table 1 shows some results that compare the

achieved ﬁnal solution quality of both approaches. In

order to allow comparability of the results of differ-

ently sized scenarios, all qualities are measured as

symmetric mean absolute percentage error sMAPE =

∑

|ζ

−x

0.5·(ζ

)

. The mean absolute percentage error

allows for comparing scenarios of different size (dif-

ferent number of agents), but in our case the max-

imum negative deviation is due to the nature of the

scenario bounded by zero. Thus, standard MAPE is

biased towards higher values. For this reason, we used

the symmetric version. All results are in percent.

Table 1 shows the results for four different scenar-

ios with different numbers of agents that all control a

CHP and different problem complexities due to differ-

ent schedule lengths. The mean total number of mes-

sages for all scenarios is signiﬁcantly higher for the

privacy case. This is immediately clear, because three

new message types have been introduced. The mean

total number of decisions that all agents make, stays

more or less in the same magnitude. The result qual-

ity slightly degrades except for the unsolvable sce-

nario. Here, both COHDA versions stagnate with al-

most the same sub-optimal solution quality. For all

other scenarios, the global optimum is known to be

zero. In smaller scenarios, standard COHDA seems

to be advantageous in terms of solution quality. In

more realistically sized larger scenarios with 100 or

more agents, the degradation in solution quality for

privacy COHDA is neglectable. That is because it is

smaller than the inherent uncertainty in such scenar-

ios due to forecast errors; e.g. of the thermal demand

(Bremer and Lehnhoff, 2017c).

Table 2 shows two additional results for a mixed

(generation and consumption) scenario comprizing 10

CHP, 10 heat pumps, 3 batteries, and 2 medium sized

cool storages, as well as a pure consumption scenario

with warm water boilers instead of CHPs.

0 20 40

0.1

0.2

0.3

0.4

(a)

0 200 400

600

800 1,000

5 · 10

−2

0.1

0.15

(b)

Figure 4: Mean convergence behavior of privacy COHDA

(left) and standard COHDA (right).

In the last two experiments, we had a deeper look

into the sent messages. obviously, the total number

of messages grows signiﬁcantly compared with stan-

dard COHDA. For standard COHDA, the main im-

pacts on the number of messages used for negotiation

are based on the used topology, the message delay, the

reaction time of individual agents, and the number of

agents. Research on these impacts can be found in

(Oest et al., 2021; Anders et al., 2012; Hinrichs and

Sonnenschein, 2014). This overall weak quadratic be-

havior is also present in privacy COHDA as we use

the same basic principles. Additionally, we can ob-

serve two effects that are due to introducing privacy.

First, we can observe a change in the distribution

of messages over time. Figure 4(a) shows the distri-

bution of messages over time for standard COHDA.

The number of messages grows for some time and

ceases eventually after a single peak. The distribution

over time for privacy COHDA differs in two charac-

teristics (cf. Fig. 4(b)). First, after the main peak,

the duration until ceasing is way longer. Introduc-

ing a distributed calculation of the sum of schedules

Privacy-Preserving Self-Organization in Distributed Energy Scheduling

259

45 50 55 60 65

·10

number of SV

messages

Figure 5: Relationship between the number of agents and

the distribution of the amount of messages that are sent dur-

ing negotiation.

obviously results in a way longer phase of propagat-

ing an already found optimal solution to all agents,

what is necessary for termination (Hinrichs and Son-

nenschein, 2017). Secondly, we can observe several

smaller peaks after the ﬁrst one and thus a damped os-

cillation behavior. The reason for this is so far unclear.

Nevertheless, as COHDA shows an anytime behavior

after some initial warm-up phase (Hinrichs and Son-

nenschein, 2017), the process could be stopped earlier

with still having a feasible (sub-optimal) solution in

case a deadline is approaching. The conditions for the

anytime property still hold for the privacy-preserving

version, so we can stop even earlier – compared to the

total length of the process.

The growth in the number of messages that is in-

duced by the number of agents grow quadratically

(with R

≈ 0.9) what can be seen in Fig. 5 for a

CHP example with 5 to 75 agents. This is quite ob-

vious as for each new agent one additional schedule

has to be summed up and this additionally involves

all present agents. On the other hand, the optimiza-

tion problem that is solved with COHDA also has a

profound impact on the number of exchanged mes-

sages. Some example analysis on other use cases

with partly worse growth behavior can be found in

(Volkova et al., 2019; Buhl et al., 2017; Bremer and

Lehnhoff, 2017a; Radtke et al., 2023).

5 CONCLUSION

Multi-agent systems in the smart grid domain in

which individual agents represent the interests of dif-

ferent (private) operators of energy resources in a

distributed coordination approach for joint planning,

must treat private data with care. This data should

only be disclosed sparingly (ideally not at all) in order

to achieve broad acceptance of such mechanisms. We

presented the integration of secret sharing into dis-

tributed, multi-agent based energy scheduling as one

building block for a privacy-preserving smart grid.

With the presented approach, it becomes possible to

achieve an optimal planning of the energy generation

within a group of distributed energy resources in a

distributed way without a need for disclosing infor-

mation on one’s own production schemes. The evalu-

ation results are promising for up to medium sized vir-

tual power plant or energetic neighborhood scenarios.

For larger scenarios, a research path for accelerating

negotiation could be sketched as a hierarchical multi-

part computation of the sums in order to reduce group

size and thus communication trafﬁc. In this way, pre-

dictive scheduling with integrated secret sharing is a

suitable approach for achieving privacy through data

sparsity in an important class of algorithms in the fu-

ture smart grid.

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