Strategic investments and regulatory framework for distribution system planning under uncertainty An option game approach M. Osthues; C. Rehtanz
G. Blanco
TU Dortmund University Institute of Energy Systems, Energy Efficiency and Energy Economics Dortmund, Germany
[email protected]
Universidad Nacional de Asunción Facultad Politécnica Asunción, Paraguay
[email protected]
Abstract—Efficient operation and well-timed investments have become a main target of the distribution system operator to cope with the increasing cost-pressure due to deregulation of the electricity sector. Distribution system planning is no longer just subject to an optimal allocation of costs for given planning horizon, it has evolved to a complex decision making process: Choosing among strategies, on the one hand to provide flexibility to react on future development, on the other hand to anticipate crucial decisions made by the Regulator. Real option evaluation has increasingly received research interest as it enables to hedge risks due to an unfavorable development. Strategic decisions, that foresee the optimal decisions of third parties, have also become an object of research in applying game theory on the planning problem of the system operator and the regulatory agency. This paper presents a so-called option game approach combining both real options and game theory for an optimal decision policy for the distribution system operator. A mathematical formulation of the planning problem including maintenance and investment projects is given and exemplified by stylized study case. Keywords-Distribution systems; Power system planning, Proactive network planner; Game theory; Flexibility; Uncertainty; Real options; Regulation; Regulatory risk
I.
INTRODUCTION
The increasing competition in the electricity sector has significantly influenced planning and operation of distribution systems. Pressurized by the regulation of the energy sector, distribution system operators (DSO) are forced to increase efficiency in maintaining existent facilities and investing appropriate to the expected future requirements. Besides, the Regulator has to set suitable incentives leading to efficient cost allocation at an adequate level of security of supply. Both, DSO and Regulator, have different roles and objectives leading to decisions that are closely linked. The DSO follows a strategy that maximizes its profits [1]. The Regulator aims efficiency in maximizing social welfare and allowing a certain level of revenues to make grid operation profitable for the investor. Both decisions are linked by the DSO’s regulated revenues and the efficient allocation of costs determining the regulated revenues.
Efficiency is to provide a sustainable network service at a correct timing of Operation and Maintenance projects (O&M) as well as the adequate investment plan for the distribution grid. In the distribution system planning procedure both types of activities are regarded separately as they are linked to different planning horizons. Whereas O&M aims on improving the asset condition in short-term scale, grid expansion investments are assessed by estimating future scenarios. Both planning procedures have in common that they deal with uncertain data. The uncertain unfolding of some key variables certainly affects the decision making process. Therefore, assumptions about that variable evolution must be made in order to deal with such uncertain information. Under traditional expansion planning, grid structure has been optimized just using expected values representing the weighted outcome of all possible scenarios (probabilistic choice). As a consequence, this approach often neglects future consequences on the chosen “optimal” investment plan. As stated in [2], power system planning becomes a matter of decision making and not of optimization because each decision bears some risk. Different general strategies can be used for dealing with risks, either ignoring risks, hedging against uncertainty by investing in projects that are robust and/or flexible [3]. Robust projects reduce regret (possible losses) if a future outcome will not occur as it has been foreseen. However, robust plans might implicit cost intensive investments that are not always affordable. On the other hand, flexibility is the ability to adapt on changing conditions at a low cost. In this sense, making optimal decisions under uncertainty means to develop a strategy that decreases losses due to a possible consequence of an unfavorable development (downside risk) and while keeping open the opportunity to seize extraordinary profits if future might develop in a preferred way (upside risk). In this context, the flexibility in the decision making process plays an important role for irreversible investments under an uncertain environment which should be quantified [4]. In this context, real options evaluation techniques have become a powerful method to
make decisions under uncertainty. Based on financial options, real options give each investment a value for the right of a given action in the future. Hence, they advise the decisionmaker to invest now or not, considering future scenarios and possibilities that come along: for instance, to defer the investment decision (execute a deferral option) and see if the future develops favorable [5]. The application of real options on the transmission and distribution planning has already been a remarkable approach to reduce risks. In [6], the option value to invest in a FACTS device (including the option of a relocation) instead of choosing a transmission line first have been evaluated and considered in the decision making process. In addition, it has been shown in [7], that a solution to invest first in flexible DG-units bears less risks than have invested in a distribution feeder first (sunk investment) if future develop unfavorable. As pointed out so far, uncertain information is an important driver for the decision making process. But a further key issue in a competitive market is given by the interactions of decisions and the decision makers’ strategies. In [8], [9] and [10], a game theory approach is applied to evaluate strategic transmission investments considering the impact on generator’s capacity expansion and its consequence on strategic bidding and electricity market price. This paper expands the real option approach for the distribution system planning problem using the game theory approach to model strategic behavior between the DSO and the Regulator. First, Section II outlines the general idea of the option game approach. It provides the concept and a mathematical formulation of the optimization problem. Finally, a stylized study case is given in Section III to illustrate the general idea, followed by a conclusion in Section IV to give an outlook for further application of this approach. II.
THE OPTION GAME APROACH
A. Concept The concept of the model is shown in Fig. 1. The DSO performs O&M and reinforcement investments for a planning horizon. In a revenue cap regulation framework, the Regulator sets the allowed revenues for each regulation period taking decisions about DSO’s efficiency. Hence, an (RPI-X)-rule is applied that controls the allowed revenues to encourage efficient network operation and investment behavior. DSO‘s planning horizon Regulation period 1
Regulation period 2
…
time Negotiation (GAME) 1. DSO takes decisions 2. Regulator sets Revenues for next regulation period
Fig. 1.
Timeline for decision making
Before each regulation period, both parties negotiate about the revenues for the next period. The negotiation is assumed as a
strategic game in that each player (DSO & Regulator) anticipates the rational reaction of the other player as a best response for a decision making. Strategic interactions between both players result from different objectives, maximizing revenues (DSO’s objective) and maximizing the social welfare in terms of cost efficiency and quality of supply (Regulators’ objectives). Each objective is depicted by a mathematical formulation evaluating the economic value of each O&M schedule, investment project and regulatory framework. As pointed out before, uncertainty has an impact on the timing of decisions. If uncertainty grows, project risks increase. The DSO is not encouraged to invest in capital intensive projects expecting a cap in future revenues if the project will not be efficient. On the other hand, the Regulator has to set proper incentives so that the DSO would operate efficiently in terms of social welfare. According to the game theory, the rules of the game have to be defined: number of rounds, order of the actions and the available information. This means that with each additional regulation period, the game becomes more complex as the number of strategy combinations increases. Furthermore, the results depend on the order of the actions (simultaneously or sequentially) and the knowledge of information between the players. In the proposed approach, actions are made sequentially: For each play, the DSO and Regulator negotiate about the DSO’s revenues for the next regulation period. At this point, decisions are based on the value of flexibility. They are treated as a deferral options so that its execution depends on the evolution of uncertainty. Hence, the DSO obtains the option to defer the investment decision to the next regulation period in order to wait-and-see for an increase of efficiency resulting in higher regulated revenues. In the order of decision making, the DSO first decides to invest or to wait with the investment. Secondly, the Regulator reacts on the DSO’s first move by setting the X-factor. A sequential move is equal to the so called leader-followers model described (s. Subsecion C). The leader moves first assuming its follower’s best response in the second move. Applied to this game, the DSO takes its decision knowing the Regulator’s best reaction. As a result, the game can be solved by dividing it in subgames and determining a solution by a backward induction procedure. Firstly, an equilibrium has to be found for the DSO’s decision (invest or defer/wait) by analyzing the Regulator’s best response (X-factor that maximizes the Regulator’s objective function). It is assumed that the Regulator have knowledge about the effect of DSO’s decision on its objective value. Afterwards, the DSO determines its best decision anticipating the Regulator’s best reaction. Depending on the project’s option value, he decides to invest or to keep up flexibility by executing the deferral option. In the game, each player chooses a strategy that performs better against other strategies. As a result they do not choose that strategy that maximizes their payoffs in the pool of all possible strategies, they choose a strategy that maximizes their profits knowing that the other player is acting in that way until no player has an incentive to make an unfavorable decision
leading to lower payoffs. So each player plays a strategy that is a best response on the other player’s strategy and vice versa. This strategy combination is called Nash Equilibrium and is, if it (one, one or more) exists, the outcome of each game and determines the players best strategic behavior [11]. B. Objective functions The DSO intends to maximize its profits in order to satisfy its shareholders. The profits result from the regulated revenue and the total costs : (1)
The Regulator allows the DSO a financial return on the asset base , the investment costs and operational costs depending on the Retail Price Index RPI, an extra premium P and an efficiency factor for the regulation period . If a and project is carried out or not, the values of asset base investment (expansion) projects as well as maintenance a value between 0 and 1. The depreciation value are and and the asset depreciation considered by its costs , the O&M costs by the number of each project and range T its specific costs . ⋅
⋅ 1
,_
(2)
, _
(3)
1, … 1, … 1, …
∈ 1, ∈ 0,1 , ∈ 0,1 , _
,
,
,
⋅
∀
,
∈
⋅
1 1
0,10 ,
0, 0.1, … 0.9, 1
1
(9)
1
Step 1: DSO takes decision about investment and , O&M projects Step 2: Regulator decides about efficiency factor
The game is solved by a backward induction procedure. Each in step decision of the DSO in step 1 leads to a subgame 2 and a subgame perfect Nash equilibrium of each subgame ∗ , ∗ that is the Regulator’s best response ∗ to . ∗
2:
,
∗
,
,
,…,
,
,…,
⋀
,…,
(10)
,…,
Subsequently, DSO takes the optimal decision anticipating the Regulator’s best response in step 2. (6)
∗
1:
∗
|
(8)
C. Leader-followers game It is assumed that both players have a common knowledge about the rules and the information that is required to calculate the objective value. Hence, information asymmetry and the influence on the strategic behavior are not modeled. The decision process follows a sequence of two steps.
0
1
,
10
1
(4)
The DSO tries to maximize profits for the planning horizon, a number of regulation periods. Its decision variables are and operative given by the investment projects activities . The present value is calculated by a constant discount rate for each year of the regulation period. ,
∈
(5)
,
⋅
The profits are reduced by the costs and as well as due to a low Quality divided in reliability costs influenced by the asset condition (failure rate, average outage time) and costs due to energy deficit . First value is calculated by a reliability model. Second one depends on the punishment by the Regulator. ,
are regarded in a regulation framework. Finally, the social welfare determined by (8) corresponds to the difference multiplied with between the and market price level the electricity demand minus the total expenditures for the distribution system . As already mentioned, the Regulator for influences DSO’s decision by choosing the X-factor each regulation period. The DSO’s costs are essential for the social welfare, therefore the X-factor has an indirect influence on it as it depends on the DSO’s reaction to invest or not. Furthermore, the development of the electricity demand plays an important role in assessing the value for the consumers and the efficiency and risks that come along with the execution or omission of investment and O&M projects.
(7)
The Regulator’s objective function represents the social welfare for the considered distribution system consisting of the consumer surplus (willingness to pay the Value Of Lost Load – ), the producer surplus (producing at a price level) and the grid operator’s revenues. In this approach, the producers generates electricity at equal marginal costs. Hence, its surplus becomes zero and only the consumer and the DSO This work was supported by the Deutsche Forschungsgesellschaft (DFG), project reference number RE 2930/1-2.
∗
, ,
∗ ∗
…,
∗
,
∗
,…
∗
∗
,
(11)
A multi-period-optimization problem emerges, when the game , … ). Additionally, when is played several times ( , decisions are made, future payoffs have to be estimated due to an uncertain evolution of state variable . Regulator’s best response ∗ on DSO’s decision , leading to expected value ∗ for time and is determined by: ∗
∈ ∈
,
,
∗
,
,…, ,…,
,…, ,…,
,
,
,
,…,
,
,
,
,…,
,
,
,
(12)
For the multi-period-optimization problem, the value of each ∗ decision in period depends on the expected profit in ∗ in and the expected equilibrium :
,
,
, , ∗
DSO’s best response ∗ in step 2 leads to ∗
,
∗
,
∗
, ∗
,
,
(13)
∗
anticipating Regulator’s best decision
∗
,
, ,
,
,
∗
,
,…,
,
,
∗
,
,…,
,
,
∗
For the decision making in , the DSO has got the by a information about a future demand growth in probability of 10% (s. Fig. 3). If demand increases, another distribution line (line2) is necessary to serve the demand at peak load time, otherwise MineCO has to utilize the DG-unit for electricity generation. 3.6
peak load 3.4
(14)
,
state2
3.2
p
state1
3
,
,
,
,
∗
III.
∗
,
,
(p-1)
,
∗
,
Demand [MW]
,
∗
(15)
∗
STUDY CASE
2.8
2.6
2.4
base load state2
A. Description For the sake of understanding, the proposed option game approach has been applied on a stylized numerical example focusing on the understanding of the decision making process. Hence, technical models (i.e. load flow calculations) have not been implemented yet and therefore a detailed description of the technical planning procedure is missing. A mining company (MineCo) located in a rural area connects to the distribution grid in order to benefit from the electricity at the spot market to meet its demand. Before the price planned connection line1 is erected, electricity has been generated using a distributed generation unit (DG-unit) at a . The test network is assumed higher generation cost level as a connection between the substation S1 and MineCo at S2, a distance of 1km and a physical flow (s. Fig. 2). spot market S1
F12
MineCo S2
planned line1
load d
not confirmed line2 DG-unit 5 km
Fig. 2.
Sample network of MineCo’s grid connection
The operation time of MineCo is set to T=10 years and the decision about the investment in line1 is done one year before, hence in . The demand of MineCo is 2.0 MW in base and 3.0 MW in peak hours (ratio base/peak load 1.5). Peak hours will occur one hour per day load of 0.04 ). The technical and financial data of the (ratio planned distribution line is shown in TABLE 1. Due to the , the DSO foresees to erect maximal transferred load in line1 with a capacity of 3.0 MW.
2.2
p
state1
2
(p-1) 1.8 -2
-1
0
1
2
3
4
5
Regulation Period 1
6
7
8
9
10
Regulation Period 2 Time [year]
Fig. 3.
Base and peak load scenarios for time torizon T
Following the approach Section II, DSO’s revenues are determined by the Regulator using following framework:
The DSO’s revenue are fixed for each regulation period A regulation period lasts five years Two regulation periods are regarded, T and T Negotiation one year before each period Objective is network efficiency, adequate quality of supply at adequate costs; DSO must be able to finance its operation and investments (positive revenues) The DSO invests in line1 and has got the option to invest in and line2 or to defer the investment for each decision point . The time of erection is assumed with one year so that the enhancement is installed in the next regulation period or . The DSO chooses a strategy ( ∈ , ) that refers to two Options, either to invest in line2 (strategy or to to the negotiation in . This results defer the investment ( for each regulation period : in profits
5
⋅ 1
⋅ 1
&
1
⋅ 1
(16)
&
1
TABLE 1 TECHNICAL AND FINANCIAL DATA OF DISTRIBUTION LINE
Line capacity MW Length km Life time a Failure rate f/year ⋅ km Average outage time [h/outage] Investment Costs [€] Operational Costs [€/a]
&
3.0 5.0 10 0.2 2.0 30000 3000
and O&M Costs of the The investment costs & distribution line are given in TABLE 1 as well as the regulatory and economical parameters in TABLE 2. The economic value of the investment at the end of the each regulation perifor od is determined by the sum of each residual value each investment ∈ 1,2 and the corresponding usage time , related to the economic life time , :
,
⋅
,
(17)
,
TABLE 2
p p RPI
.
&
REGULATORY AND ECONOMICAL PARAMETERS
Investment’s rate of return [%] X-factor (Decision variable) [%] DSO’s discount rate [%/a] Market price [€/MWh] Fuel price DG-unit [€/MWh] Retail price index [%/a] Value of Loss of Load [€/MW]
pointed out, technical models have to be used for a realistic investigation. . ⋅
⋅
⋅
⋅
,
,
&
∀
,
∈
| 10,5 ,
,
0.
The Regulator’s payoffs for each regulation period given by ∑
⋅ ,
&
⋅ 8760 .
,
,
⋅
,
⋅
,
,
1
2
2
(21) (22)
⋅
⋅
⋅ 8760
Invested
G3
Not Invested
G4
Invested
G5
Not Invested
time
(19)
(20)
⋅
G2
are
In this example, maximal transferable capacity exceeds the if state2 occurs and the DSO has not line capacity invested in line2. Then, the energy deficit is determined by: ,
G1
T1
(24)
Previous decision in t-1
(18)
The quality costs , are influenced by the DSO’s decisions and the uncertain load development. They depend on , and costs due to energy not system reliability costs , . deficit For the first one, an evaluation technique for radial systems is applied using the three basic reliability parameters (s. TABLE 1), failure rate or the value for a set of series components depending on the investment strategy , the average outage [12]. The time and the expected annual outage time expected ENS per year is dependent on the occurrence of base and peak load demand and the ratio between peak and base load level. It is assumed, that the DSO has to pay compensation to MineCo according to the damage for each time of failure and state according to the VOLL. ,
1,5
The structure of the game is given in Fig. 4. As described in II, the optimization problem is solved by backward reduction. Because uncertainty is dissolved in , the payoffs for the second regulation period only depend on the decision made in . Hence, four games have to be solved in (s. Fig. 4), for a load increase and it has been invested (G2) or not (G3), or without a load increase (G4 and G5).
15 100 – 50 2 30 30 2 80
According to the DSO’s decision making, the Regulator sets the X-factor between 1 and 0.5 choosing strategy , considering that the DSO must be able to finance its operation and investments: ∈
∈ |
Fig. 4. Game structur and number of subgames
If the outcome in G1 will be that the DSO invests, the option to defer the investment is lost and the Regulator can set the efficiency factor for the second regulation period (G2 and G4 in ). Depending on the development of load, he takes its decision upon the payoffs in G2 or G3. If the DSO decides in to defer the investment, its options in the second negotiation in are again to invest or to wait depending on the load increase (either he will be in G3 or G5) and the impact on the payoffs. According to Section II, each game leads to a leader-follower game illustrated in Fig. 5. In each game, after payoffs have been calculated, the equilibrium is determined applying (13) and (14). Finally in , the expected payoffs for each strategies are calculated (s. (25)) including the expected payoffs in and the equilibrium outcome in . Finally DSO’s following optimization problem for the first regulation period is solved: ∗
,
∗
, 1
∗
(25)
,
⋅ ⋅
(23)
In this case, it is assumed that the deficit can be generated by a . Of course further treatments of the DG-unit at price level punishment are adaptable. In any other cases, becomes zero. As technical models have not been implemented yet, the influence of maintenance on system reliability has to be assumed. Depending on the increase of X-factor the DSO will decrease maintenance costs . For this & leading to an increase of reliability costs example, an exponential characteristic is assumed. It has to be noted, that these functions are constructed to implement the maintenance projects in the decision making process. As
T2
,
∗
∗
∗ ∗
,
,
∗
∗
Regulator
DSO
Fig. 5. Structure of the leader-followers game
B. Results As mentioned before, the option to defer the investment is lost if the DSO decides to invest in . For G2 and G4, the regulator’s best response is to lower the efficiency to 0.8 (s. TABLE 4) and 0.7 (s. TABLE 4). TABLE 3 Results of game G2
Strategy Defer
Π ϕ
1 39.668 5.906E3
Invest
X- factor 0,9 0,8 0,7 0,6 33.645 20.177 12.164 27.219 5.907E3 5.908E3 5.907E3 5.906E3 Invested in
(option lost) (Decision in
, state2, invested in
)
If the DSO does not invest in and state2 occurs, the equilibrium would be to invest and to receive an efficiency of 0.9 in order to reduce quality costs. This state corresponds to the objectives of the regulator who benefits from the additional investment in line2. TABLE 4 Results of game G4
Strategy Defer Invest
Π ϕ Π ϕ
1 -167 5.900E3 9.668 5.876E3
X- factor 0,9 0,8 0,7 -5.499 -12.536 -22.364 5.898E3 5.895E3 5.888E3 -2.781 -9.823 3.645 5.877E3 5.877E3 5.877E3 (Decision in
0,6 -36.767 5.877E3 -17.836 5.876E3
, state2, not invested in
)
If no load increase occurs (state1), it is more favorable for the DSO to defer the investment and to receive an efficiency factor of 1. An investment would lower the profits because of the high investment costs and the fact that the Regulator , the profit decreases its efficiency. For the final decision in of each pair of decisions (invest or wait, X-factor) have to determined considering the payoffs in both regulation period. The payoffs for all strategies are shown in TABLE 5. According to the payoffs, the DSO is incentivized to defer the investment because its payoffs are higher as if he would invest. The flexibility (option to “wait-and-see”) has got a value of 140€ (1.996€-1.856€). TABLE 5 Final results of game G1
Strategy Defer Invest
Π ϕ Π ϕ
1 1.996 8.239E3 1.856 8.193E3
X- factor 0,9 0,8 0,7 -2.562 -8.325 -16.050 8.238E3 8.236E3 8.232E3 -4.013 -10.184 -16.806 8.194E3 8.195E3 8.195E3
0,6 -26.988 8.224E3 -24.125 8.195E3
(Decision in
IV.
)
CONCLUSION
In this paper an approach has been presented to model strategic interactions between the distribution system operator and the regulator. This so-called option game approach provides strategies of both parties deciding to invest or not (DSO) and
to set the efficiency for the DSO’s network operation and investment planning (Regulator) under uncertainty. The proposed methodology enables an evaluation of the DSO’s asset base and investment portfolio combining O&M and grid expansion in one optimization problem. For the DSO, the mathematical formulation of the optimization problem of the decision making process in a revenue cap regulation under uncertainty is presented. The benefit of the approach has been illustrated by a numerical example showing that for an expected probability of load development it is more favorable to exercise the flexibility option to defer the investment to the next regulation period (“wait-and-see”) than to invest and to take the risk that a development might turn out the investment being inefficient. Generally, it is a first approach to describe the distribution system planning as a decision making process among different market participants. Worked out the proposed mathematical solution, next steps are to implement a reliability model for a realistic evaluation of quality of supply and to enhance the decision making variables for the DSO (maintenance and investment) and for the Regulator (rate of return and X-factor). Finally, the approach will be applied on a complex distribution system. REFERENCES [1]
K. S. Alvehag, “The impact of risk modeling accuracy on cost-benefit analysis of distribution system reliability,” presented at the Proceedings of the 17th Power System Computational Conference (PSCC), 2011. [2] V. Miranda, “Why risk analysis outperforms probabilistic choice as the effective decision support paradigm for power system planning,” IEEE Transactions on Power Systems, vol. 13, no. 2, pp. 643-648, 1998. [3] V. Neimane, “On development planning of electricity distribution networks,” vol. PhD, p. 228, 2001. [4] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty. Princeton University Press, 1994. [5] S. Olafsson, “Making Decisions Under Uncertainty — Implications for High Technology Investments,” BT Technology Journal, vol. 21, no. 2, pp. 170–183, Apr. 2003. [6] G. Blanco, F. Olsina, F. Garcés, and C. Rehtanz, “Real option valuation of FACTS investments based on the least square Monte Carlo method,” IEEE Transactions on Power Systems, vol. 26, no. 3, pp. 1389-1398, 2011. [7] E. Buzarquis, G. A. Blanco, F. Olsina, and F. F. Garcés, “Valuing investments in distribution networks with DG under uncertainty,” in Transmission and Distribution Conference and Exposition: Latin America (T&D-LA), 2010 IEEE/PES, 2010, pp. 341-348. [8] M. R. Hesamzadeh, N. Hosseinzadeh, and P. J. Wolfs, “A leaderfollowers model of transmission augmentation for considering strategic behaviours of generating companies in energy markets,” International Journal of Electrical Power and Energy Systems, vol. 32, no. 5, pp. 358367, 2010. [9] E. E. Sauma and S. S. Oren, “Proactive planning and valuation of transmission investments in restructured electricity markets,” Journal of Regulatory Economics, vol. 30, pp. 358-387, Dec. 2006. [10] J. P. Molina, J. M. Zolezzi, J. Contreras, H. Rudnick, and M. J. Reveco, “Nash-cournot equilibria in hydrothermal electricity markets,” IEEE Transactions on Power Systems, vol. 26, no. 3, pp. 1089-1101, 2011. [11] P. K. Dutta, Strategies and Games: Theory and Practice, 2nd ed. MIT Press, 1999. [12] R. Billinton, Reliability evaluation of power systems, 2nd ed. New York: Plenum Press, 1996.
