Power System Optimization Problems: Game Theory Applications

Sudha Balagopalan and Ravishankar S.

Vidya Academy of Science and Technology, APJ Abdul Kalam Technological University, Thrissur, India

Keywords: Game Theory, Power System Optimization, Pricing, Graph Theory, Resource Allocation.

Abstract: We look at the conflict situation in power system problems from an optimization perspective and use Game

Theory (GT) concepts for modelling and solving the problem. In order to model the conflicts effectively, we

first identify the players, the optimizing quantity and the optimizing platform. This paper details two power

system problems and present a case study. We also identify two more areas where the same principles may

be applied. Though our work focuses on Cooperative Game Theory (CGT), an extension to the Non-

Cooperative Game Theory (NGT) platform is possible. Since GT is more relevant to a market structure, we

use market engineering principles including multilateral trades, differential pricing, inverse elasticity rule,

graph theoretical allocation, etc. as tools for organizing the optimization process. A useful addition for

inducing stability in the decision making process is the concept of ‘Power Vectors’ borrowed from sports and

game parlance for ceding players. Results are encouraging, with a transmission loss reduction of more than

70% in a five bus and 40% in a 24 bus system. We conclude that both versions of GT, the CGT and NGT are

powerful tools for optimization in a practical scenario with conflicts and contradictory incentives.

1 INTRODUCTION

Many power system problems are conflict ridden

requiring application of Game Theory (GT) concepts

to mathematically model the complexities and

optimize the ‘live’ variables. Conflict situations

warrant the need to compete or cooperate/ negotiate

and accordingly choose strategies to maximize

benefits via a rational decision making. In GT, the

decision variables contributing to the benefits are

‘live’ and evolve continuously based on strategies.

This decision evolution procedure in a Cooperative

Game Theory (CGT) model uses coalitions of players

to maximize social welfare. Coalitions to reap more

gains and share benefits are outcomes of shared

information. However striking a discord and falling

out of coalitions before the grand coalition is formed

is a disruptive eventuality to be addressed in such

scenarios. If decision making is simultaneous,

without sharing of information, the operational

structure is non-cooperative (NGT) and the force-

majeure is the competition embedded in the game.

A distinctive feature of CGT is the ‘characteristic

function’ which is maximized via formation of

coalitions. Since the objective is conflict resolution,

all perceived road-blocks or desirables may be used

to embellish the characteristic function. Moreover,

the strategies focus on maintenance of accord among

coalition partners via an acceptable pay-off or sharing

of benefits. This is to prevent the partners leaving the

coalition for greener pastures at any point of time.

Similarly for pay-offs in NGT, the strategies opted by

the players are with the objective of achieving

equilibrium, also called the Nash equilibrium. This is

a ‘minimax’ solution, since each player minimizes the

maximum payoff possible for other players if the

game is zero-sum; they simultaneously minimize

their own maximum loss. Thus situations with many

complex options and each with several outcomes are

most suitable for GT based modelling. The

assumption and/or the hitch is that only rational

decisions are taken by the players and nothing is left

to chance.So, GT concepts can be applied to model

engineering problems with conflicts which can be

resolved by adopting strategies that optimize the

‘live’ design variables.

On examining reported works on applications of

GT concepts to power system problems, it is noted

problems in transmission expansion, loss allocation,

demand/ response management, pricing, smart grid

applications, renewable energy sizing, distribution

networks, etc. are prominent. (Sore et.al 2006, Zhu

et.al 2012, Kreyac et.al 2013, Mediwaththe, 2017,

Mekontso et.al, 2019, He et.al 2019, Chen et.al 2017,

Balagopalan, S. and S., R.

Power System Optimization Problems: Game Theory Applications.

DOI: 10.5220/0009170302550262

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 255-262

ISBN: 978-989-758-396-4; ISSN: 2184-4372

255

Keren 2017, Contreras 1997) In all cases, two

influencing factors observed are: Elasticity of

demand versus price and incentive based commerce.

Both factors make GT option possible in complex

power system parlance. Elasticity is proposed to be

designed and injected into the 4 problems presented

here. And a differential pricing structure for power

system variables like energy, demand blocks, open

access of transmission corridors, etc is adopted to

stimulate the incentive/ disincentive ridden

commerce. Thus the four problems are appropriately

modified and attendant issues addressed as follows.

2 GAME THEORY IN POWER

SYSTEM OPTIMIZATION

Some pertinent questions on GT based optimization

are raised here to define the power system problems

presented, model them and identify the appropriate

solution approaches. The conflicts associated with

specific problems give the impetus for the approach.

 How can the conflicts be represented to apply GT

concepts to arrive at workable solutions?

 How do the conflicts suggest theplayers?

 What would be the decision/ design variables and

how do they evolve and stages thereof?

 What are the strategies that influence the

evolution of the best of solutions?

 What is the level of information availability and

sharing, for designing strategy sets?

 How is the characteristic function formulated?

 What is the outcomeof sharing the benefits?

Optimum power flow in corridors and allocation

of transmission price are modelled here. Two recent

reported works, power system islanding and then

restoration, both with conflicting requirements, are

presented as amenable to GT applied model with

evolved solutions based on incentives and

disincentives. The conflict modelling, both in the

CGT and NGT platforms, are projected here.

2.1 Optimizing Power Flow in the

Power Corridors

Some conflicts to be resolved in restructured power

markets requiring modelling via GT concepts follow

1. Due to uneven generation and usage pattern in

grids, quantum of power moved over transmission

lines is large. In some corridors power shuttling is of

the order of 3-5 times the total transactions, causing

unacceptable congestion, voltage drops and power

losses in lines unless optimal transactions are made.

2. An otherwise useful Optimal Power Flow (OPF)

analysis tool has little significance in an electricity

market which has distributed and closely guarded

information. Also,OPF focuses on influence of

generators on energy prices to derive line impacts.

These impacts as feedback inputs cannot target

control of abuse of the power lines. This is especially

true due to distributed ownership of generation and

transmission assets and their conflicting incentives in

an electricity market.

3. In an electricity market, generating companies

(GENCOS) do not reveal sales data and capture

maximum power portfolios resorting to even profit

cuts from energy prices. Then more distribution

companies (DISCOS) buy cheaper energy leading to

congestion and other problems on the network; thus

the end users make profits while the afflicted party,

i.e. the Transmission Provider (TP) provides access

for both use and abuse of the network.

4. However, power flows obey Kirchhoff’s laws

only and no contractual laws. While in the erstwhile

system, roles and responsibilities and a centralized

authority were assigned, an electricity market is ‘free-

for-all. Then TP is the only entity who can exercise

control on the ‘runaway’ on lines. Then TP should

optimize trades of the GENCOs and conduct least

loss iteration. However, too many rules of the road

like curtailment, loading vector etc. cannot be laid

down by the TP since competition is hampered, the

reason for the development of electricity markets.

Thus, though optimization has applicability as a tool,

the implementation of its findings needs other market

engineering tools as in GT concepts.

2.1.1 GT Modelling of Power Transactions

Some choices are made to model the problem in the

GT platform. (Varaiya 1997, Sudha 2011)

1. A multilateral market structure is chosen over the

pool. Thus a cartel, prone to dangerous power and

market games and commercial considerations

outweighing engineering requirements is averted.

Since no reliable cost-benefit data is revealed by the

market agents a transaction model independent of the

economic data of the end-users is the best.

2. The distinctive entities are thus the GENCOs,

DISCOs and the central TP. The GENCOs have

influence over determining the Energy Charges (EC).

The TP is given the prerogative over the construct of

the Transmission Service Charge (TSC). Finally the

choice of determining who to buy from and in what

ratio will lie with the DISCOs.

3. To implement least loss formulation, an appeal to

the selfish profit motive of the end-users is the game

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

256

plan. This encompasses them in a game to collude or

cooperate so as to minimize the impact on the grid. In

cooperative games, modus operandi includes both

collusion and cooperation and so is inferred as the

best choice. This is because collusion is not

detrimental to the ultimate aim of least loss and other

impacts on lines as evinced by the TP, but serves only

to increase the cohesion between the cooperating

agents. Another reason for choosing CGT is that GT

is one of the strongest tools of market engineering, of

which CGT uses both the nuances of coalition- threat

(when with others) and promises (when with oneself),

leading to more stable coalitions. If maximum

benefits are given to the grand coalition, when all

loads are transacted, then the combinatorial process

of coalition formation is faster and this thought

process is employed.

4. DISCOs play the game because they can form best

coalitions, being privy to all local information. This

helps to identify partners causing counter-flows and

yet enhance their trades in a multilateral set-up.

5. Four phases of CGT are viewed here in a

multilateral trade structure. 1. DISCOs derive Local

information and based on merits become decision

authorities. 2. TP computes and broadcasts Central

information vital for the next phase using the

communicated trades. 3. In the Common Information

Derivation and Negotiation phase DISCOs divulge

information conducive to trades with some trades

dropped, some increased, exchanged or even shared

for individual, coalitional and group benefits. In this

manner a set of stable coalitions or the grand coalition

are formed in Phase 4 and the result is committed and

accordingly scheduled.

The TSC is next designed as the characteristic

function, which dictates decisions and negotiations.

2.1.2 TSC - Characteristic Function

The design of the TSC answers three questions:

1. What is the benefit of cooperating in coalitions?

2. In an asymmetric environment how do the agents

locate reliable / fruitful partnerships?

3. How can agents ascertain that what is bought is

what is got since electricity is fungible?

TSC is designed (Sudha 2011) considering some

entity interactions andbenefits of co-operating and

honouring transactionsas the characteristic function.

a. Instead of passively providing open access, the TP

manipulates the situation such that each DISCO is

forced to compare TSC with EC and arrive at a

compromise solution. The design should penalize all

unacceptable transactions and be the lowest for a least

loss formulation for a set of loads at any time.

b. Since the DISCOS and GENCOs have

contradictory intentions and strategies only DISCOs

are chosen to play. On comparing EC and TSC, the

DISCOs shift their contracts to more profitable

GENCOs. Since GENCOs do not control both EC and

TSC, market is not skewed. DISCOs are accountable

for the loss and play to reduce it.

c. Market engineering principles are beacons in the

design of prices since perception exists that electricity

sector has little elasticity. Experimenting with

differential pricing and penalized deviations from

least loss condition, elasticity of transmission and an

empowered design of TSC was achieved.

The features in the TSC enable the DISCOS and

their coalitions to identify scope for partnerships with

an understanding on how to share the TSC. In an

information asymmetric market only a few sources

are reliable. Therefore the concept of power vectors

from graph theory has been developed and put to use

in three phases of the game. The main idea of power

vector is that a node derives maximum power from

subservient nodes and their chain of successors.

Using power vectors, if DISCO A say appraises

DISCO B as having a high strength it means that B

makes a good candidate for causing counter-flows in

relevant lines via B’s trades. Also, the benefits for

agent A continue to increase if B’s trades enhance

further. But the appreciation needs to be mutual for

any meaningful dialogues between the two agents in

the common information phase.

In short the players of the CGT based game are

DISCOs; the design variables are the trades with

minimized TSC. Strategies of DISCOs identify

suitable coalitions causing counter-flows in lines.

Information asymmetry is resolved through coalition

derived information. Stability for the solution is

ensured via power vector governed pay-off design.

The TSC, as the bone of contention gives the design

of payoff vector for forming coalitions. If feasible, it

must lie in the solution space of the game.

2.2 TSC Sharing Problem

To match power allocated by GENCOs with trade

contracts, fungible nature of electricity is a deterrent.

In a trade based market, it is distressing that after all

the haggling and negotiations between DISCOs and

GENCOs, power flow from a specific generator to a

precise load is affected by other trades on the grid,

even gaming. In short in an electricity market with an

assembly of trades, what is bought is not what is got;

negotiation entered into by the agents is based on

pseudo- trades. Since all trades mix on the grid

system, contracts exist even if there is nil or minimal

Power System Optimization Problems: Game Theory Applications

257

allocation from a generator and impact on the lines

still have to be accounted for. Game and Graph theory

are used to resolve such ambiguities.

2.2.1 CGT and Transmission Pricing

CGT comes hand in phase3; when two DISCOs or a

conglomerate consider further mergers, after

resolution using graph theoretical allocation. The

resource allotment enables the partners to negotiate

further, contract more profitable trades at lower grid

impact and by cooperating, a lower TSC. Moreover,

divergence from a common understanding is tackled

using Ramsey pricing rule. Thus, via best trades at

every coalitional step, the grand coalition is reached.

Diametrically opposite to the existing power

system, the chain of events in the electricity markets

leading to power flow starts from the other end of the

spectrum i.e. trades are contracted and thereafter there

is a confluence of trades on the transmission lines. So,

first demand and supply is visualized as a set of

trades, then assembled via a combinatorial process.

Trades merge with an eye on TSC, two at a time in

the first iteration, all the time going for the least loss

trades. In subsequent iterations the fused trades

continue to coalesce till the grand coalition or a set of

stable coalitions is reached. The attainment of the

solution, i.e. the payoff vector is the proof of the

feasibility of the proposed model. Of the several

techniques, the marginal vector, if it can be obtained

is the stable and unique solution and is strived for.

2.2.2 Stable Coalition for Sharing TSC

1. The construct of TSC and the elastic curves lend

convexity and superadditivity (Herings et.al 2006,

2007) to the game, resulting in an economically

feasible payoff vector existing in the solution space.

Convexity implies that more benefits accrue as more

agents join the coalition. Hence, the TP reveals the

maximum benefit only, which a grand coalition alone

can earn. At intermediate steps the perceived

minimum of TSC is used for evaluating the coalitions.

ThusTP dispatches all loads or assembles all the best

trades contracted i.e. schedule generators to meet all

loads and losses.

2. Permutational convexity in a game implies

incentives for including more, higher ranked players

in the coalitions for a specific ranked permutation.

Also with a permutationally consistent power vector

(rank order echoes the power vector) the socially

stable solution core has the marginal vector, a highly

desirable unique payoff vector or solution. Social,

technical, commercial, considerations can rank the

players and a combination of these is proposed as a

power vector, also for checking consistency.

3. A permutationally convex game is designed with

a compatible power vector to ensure the presence of

the sole marginal vector in the socially stable core.

(Permutational compatibility ensures that no agent

can hijack the game in her favour.) The locally

computable power vectors and the derivation of

hierarchy proposed are workable and compatible with

all the steps taken so far and hence the solution is

rationally true and acceptable.

4. Certainly, though not unique, a valid solution

space is realized. It was checked for balancedness

using power vectors of coalitions that sustain these

payoff vectors. All these steps assure a socially stable

core which is needed for stability an absolute

necessity to prevent anarchy on the network.

2.2.3 TSC and Power Vectors

For a network with n nodes, L lines, line flow z, and

line loss q, if weights for penalizing loss, sum of

power flowing in all lines and flow in congested lines

are a ($/MW

h), b and d ($/MWh) respectively and

embedded cost is cin ($/hr.), then the price function

p(q)in $/hr is













∑







∑





(1)

To measure power of players (Herings et.al 2001)

ascribe to a node, power, from both the number as

also power of its successor. Let



be the collection of

irreflexive digraphs on the vertex set

{1, 2 , , }Nn

with (, )ij N N



 denoting the arc



, ((i,j) є A

(node i dominates j). The positional power function is

the function

R which maps each





() ( )

AITs





(2)

Here

is the adjacency matrix of

, with the

ij entry



if (, )ij is an arc of

and

otherwise;

is the score vector giving the number of

successors of each node. The TSC is used as the

characteristic function and power vectors are used to

form coalitions and also to design the socially stable

core as shown in the case study in section 3.

2.3 Islanding of a Power System

The problem of optimized islanding of power system

using GT concepts, on fault clearance is briefed.

1. The conflicts to be addressed are: number and

extent of islands, coherency of generators within, and

the power corridors to be evacuated, priority of loads.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

258

2. The players are the GENCOs (of islands based on

geographical proximity/ parts thereof).

3. The characteristic function is to be derived as a

quadratic, penalizing the most unwanted outcomes of

islanding such that an elastic curve can be derived.

4. The phases of coalition formation based on a

power vector design which shows the technical

features of the consequence of islanding.

5. The solution with pay-off vectors.

Depending on the catastrophic event preceding it,

the restoration process also is to be optimized.

Figure 1: A 5 bus system.

2.3 Power System Restoration

1. All agents will seek priority in getting the power

restored, because of the enormous economic and

social implications with aligned conflicts.

2. The islanded areas and some privileged loads will

be the players of this optimization game.

3. To use NGT and arrive at a Nash equilibrium.

4. The solution is to be iteratively inducted with the

outcome of optimal restoration vector as pay-off.

3 A CASE STUDY

A case study for a 5 bus system for the first two

problems in optimization is given below. Briefly the

following phases are explained:

1. A set of .multilateral trades are derived from the

problem using graph theoretical allocation (Table 1)

(Wu, 2000, Varaiya, 1999, Penh et.al 2002).

2. Next is the local phase computations and

derivation of all required information by all DISCOs.

(Sample cases of 2 DISCOs deriving power vectors

for their own trades and other useful data for

negotiations are given in Table 2, 4).

3. The coalition move in the direction of data related

to optimal trades using power vectors (Table 2, 3, 4,).

4. Optimal trades are derived & scheduled and

committed at the best total TSC (Table 4).

5. Sharing of the benefits in the final phase, which

proves the feasibility of the method and is reflected in

the final core is derived. (Fig. 3). (Appendix).

Table 1: Division of power demand and flows into trades.

Disco on buses Genco on bus 1 Genco on bus 2

No: (Demand) Load (MW) Load (MW)

2 (20MW) 13.623 6.376

3 (45MW) 39.544 5.456

4 (40MW) 30.463 9.536

5 (60MW) 41.374 18.628

Total load 129.74 40

Line loss 4.77MW

Sum of power flow i

all lines

262.6 MW

Table 2: Local Information computation: Power Vectors.

Disco2 buying 20MW from

Genco1

Disco3 buying 45MW

from Genco4

Bus No:

Lines with

Power

vector

Lines

with t

Power

vector

Ref 0-1 1 .2401 0-4 1 .2736

1 12,1-3 2 .4402 1-3 1 .2021

2 2-0 1 .2067 1-3,2-3 2 .4024

3 2-3,3-4 2 .4347 3-0 1 .2123

4 2-4,4-5 2 .4013 From 4 3 .6414

5 2-5 1 .2012 2-5 1 .2338

Table 3: Coalition Formation using Common Information.

Bus

Power

Vector

Graph theoretical allocation of load

Ref 0.5187 Load Gen. 1 Gen. 4

coalitio

n {2,3}

1 0.4528 20MW 13.322 6.678MW

2 0.4620 45MW 5.886MW 39.11MW

3 0.2532 40MW 0 40MW

4 0.6598 Loss .231MW -

5 0.2437 Total 19.439 85.798

Table 4: Optimal trades as derived in the Central

Information Derivation Phase.

Optimal Trades in MW

bus(demand) Load- Genco 1 Load- Genco 4

2(20) 13.846 6.154

3 (45) 5.002 39.996

4 (40) 0 40

5 (60) 23.654 36.546

Total 44.102 122.696

Loss 1.6 MW

Sum of power flow in all lines: 163.7 MW

4 CONCLUSIONS

It is mooted that game theory offers a very suitable

platform to model complex situations in power

system optimization problems. The whole idea of GT

Power System Optimization Problems: Game Theory Applications

259

concepts encourages choices and hence is fertile

ground for having different perspectives for deciding

the players of the game. The benefits nor its

maximization process is narrow framed and offers

plenty of research opportunities. The three phases of

CGT was demonstrated to successfully coordinate

multilateral trades using two tools, a suitable TSC and

power vector and that the Socially Stable game is

instrumental in ensuring stable trades. The case

studies on 5 bus and 24 bus (not shown here) power

systems reveal the following advantages.

1. In a 5 bus, 169.74 MW demand system with a loss

of 4.44 MW and a total power shuttling over the lines

of 262.6 MW is optimized to a power system with 1.6

MW loss and a total power of 163.7 MW shuttling on

the lines.

2. A 24 bus system with a demand of 1219 MW, and

36.355 MW loss optimizes to 15.43 MW loss and

power shuttling dropping from 3825 to 2805 MW via

GT concepts.

All contributions to the process are based on

market engineering techniques which are more

applicable, suitable and acceptable.

In Figure 2 is given one such contribution where

a coalition based optimization is visualized as a step

in the negotiation phase.

Figure 2: Least loss iteration by coalition {2,3}.

Another contribution is indicated in Figure 4

where a sample of a TSC designed in a novel manner

such that the elastic nature is utilized by the DISCOs

for least loss iteration.

The derivation and adaptation of such vectors at

each stage of the GT based optimization is another

contribution, especially since it has been imported

from the sports and games field to cede players. Here,

the powerful use is for deciding by the agents,

initiating the trades, the best partner to obtain counter-

flows and thus reduce TSC as the partnership deal

between the coalition partners.

The inherent choice factor, its capacity to promote

competition and scope for negotiation and extraction

of hidden information, resolve the uncertainty factor

in an information asymmetric complex scenario. In

conclusion it can be said that the biggest engineering

advantage of GT is that solution of the problem

becomes a common agenda and a unifying force, even

in a profit motivated milieu, where commercial

considerations overrule engineering requirements.

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APPENDIX

Figure 3: Socially Stable Core for a 5 bus TSC sharing

game.

Table 5: Optimizing trades derived in, local and negotiation & common Information Derivation Phase.

Buyer

Disco

Seller Genco Power Trade in MWline loss ∑z or ƒ

Z or ƒ

in 1-3 Z or ƒ

in 4-5

2 1 20 .0674 25.43 3.143 .5714

2 4 20 .0869 36.19 3.429 3.619

3 1 45 .6017 93.86. 16.71 5.143

3 4 45 .1794 53.79 1.929 1.714

5 1 60 1.571 139.9 12.86 18.1

5 4 60 1.307 117.9 6.857 27.24

{2,3} 1 65 .8954 108.4 19.86 4.571

{2,3} 1,4 (20,45) .2314 69.07 5.072 2.286

{2,3) 4 65 .4417 82.81 13.29 1.524

(2,3} 4,1 (20,45) .3485 85.83 1.5 5.333

{2,3}- (G1-13.32& 5.89 to 2&3),

(G4-6.68 & 39.11 to 2&3)

.2312 69.02 4.812 2.406

{2,5} 1 80 2.016 158.7 16 18.67

{2,5} 1,4 (20,60) 1.43 132.5 3.714 27.89

{2,5} 4 80 1.567 132.9 9.429 21.71

{2,5} 4,1 (20,60) 1.742 146.9 10.29 30.86

{2,5}- (G1-12.73&21.91 to 2&5)

(G4-7.27 & 38.09 to 2 & 5)

1.3622 124.1 1.095 25.58

{3,5} 1 105 3.099 194.5 29.57 12.95

{3,5} 4,1 (45,60) 1.622 150.6 14.79 19.81

{3,5} 1,4 (45,60) 1.415 145.3 9.857 22.1

{3,5} 4 105 1.651 154.1 4.929 28.95

{3,5}-(G1-5 & 25.76 to 2 & 5 ) (G4-40 & 34.24 to 2

&5 )

1.3512 141.3 5.177 24.27

{2,3,5}-(G1-13.85,5,23.45 to 2,3&5 ) (G4-

6.15,40,36.55 to 2,3&5)

1.604 163.7 5.607 26.1

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Figure 4: Elasticity curve- Demand Vs TSC for 165MW met by generation at bus 1&2.

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