Production Equilibrium in Cooperative

Smart Hybrid Renewable Minigrids

Jayaprakash Rajasekharan and Visa Koivunen

SMARAD CoE, Department of Signal Processing and Acoustics, Aalto University, P.O.Box 13000, FI-00076 AALTO, Finland.

Email: {jayaprakash.rajasekharan.visa.koivunen}@aalto.fi

Abstract-In this paper, we propose a framework for modeling the production of smart hybrid renewable energy minigrids as a production possibility curve. Modeling the individual and joint production possibilities of hybrid renewable minigrids demon­strates the absolute and comparative advantages in terms of en­ergy production that each minigrid has depending upon weather conditions and technology employed. Comparative advantage in a particular technology leads to specialization in production that guarantees mutually beneficial trading possibilities for the minigrids to mitigate the effects of weather variations on energy production. The optimum point of production or the production equilibrium that forms the basis for mutually beneficial trading is arrived at by maximizing the joint production possibility function subject to certain constraints. These constraints are given by consumer theory that prescribes various utility func­tions for optimizing the total energy production and modeling weather/climate variations. We graphically illustrate the com­putation of production equilibrium under various constraints such as output maximization during sunny or windy weather conditions. Once the total amount of energy production as stipu­lated by the production equilibrium has been decided, minigrids can trade towards an optimal level for their consumption. The terms of mutually beneficial trade are dictated by the marginal opportunity costs of energy production for each minigrid. Finally, we formulate the energy trading price ratio which provides the minigrids with an incentive to cooperatively exchange energy.

Index Terms-Smart grids, energy trading, production equi­librium, hybrid minigrids, cooperative game theory.

I. INTRODUCTION

Minigrids are defined as the interconnection of small, modu­lar generation sources to low voltage AC distribution systems [I]. Minigrids typically supply multiple users and although they can operate autonomously, they may be interconnected with (or be part of) the distribution grid of the local utility company [2]. Renewable minigrids may be powered by solar photovoltaics, wind turbines, hydro turbines, or a combination of these technologies. Hybrid renewable minigrids combine different energy sources to provide competitive advantages over minigrids using a single technology [3]. The combination of multiple renewable energy sources could serve as a low­cost solution for rural areas [4]. It brings robustness to the distribution of energy since the benefits and advantages of each technology complement each other. Moreover, it also brings up issues of system control, coordination, sustainability and the role of local utility companies. For example, a mix of renewable energy sources can accommodate seasonal climatic

resource fluctuations, with solar PV collectors complementing wind energy during months with less wind, or picking up when hydro generation drops during the dry season [5]. Where daily energy variations are concerned, solar energy typically has a production peak around noon, while wind power fa­cilities can operate whenever the wind is blowing. Hybrid minigrids may also be served by utility companies that may exogenously provide them energy. Diesel generators could also be used to tide through periods of insufficient production. Additionally, minigrids may have energy storage capabilities that add stability to the system by storing the energy for peak consumption periods when there is insufficient production from the renewable sources. Energy storage provides flexibility in terms of energy trading and pricing options [6]. Thus, the production possibilities of a minigrid need not be restricted to its renewable energy sources, but may also include other sources of energy supply.

Though energy production from renewable sources varies significantly depending upon weather/climate variations, local energy demands must be met by minigrids. One possibility is to oversize [7] the hybrid minigrid such that impact of weather/climate variations is smoothed out to a large extent, but this comes with greatly increased capital cost and wastage of energy that will be produced in surplus for most of the time [8]. Another viable option is to trade energy, i.e., to buy energy from another minigrid during insufficient production periods and to sell energy during surplus production periods. Coop­erative game theory can be used to model energy exchange between minigrids [9] and arrive at an optimal production equilibrium that provides the basis for trading energy. In order to achieve this, it is necessary to know the production possi­bilities of minigrids, i.e., the various combinations of amounts of energy that can be produced by a minigrid using its energy sources. The key question here is, how to model the production possibilities of hybrid renewable energy minigrids by taking weather conditions into account such that a production equilib­rium is achieved while also giving incentives to cooperatively trade energy that is beneficial to both minigrids? Consumer theory [10] can be used to formulate utility functions that capture the variations in weather/climate and also model the constraints for optimization of joint production.

In this paper, we model the individual and joint production possibilities of hybrid solar-wind minigrids with different

978-1-4799-3001-2/14/$31.00 ©2014 IEEE

capacities using a production possibility curve. Weather and climate variations give distinctive advantage (both absolute and comparative) to one minigrid over another in terms of energy production. Comparative advantage in one type of energy production technology leads to specialization which provides an opportunity for trading among minigrids to mitigate the effects of weather variations on energy production. Using con­sumer theory, weather/climate variations are modeled as utility functions. We compute the production equilibrium that forms the basis of mutually beneficial trade as the optimum point at which the joint production possibility curve is tangential to the indifference curves of the utility function. Minigrids can trade towards an optimal energy level for their consumption once the total amount of joint energy production is given by the production equilibrium.

The rest of this paper is organized as follows. We describe the individual and joint production possibilities of hybrid renewable minigrids along with their advantages and special­ization in detail in section II. Consumer theory that is used to arrive at the production equilibrium is described briefly in section III. The production equilibrium is computed and graphically illustrated in section IV along with the terms of mutually beneficial trade. Section V concludes the paper.

II. HYBRID MINIGRIDS

In this paper, we study a simple set-up where we con­sider hybrid minigrids with two renewable energy produc­tion sources, namely, solar and wind generators. However, additional sources of energy supply from utility companies, diesel generators, storage devices and other combinations of renewable energy production sources can also be studied using the same model without any loss of generality. The model can be appropriately scaled up to work with production possibility surfaces in higher dimensions as opposed to the simpler production possibility curves in two dimensions (solar and wind energy) as described below.

A. Production Possibility Curve

Consider two hybrid minigrids that serve a local neighbor­hood of households. Minigrid I consists of a 100 KW solar PV generator (Psd and a 100 KW wind generator (Pw1), whereas minigrid 2 consists of a 200 KW solar PV generator (Ps2) and a 50 KW wind generator (Pw2). The production possibility set (PPS) represents all possible combinations of solar and wind energy that can be produced by two minigrids. This is illustrated using a production possibility curve (PPC) as shown in Fig. l.

The extreme points on PPC indicate the amount of energy that a minigrid can produce by relying on only one technology. For minigrid 1, the PPC implies that over a period of one hour, if it were to rely only on its PVs (sunny, but not windy), then it would produce 100 KWh of solar energy (Esd and if it were to rely only on its wind turbines (windy, but not sunny), then it would produce 100 KWh of wind energy (Ewd. Similarly, for minigrid 2, the PPC implies that over a period of one hour, if it were to rely only on its PV s, then it would produce 200

Product>::m PossiblityCurveforMiniiTK11

" 00 Wloo Ene.-gy ProouGtkln (KWhj

PrC<iJcton Possibilty Curve for Min>:Jrid 2

� � 'MndEtlergyProouctkln(K'M1)

Fig. l. Production Possibility Curves for Minigrid 1 and 2. The production possibility curve depicts all possible combinations of solar and wind energy that can be produced by a minigrid.

KWh of solar energy (Es2) and if it were to rely only on its turbines, then it would produce 50 KWh of wind energy (Ew2). If the weather is both sunny and windy, minigrid I should ideally be able to produce 100+ 1 00 = 200 KWh of energy or in other words, the PPC must extend to the north­east corner of the plot, but this does not happen in reality as it requires technological capabilities (minimizing shadow flicker, switching capabilities, etc.) that cannot be achieved by the minigrid and hence, the peak production capacity of the minigrid is usually not the sum of the production capacities of each individual technology. However, this is a lot better than the scenario where one kind of energy is produced at the cost of another as shown by the dotted blue line in Fig. 1. Any point within the PPC is achievable by the minigrid, whereas points outside the PPC are unachievable. Points lying on the PPC are efficient as it means the minigrid is attaining its peak capacity given the weather conditions.

PPCs are also useful in visualizing the marginal opportunity cost of production of one kind of energy for another. The slope of the PPC at any point measures the marginal rate of transformation (MRT) given a particular technology, i.e., the rate at which production of solar energy can be redirected into production of wind energy or vice-versa. In other words, it represents the marginal opportunity cost, ie., the opportunity cost of producing solar energy in terms of wind energy or vice-versa at the margin.

B. Joint Production Possibility

Now, let us consider the joint production possibilities of both minigrids. If both minigrids relied only on the sun, they would be able to harness 300 KWh of solar energy in an hour and if they both relied only on wind, they would be able to harness 150 KWh of wind energy in an hour. The production possibility set then would consist of all possible linear combinations of solar and wind energy in this range as indicated by the cyan line in Fig. 2.

C. Advantages and Specialization

1) Absolute Advantage: In the above example, minigrid 1 has an absolute advantage in producing wind energy and minigrid 2 in solar energy. This is quite obvious because minigrid I can produce 100 KWh of wind energy in an hour as compared to 50 KWh of minigrid 2, whereas minigrid 2 can produce 200 KWh of solar energy in an hour as compared

Joint Production Possibilities 300�----__ _

I

-PPC1

I

l c: 200 o n " " e D­>-e> Ql

-PPC2 -JPC with specialization -JPC without specialization

r Gains from Trade

� 100 -----9 _________

(;; o (f)

50 100 Wind Energy Production (KWh)

150

Fig. 2. Joint Production Possibility Curves for Minigrids. The individual production possibility curves for the two minigrids are given in green and blue. The joint production possibility curve in cyan depicts all possible combinations of solar and wind energy that can be produced jointly by both minigrids. Minigrid 1 has an absolute advantage in producing wind energy and minigrid 2 in solar energy. Based on marginal opportunity costs of energy production, minigrid 1 has a comparative advantage in producing wind energy and minigrid 2 in solar energy. Comparative advantage leads to specialization that causes the joint PPC of minigrids to be pushed outward to maximize production and attain productive efficiency as indicated by the red curve.

to 100 KWh of minigrid 1. However, absolute advantage does not guarantee that trading energy will result in a mutually beneficial scenario for both minigrids.

2) Comparative Advantage: From Fig. 2, comparing the marginal opportunity costs of production of solar energy in terms of wind energy or vice-versa for both minigrids gives interesting insights. A minigrid is said to have comparative advantage in a technology if its marginal opportunity cost of production using this technology is lower than in other minigrids. For minigrid I, the opportunity cost of producing 100 KWh of solar energy is 100 KWh of wind energy, whereas for minigrid 2, the opportunity cost is around 40 KWh of wind energy. Thus, minigrid 2 has a comparative advantage in solar energy production. Similarly, for minigrid 1, the opportunity cost of producing 50 KWh of wind energy is around 90 KWh of solar energy, whereas for minigrid 2, the opportunity cost is 200 KWh. Hence, minigrid I has a comparative advantage in wind energy production. When specialization takes place according to comparative advantage, both the minigrids can be made better off by trading.

3) Specialization: As mentioned above, it is comparatively advantageous for minigrid I to produce wind energy and for minigrid 2 to produce solar energy. Therefore, minigrid 1 and 2 may want to specialize in wind and solar energy production respectively. By specializing in a production technology, the joint PPC of minigrids can be pushed outward to maximize production and attain productive efficiency as indicated by the red curve in Fig. 2. We also see that specialization results in gains (achieved by means of trading) for both the minigrids in terms of the total possible energy production. When minigrid

I specializes in wind energy production, it does not mean that the minigrid does not produce any solar energy, instead, it means that minigrid I is operating at a point on its PPC such that the marginal opportunity cost of producing solar energy is minimized and that of wind energy is maximized.

The optimal point of production or production equilibrium for the minigrids is given by consumer theory.

TIT. CONSUMER THEORY

Consumer theory dictates the constraints under which opti­mization of joint production takes place in order to arrive at the production equilibrium. The constraints could be determined purely by operational principles or by weather conditions. These constraints are modeled as utility functions that help in visualizing the nature of the optimization problem. Consumer theory is also useful in modeling the preferences or choices of minigrids while acting as a consumer during trading. The utility function used for optimization varies based on the weather conditions, minigrid specialization and the demand and supply dynamics that drive the energy market.

A. Utility Functions

Preferences are used to model the way rational users consume and make choices. Preference relations are defined in a continuous energy space and are monotonic, meaning minigrids always prefer more energy to less (both in terms of production and consumption). Formulating appropriate utility functions that reflect the preferences of minigrids with regards to solar and wind energy is of vital importance in modeling a production economy. Utility functions with respect to solar and wind energy are represented by three dimensional plots with each type of energy sand w on each of the horizontal axes and the utility value u (s, w ) on the vertical axis as shown in Fig. 3(a).

Fig. 3. Example of a utility function of solar and wind energy (a). Indifference curves link points of constant utility along the energy space (b).

However, utility functions are usually visualized as iso­quants or contours in a two dimensional space (referred to as the energy space) with each type of energy on each of the axes and the contour lines linking points of equal utility. Any point on the energy space is referred to as a bundle and the contour lines reveal the associated utility, that is, how much a minigrid values a particular combination of solar energy and wind energy that it may produce. The constant utility contour lines are known as indifference curves as they link points of equal preference or in other words, linking bundles that are indifferent as shown in Fig. 3(b). Monotonic preferences have

downward sloping indifference curves that do not cross each other and moving in a north easterly direction in the energy space moves to more preferred indifference curves. Convex preferences imply that the indifferent curves get flatter as we move along them.

Marginal utility of a minigrid is defined as the partial derivative of the utility function u( s, w) with respect to one type of energy. It measures the rate at which the utility of a minigrid increases as we increase the amount of one type of energy that the minigrid is capable of producing.

rnus(s, w) = 8u(s, w)/8s,

rnuw(s, w) = 8u(s, w)/8w.

(1)

(2)

The marginal rate of substitution (MRS) of one type of energy for another is defined as the rate at which solar energy can be traded for wind energy while keeping the minigrid indifferent. The MRS between energies is equal to the ratio of the marginal utilities of those energies, or in other words, the slope of the indifference curve. Therefore,

8u(s, w)/8s MRS(s,w) =

8 ( )/8 ' u S,w w (3)

Some common utility functions used in consumer theory as shown in Fig. 4 are :

u(s,w)=s+w,

u(s, w) = rnin(s, w),

u(s,w) = sa *wl-a.

(perfect substitutes)

(perfect complements)

(Cobb-Douglas utility)

Perfect Substitutes Perfect Complements

f� "" f� Wind Energy Wind Energy

(4)

(5)

(6)

Fig. 4. Examples of various utility functions. (a) Perfect Substitutes, (b) Perfect Complements, (c) Cobb-Douglas utility function with ex = 0.5.

If a minigrid values both types of energy equally and does not care about of how much of each energy it has, but is concerned only about the total amount of energy, then the perfect substitute utility function captures this preference aptly, that is one type of energy can be substituted for another. If a minigrid values one type of energy with a certain minimum constraint on the other type energy, this preference is reflected in the perfect complement utility function. However, if a minigrid values a certain share of solar energy in relation to wind energy, the Cobb-Douglas utility function is useful in modeling these preferences as it has desirable properties such as continuity, monotonicity and convexity ensuring the existence of an equilibrium in the market.

IV. PRODUCTION EQUILIBRIUM AND TRADING

Formally, a production economy [11] can be modeled as a cooperative game with,

• A set N of all minigrids in a small locality. • A positive integer I, which is the number of goods that

is being produced in the market. Here, 1 = 2, as the minigrids produce only solar and wind energy, but a higher number may be studied when considering energy supplied by utility company, energy from storage systems, energy from diesel generators and other renewable energy production possibilities.

• For each minigrid i E N, a vector Ei E R�, which is the endowment of minigrids i, i.e., the solar energy E; and wind energy E� capable of being produced.

• For each minigrid i E N, a continuous, non-decreasing and concave production function Ii : R� --+ R+.

Production equilibrium is defined as the point on the joint PPC that maximizes the energy production of the minigrids subject to certain constraints. As producers, minigrids may simply want to maximize their total energy production or depending upon the weather, maximize the energy production from a particular technology. Various utility functions that reflect these preferences, serve as constraints to arrive at an optimal equilibrium point on the joint PPC. We consider a few examples to graphically illustrate these concepts.

A. Output Maximization

Assume that the minigrids wish to maximize their energy production. The perfect substitutes utility function, u(s,w) = s + w, is an apt choice for finding the equilibrium point on the joint PPC curve as it maximizes the sum of solar and wind energy produced without taking into account how much of each is produced. The production equilibrium occurs at a point where the indifference curves of the perfect substitute utility function are tangential to the joint PPC curve as shown in Fig. 5. In this example, the production equilibrium occurs at (Es*,Ew*) = (281.25,75) KWh, i.e., 281.25 KWh of solar energy and 75 KWh of wind energy for a total of 356.25 KWh of energy. This is the maximum possible combined energy production of the two minigrids.

B. Output Maximization under Weather Constraints

Assume that minigrids wish to maximize their energy pro­duction with stress of one type of energy depending upon the weather conditions. The Cobb-Douglas utility function, u(s, w) = sa * wI-a can be used to maximize the energy production from a particular technology. Let p( s) be the proba­bility that the weather is sunny and p( w) be the probability that the weather is windy. These probabilities can be incorporated into the parameter Q in the Cobb-Douglas utility function and be used to prefer one production technology over another. Let,

p(s) p(w) Q =

p(s) + p(w) and 1

-Q =

p(s) + p(w) (7)

Thus, on sunny days, the production equilibrium tends towards solar energy and on windy days towards wind energy. For

300------.:::::::::,� -- 281.2

l j 200- � e

D->-e> Ql c

W ro 100 o (f)

Production Equilibrium

75 100 Wind Energy Production (KWh)

-Joint PPC

@Indifference Curves

150 200

Fig. 5. Production equilibrium for minigrids under the maximum total energy production constraint. The production equilibrium occurs where the joint PPC curve is tangential to the indifference curves of the utility function. Perfect substitute utility function is used here to maximize the total energy production.

example, on a bright sunny day without much wind, 0: = 0.95 and 1 - 0: = 0.05. The production equilibrium occurs at (Es*, Ew*) = (296.93,50) KWh as shown in Fig. 6.

c a U " 200 "0 e

D­>-e> Ql c

W ro 100 o (f)

Production Equilibrium

-----�-- -

50 100 Wind Energy Production (KWh)

-Joint PPC

®Indifference Curves

150 200

Fig. 6. Production equilibrium for minigrids under sunny weather conditions. The production equilibrium occurs when the joint PPC curve is tangential to the indifference curves of the utility function. Cobb-Douglas utility function with a = 0.95 is used here to maximize the solar energy production.

Similarly, on a cloudy windy day without much sunshine, 0: = 0.05 and 1-0: = 0.95. The production equilibrium occurs at (Es*, Ew*) = (50,143.31) KWh as shown in Fig. 7.

It must be noted here that the production equilibrium does not stipulate as to which minigrid produces how much of each kind of energy. It merely reflects an optimal production possibility under a certain constraint such as maximizing total energy production under different weather conditions in this case. The amount of solar and wind energy that will be individually produced by the minigrids depends upon how much and at what price they are willing to trade.

Production Equilibrium

3001-----1-=

-Joint PPC

@Indifference Curves

c a U " 200 "0 e

D­>-e> Ql c

W ro 100 o (f)

50

50 100 1433150 Wind Energy Production (KWh)

200

Fig. 7. Production equilibrium for minigrids under windy weather conditions. The production equilibrium occurs when the joint PPC curve is tangential to the indifference curves of the utility function. Cobb-Douglas utility function with a = 0.05 is used here to maximize the wind energy production.

C. Trading

Once the total amount of energy production as obtained in the production equilibrium has been decided, minigrids could trade towards an optimal level for their consumption. In order for trade to take place, each minigrid must benefit. A minigrid will not be willing to trade any kind of energy that would cost it more than what it could produce itself. The highest and lowest prices at which each kind of energy will be traded is determined by the individual minigrid's marginal opportunity costs for each kind of energy. The terms of trade would include the rate at which one kind of energy trades for another. Minigrid 1 will not be willing to trade wind energy at a rate that is lesser than its marginal opportunity cost of producing wind energy. Similarly, minigrid 2 will not be willing to buy solar energy at a rate that is higher than its marginal opportunity cost of producing solar energy. In other words, both minigrids would benefit if trade takes place in between these marginal opportunity costs, i.e.,

!J.Esl -- < !J.Ew1 -

d. . . !J.Es2

Tra mg Pnce RatIO ::.; � . D.Ew2

(8)

The actual prices at which trading takes place will be deter­mined by the prevailing market prices for energy consumption and distribution in addition to the running costs for energy production incurred by the minigrids. Once the production equilibrium is achieved, the terms of trade provide a range or bound for the trading prices in terms of the marginal opportunity costs of production within which energy exchange is beneficial to both minigrids.

V. CONCLUSIONS

We propose a cooperative framework for modeling the individual and joint production capabilities of hybrid renew­able energy minigrids as a production possibility curve. In particular, we study solar-wind hybrid minigrids with different

capacities, but it is also possible to consider other sources of energy supply such as energy provided by utility company, en­ergy from storage systems, energy from diesel generators and other combinations of renewable energy production sources. Weather and climate variations give distinctive advantage (both absolute and comparative) to one hybrid renewable minigrid over another in terms of energy production. We show that comparative advantage in one type of energy production technology leads to specialization in that technology which guarantees that all the minigrids can be made better-off by trading to mitigate the effects of weather variations on energy production. We graphically compute the optimum point of pro­duction or the production equilibrium that forms the basis of mutually beneficial trade. The necessary constraints required for optimization are aptly captured by utility functions that also reflect the changes in weather conditions. The production equilibrium is shown to be the point at which the joint production possibility curve is tangential to the indifference curves of the utility function. The production equilibrium may be computed using a Cobb-Douglas utility function under sunny and windy weather conditions. Once the total amount of energy production as stipulated by the production equilibrium has been reached, minigrids could trade towards an optimal level for their own consumption. The terms of trade provide a range for the trading prices in terms of the marginal opportunity costs of production within which energy exchange is beneficial to both minigrids.

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