A Principal-Agent Approach to Transmission Expansion-Part II: Case Studies

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A Principal-Agent Approach to Transmission Expansion  Part II: Case Studies Juan D. Molina, Graduate Student Member, IEEE, Javier Contreras, Senior Member, IEEE, and Hugh Rudnick, Fellow, IEEE  Abstract—This paper is the second of a two-paper series and presents a model to assess and promote investment projects defined in a plan of expansion of the transmission. We propose a model that consists of three main elements: valuation of a project based on the design of a linear contract, a principal-agent model to assess the optimal effort of an agent and the right-of-way negotiating cost. We also define a model to evaluate bids by the agents. The value of the project depends on the number of competitors, the incentives to invest, and the right-of-way costs. The right-of-way cost is approached from the perspective of a bilateral bargaining problem. We analyze two case studies. The first is a plan to expand the IEEE 24-RTS system. The second is based on the expansion plan of the Central Interconnected System (SIC) of Chile. The results show that the principal-agent model obtains the real costs of the bidders and creates incentives for disclosure of information. This creates optimal offers that depend on the incentive generated by the social planner. Index Terms—Transmission expansion planning, game theory, design mechanism, principal-agent, incentive and bidding contract.

NOMENCLATURE A. Indexes j Index of projects. n Index of investors. o Index of land’s owners. Index of mandatory allocation rules. M Index of bargaining solutions between agents n and RoW o. Z Expansion plan done by a centralized planner. Set of pairs of payments where the players act S cooperatively. B. Decision Variables Expected value of the winning bid for project j V jw [M$]. bw j

Expected winning bid for project j [M$].

This work was supported in part by CONICYT-Programa en Energías 2010, Pontificia Universidad Católica de Chile, Fondecyt, MECESUP(2), Transelec and the Ministry of Science and Innovation of Spain grant ENE2009-09541. J. D. Molina and H. Rudnick are with the Electrical Engineering Department, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile (email: [email protected]; [email protected]) J. Contreras is with E.T.S. de Ingenieros Industriales, University of Castilla − La Mancha, Campus Universitario s/n, 13071 Ciudad Real, Spain (email: [email protected])

b nj

Expected investor’s bid for project j [M$].

c nj*

Expect optimal cost for an investor n in project j [M$]. Expected right-of-way cost of the bargaining solutions of project j between agent n and o [M$]. Payment pair when agents n and o act noncooperatively (disagreement point).

x RoW j

d nj,M I nj

Expected income determined by Tjn – gjn.

g nj

Effort cost function of investor n determined by 0.5 jn (ejn)2.

e nj

Effort value to reduce the cost of project j [M$].

T jn

 nj

Linear payment function Tjn expressing the monetary transfer to the investors n determined by jn + jn  jn. The degree of risk assumed by the investor n (if it does not assume any risk, jn = 0 and if it assumes all the risks, jn = 1).

 nj

Profit function determined by jnejn + wjn.

j

Cost-share factor where the linear contract is of the incentive-type.

C. Random Variables  nj

Unpredictable cost of project j [M$].

w nj

Random variable of profit function, wjnN[0, (jn)2].

D. Constants c nj

N nj  nj

 nj 2  nj  nj, o

Reference value of project j [M$]. Number of investors that participate in the tender of project j. Effort coefficient reflecting the ability of the agent n to build a project j. The positive coefficient of effort cost of investor n to project j. Variance of the project valuation j by investor n. Degree of risk aversion of investor n for project j. Negotiation factor between agents n and o.

z nj

Minimum profit or income to reinvest in project j by investor n.

 nj

Fixed transfer in linear payment function Tjn.

2

R

II. INTRODUCTION

estructuring the electricity sector in various countries around the world has been characterized by the separation of the generation, transmission, and operation of the energy market. The evolution of energy markets shows that the interests of agents often conflict in determining the optimal transmission system expansion. This varies depending on the type of market structure and/or regulatory framework for expanding the transmission system. From the economic point of view, oligopolistic models are able to describe these issues. For example, in the case of interconnected systems, the capacity of a transmission line decreases with the increase in the injection capacity of generation, which creates conflicts in the planning of the design capacity of transmission lines and the indivisibility of investments. In turn, from the social point of view, the capacity of a line should be maximized, which leads to a conflict of interest between the generators. In addition, the behavior of the transmission owner, dependent on the payment of congestion, shows a low-capacity response. In practice, neither the lines are funded strictly by these rents nor can a minimum capacity rent fully finance their costs. Additionally, it creates a dilemma in regard to whether the market structure reflects the necessary investments in capacity and the timing thereof, that is whether the expansion should be proactively done [1]. It creates a problem between the anticipated expansion of transmission lines and the inclusion of new generation. This kind of behavior brings a high degree of uncertainty because, although from the social viewpoint it is more efficient, anticipation carries a cost and risk as to whether or not to build power generation [2][3]. Moreover, increasing demand as well as the reduction of land for the construction of transmission projects has generated conflicts between the owners of land and transmission companies. Defining the true cost of land is a negotiation process that depends on each partner and the rules that define each market. On one side, the company wants to pay as little as possible, but, on the other side, the owners want to receive as much as possible, even delaying or preventing the construction of a project in order to make it extremely profitable. The paper is organized as follows: section III presents the principal-agent process to assess and encourage investment projects defined in an expansion plan. Sections IV and V present the final expansion plans of two case studies, one from the IEEE 24-RTS and the other one from the SIC Chilean System. For the generation of expansion plans we use a multiobjective approach [4]. Both sections show the results regarding the right-of-way cost, the valuation of the effort, the results in terms of costs and optimal bids, bid incentives and the final valuation of the expansion plans. Conclusions are shown in section VI. III. PRINCIPAL-AGENT MODEL The valuation of a project is defined in terms of the costs and activities that maximize its profit. The optimal costs and activities will depend on how effective is the one who performs them. This type of problem is called the agency problem, which defines a principal, the owner or the social

planner of a project, and the agent that performs the activities to produce a project or service. This agency problem requires incentives to perform as planned and the design of incentives is an important factor, since these determine the behavior of the agent. Another important factor in a contract is the existence of a competitive environment. Although the winning bid represents the lowest cost value, incentives should be implemented to promote competition among the bidders in order to define an efficient value of the project. Here are the foundations of mechanism design, the principal-agent model and competitive bidding for a contract to carry out a project. We define a transmission game between a central planner or regulator (principal) and investors (agents). The transmission game model proposed produces the optimal value of a project j  J. The model considers hidden costs and right-of-way negotiation costs for a more accurate valuation and to avoid delays in project implementation. The project is part of a plan or plans of expansion defined by a central planner. This is obtained from a scheduling algorithm [4] or a predefined plan. The model includes three main elements: the negotiation cost of the rights of way, the optimal cost of the project and the competitive process to obtain the allocation of the project. The valuation of the contract is subject to moral hazard, risk cost and the number of bidders. The optimal bid in a first-price auction (tender with the lowest Net Present Value, total value or annuity) will depend on incentives and project risks. The optimization problem to encourage investment by the central planner can be formulated as (See section V-D of Part I): max

∙

∙

1

1

(1)

;∀

subject to 1

min ∗

∙

∗

opt

,

,

∗ ,

,

;∀ ;∀

max

∈

∙

∙

∙ ∙

∙ ∙

(2)

,

(3)

,

(4)

,

; ∀

∙ ∙ ∙

; ∀

,

∙

(5)

,

∙ 2

∙ 0.5

;∀

∙ 2

∙ ∙

,

;∀

,

;∀

,

(6)

(7)

(8)

The optimization defines an expected value of the transmission expansion plan with project j  J and investors n  N. The investor makes an offer that maximizes its profit and the land’s owner maximizes its profit. The optimal value of

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project, Vjw, depends on the cost-share factor j and the number of participants in the tender (1). Note that bjw is the lowest bid, bnj (the central planner only knows the bids). The offer bnj depends on the expected optimal cost, cjn*, and the number of bidders participating in the tender (2). The expected optimal cost, cjn*, (3) is determined by the reference value of the project cjn, the right-of-way cost xjn,RoW (4), the unpredictable cost, φjn, and the effort value, e jn (5). The reference value includes asset cost, studies, etc. We use 10 Bargaining Solutions -BS- (proposed in Part I, section II) to define xjn,RoW (4). The disagreement point, dM, is determined by a central planner or regulator, which defines the mandatory value (we consider four possible allocation rules in section VA of Part I). φjn is a random variable that represents unpredictable costs (the expected value of jn is 0). e jn represents the monetary cost to the agent of its effort to reduce actual cost (5) (the profit maximization problem described in section III-B of Part I). Now, considering the individual rationality constraint, the agent participates in the tender if the expected income, [Ijn] (6), is such that [Ijn]  , where is the minimum profit or income to reinvest in the project, and the incentive compatibility constraint makes the agent to / ejn = 0), so that choose the effort ejn to maximize Ijn ( the constraints (7) and (8) are satisfied. The proposed methodology has four phases. The first phase identifies the expansion plan, which may be either preset or obtained through an algorithm to generate expansion plans. The second phase determines the cost or valuation made by each investor. We consider three components: asset value and project studies (reference value), expected value of negotiating rights of way as a function of the type of land owner, and evaluation of effort depending on the type of investor. In the third phase, a linear contract to promote investment as a function of the number of investors is done. Finally, the fourth phase determines the expected value Vjw of each project and the plan or plans for expansion in the study. The application of the described model in real cases is done in the following sections. IV. IEEE 24-RTS CASE STUDY This system considers a network of 34 existing transmission lines and 7 additional corridors. It is assumed that all the lines per corridor are identical and the maximum number of lines per corridor is 3. Investment costs per corridor are obtained from [5]. Data for the IEEE 24-RTS are in [5]. For the generation expansion plans we use a methodology based on multi-objective optimization with ordinal optimization and Tabu Search with path re-linking approach [4]. Table I shows the expansion plans called Elite-Pareto. Five Elite expansion plans are generated and two plans are optimal. The first one is defined by a failure cost of $0.6/kWh and the second by a $2/kWh failure cost. We define five types of bidders (n = 1, ..., 5) for each project j. The first type is called non-desired, and the latter type is the desired. According to this assumption, for each bidder, a negotiation factor is defined (0 <  n < 1), its aversion to risk ( 0 < jn < 1), the effort level coefficient (0 < jn < 1),

the variance of the project valuation (jn)2, the cost of effort coefficient (jn > 0), the right of way historical budget (% of historical budget with respect to the reference value of the project), and the asset budget (% of historical budget with respect to the reference value of the project). See Table II. The degree of negotiation of each land owner (there are four in the Opt. 1 project) is a function of the evaluation done on its own land, that is, the more the owner values its land, the greater its negotiation factor is. The valuation of the bidder is defined by a uniform distribution. Likewise, a distribution is assumed between the average value of land and the maximum value for the owner. The reference value of each project is obtained from [5]. The rate at which the valuation ranges is [0.9 1.1]. The valuation of the land corresponds to a uniform distribution function between $25,000 and $150,000 (typical real cost of land in Chile). The value of projects j represent regulated revenues, therefore, the minimum profit or income 0. to reinvest in the project is zero, TABLE I ELITE-PARETO EXPANSION AND OPTIMAL SOLUTION Plan 1 2 3 4 5 Opt. 1 Opt. 2

Corridor 7-8, 14-16, 16-17 3-9, 14-16, 16-17, 20-23 1-8, 2-8, 12-13, 14-16, 16-17 14-16, 15-24, 16-17, 19-20 10-11, 14-16, 14-23, 15-16, 16-17 14-23 7-8, 14-16, 16-17 TABLE II PARAMETERS OF THE BIDDERS Bidder n1 n2 n3 n4 0.9 0.7 0.5 0.4 n 1.0 0.8 0.7 0.6 jn 0.80 0.80 0.90 0.92 jn (jn)2 0.05 0.04 0.03 0.02 n 2.0 1.6 1.4 1.2 j RoW ref(%) 0.3 0.25 0.2 0.18 Asset ref (%) 0.9 0.88 0.89 0.86

Cost (M$) 10.4907 15.0763 19.8202 20.9492 25.8196 5.16 10.4907

n5 0.3 0.5 0.95 0.01 1.0 0.15 0.85

The value of a project is determined by the degree of negotiation of the investor. The planner should infer the level of effort, the ability to reduce costs, and to efficiently select the best type of investor. A. Right-of-way Cost To determine the right-of-way cost, xjn,RoW (4), we assume a bargaining game between the land owner and the investor. For each bargaining solution and investor there is a right-of-way cost. See Table III where the rows represent the allocation rules M and the type of bidder n. The column identifies the bargaining solutions described in section II-B of Part I. The type of mandatory valuation rule influences the final negotiation cost. The equilibrium cost of bidder n1 is the lowest for all of the cases. This happens because its negotiation factor is higher, i.e., the investor negotiates without haste and believes that the negotiation time is not relevant. In general, the rule of negotiation which has the lowest cost is M1. The other rules from highest to lowest costs are M4, M3 and M2 for the Rubinstein solution. The M4 rule is competitive since each player valuation tends to produce a

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realistic assessment of the rights of way. R4 produces the same solution for any mandatory rule. The most costly bargaining solution is R1 (Rubinstein generic) and the one with the lowest cost is R4. Remarkably, the pure Nash (PN) and Stackelberg solutions (SS) are identical, indicating that, regardless of whether the negotiation is symmetrical or asymmetrical, the equilibrium depends on the negotiation rule. They are lower than or equal to the Nash bargaining solution (NB). Berge’s solution (BS) shows that altruism, a higher payment for the right of way, happens with the most desirable investors, n4 and n5. For investors n1, n2 and n3, the valuations obtained are either lower or greater than the Nash solutions. For R1 and R3 the cost of n5 is greater than the one for n4, for R2 not necessarily, and for the other solutions, n4 > n5. TABLE III BARGAINING SOLUTIONS OF IEEE 24-RTS OPT.1 PLAN (M$) M

M1

M2

M3

M4

0

R1

R2

R3

R4

NB*

BS

PN

SS

n1

1.76

1.63

1.74

0.25

2.05

2.05

2.05

2.05

n2

2.30

1.99

1.94

1.53

2.24

2.46

2.24

2.24

n3

2.53

1.99

2.20

1.05

2.26

2.23

2.26

2.26

n4

2.67

2.02

2.36

1.19

2.30

2.51

2.30

2.30

n5

2.80

2.01

2.54

1.03

2.13

2.20

2.13

2.13

n1

1.97

1.86

1.74

0.25

2.34

2.05

2.34

2.34

n2

2.28

1.96

1.94

1.53

2.41

2.46

2.37

2.37

n3

2.59

2.08

2.20

1.05

2.43

2.23

2.43

2.43

n4

2.73

2.13

2.36

1.19

2.43

2.51

2.43

2.43

n5

2.87

2.18

2.54

1.03

2.39

2.20

2.34

2.34

n1

2.05

1.95

1.74

0.25

2.08

2.05

2.08

2.08

n2

2.07

1.69

1.94

1.53

2.39

2.46

2.14

2.14

n3

2.36

1.70

2.20

1.05

2.11

2.23

2.05

2.05

n4

2.73

2.13

2.36

1.19

2.30

2.51

2.30

2.30

n5

2.89

2.23

2.54

1.03

2.21

2.20

2.15

2.15

n1

2.38

2.31

1.74

0.25

2.93

2.05

2.93

2.93

n2

2.75

2.57

1.94

1.53

2.48

2.46

2.48

2.48

n3

2.71

2.29

2.20

1.05

2.31

2.23

2.31

2.31

n4

2.53

1.74

2.36

1.19

1.99

2.51

1.99

1.99

n5

2.64

1.59

2.54

1.03

1.48

2.20

1.48

1.48

* The Kalai-Smorodinsky (KS) and Egalitarian solutions (ES) coincide with the Nash bargaining solution (risk-neutral negotiation).

B. Valuation of the Investor’s Effort The principal-agent model allows us to infer the degree of effort (5). This level of effort represents a reduction in the cost

M 0 R1 R2 R3 R4 NB KS ES BS PN SS

n1 7.05 6.93 7.03 5.54 7.34 7.34 7.34 7.34 7.34 7.34

n2 6.89 6.58 6.53 6.12 6.83 6.83 6.83 7.05 6.83 6.83

M1 n3 6.92 6.38 6.59 5.44 6.65 6.65 6.65 6.62 6.65 6.65

n4 6.86 6.21 6.55 5.37 6.49 6.49 6.49 6.70 6.49 6.49

of the project. We can calculate what the effect of effort on the reference value of each project. This effect can be defined as efficiency “productivity factor”. The productivity factor is obtained as the difference between the reference cost (Table II) and the optimal cost of the project, cjn*. For example, if the investment has 20 years of lifetime, the annual efficiency "productivity" in terms of reduction cost are 0.131%, -0.552% -0.650%, -0.941%, -0.978% for n1, n2, n3, n4 and n5, respectively. According to the specifications adopted for each investor, n1, the non-desired investor presents a sub-effort, i.e., the cost of the project is higher than the reference cost. But n2, n3 and n4 have negative productivity factors, i.e., they have the ability to reduce costs. The factor emulates the productivity factor in the price-cap regulation “CPI-X” to subtract expected efficiency savings, X. C. Optimal Cost The optimal cost of the project, cjn* (3), is determined by the reference value, the effort factor and the negotiation cost of the right of way. Now if we consider that the reference value does not include the right-of-way cost, in the case of IEEE 24-RTS, the n1 investor has a cost of M$5.3, which is 2.62% higher than the base evaluation cost. The other investors have the following costs: M$4.59, M$4.49, M$4.19 and M$4.15 for n2, n3, n4 and n5, respectively. Considering the right-of-way cost, the results show that the n1 investor is the most expensive, but it is not necessarily true that the n5 investor has the lowest cost. According to the values described in Table IV, the lowest cost for the solutions depend on the mandatory rules M, as seen in bold face in Table IV. D. Bidding for the Contract We assume that the auction is of the first price sealed-bid type, in which the winner is the investor that bids with the lowest net present value (2) (the lifetime and interest rate defined by the regulator). The results are presented in terms of total project value. Now, considering the optimal cost described in the previous section, we obtain the optimal valuation depending on the number of bidders N, #N, where # is the number of auction participants (see Table V, where the rows are the sorted bargaining solutions and the M rules, and the columns are described in terms of the number of bidders #N and (2). .

TABLE IV OPTIMAL COST OF THE IEEE 24-RTS PLAN WITH RIGHT-OF-WAY COSTS (M$) M2 M3 n5 n1 n2 n3 n4 n5 n1 n2 n3 n4 n5 6.95 7.26 6.87 6.98 6.92 7.02 7.35 6.66 6.75 6.92 7.04 6.16 7.15 6.55 6.47 6.32 6.33 7.24 6.28 6.09 6.32 6.38 6.69 7.03 6.53 6.59 6.55 6.69 7.03 6.53 6.59 6.55 6.69 5.18 5.54 6.12 5.44 5.37 5.18 5.54 6.12 5.44 5.37 5.18 6.28 7.64 7.00 6.81 6.62 6.54 7.38 6.98 6.49 6.49 6.36 6.28 7.64 7.00 6.81 6.62 6.54 7.38 6.98 6.49 6.49 6.36 6.28 7.64 7.00 6.81 6.62 6.54 7.38 6.98 6.49 6.49 6.36 6.35 7.34 7.05 6.62 6.70 6.35 7.34 7.05 6.62 6.70 6.35 6.28 7.64 6.96 6.81 6.62 6.49 7.38 6.73 6.44 6.49 6.30 6.28 7.64 6.96 6.81 6.62 6.49 7.38 6.73 6.44 6.49 6.30

n1 7.68 7.60 7.03 5.54 8.22 8.22 8.22 7.34 8.22 8.22

n2 7.34 7.16 6.53 6.12 7.07 7.07 7.07 7.05 7.07 7.07

M4 n3 7.10 6.68 6.59 5.44 6.70 6.70 6.70 6.62 6.70 6.70

n4 6.72 5.93 6.55 5.37 6.18 6.18 6.18 6.70 6.18 6.18

n5 6.79 5.74 6.69 5.18 5.64 5.64 5.64 6.35 5.64 5.64

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TABLE V OPTIMAL BID IN THE IEEE 24-RTS OPT.1 PLAN WITH ROW (M$) M Sol 2N 3N 4N 5N Cj* R1 3.43 4.57 5.15 5.49 6.86 R2 3.08 4.11 4.62 4.93 6.16 R3 3.27 4.35 4.90 5.22 6.53 R4 2.59 3.45 3.88 4.14 5.18 BN 3.14 4.18 4.71 5.02 6.28 M1 KS 3.14 4.18 4.71 5.02 6.28 ES 3.14 4.18 4.71 5.02 6.28 BS 3.17 4.23 4.76 5.08 6.35 PN 3.14 4.18 4.71 5.02 6.28 SK 3.14 4.18 4.71 5.02 6.28 R1 3.43 4.58 5.15 5.49 6.87 R2 3.16 4.21 4.74 5.06 6.32 R3 3.27 4.35 4.90 5.22 6.53 R4 2.59 3.45 3.88 4.14 5.18 BN 3.27 4.36 4.91 5.23 6.54 M2 KS 3.27 4.36 4.91 5.23 6.54 ES 3.27 4.36 4.91 5.23 6.54 BS 3.17 4.23 4.76 5.08 6.35 PN 3.24 4.33 4.87 5.19 6.49 SK 3.24 4.33 4.87 5.19 6.49 R1 3.33 4.44 5.00 5.33 6.66 R2 3.05 4.06 4.57 4.87 6.09 R3 3.27 4.35 4.90 5.22 6.53 R4 2.59 3.45 3.88 4.14 5.18 BN 3.18 4.24 4.77 5.09 6.36 M3 KS 3.18 4.24 4.77 5.09 6.36 ES 3.18 4.24 4.77 5.09 6.36 BS 3.17 4.23 4.76 5.08 6.35 PN 3.15 4.20 4.72 5.04 6.30 SK 3.15 4.20 4.72 5.04 6.30 R1 3.36 4.48 5.04 5.38 6.72 R2 2.87 3.82 4.30 4.59 5.74 R3 3.27 4.35 4.90 5.22 6.53 R4 2.59 3.45 3.88 4.14 5.18 BN 2.82 3.76 4.23 4.51 5.64 M4 KS 2.82 3.76 4.23 4.51 5.64 ES 2.82 3.76 4.23 4.51 5.64 BS 3.17 4.23 4.76 5.08 6.35 PN 2.82 3.76 4.23 4.51 5.64 SK 2.82 3.76 4.23 4.51 5.64

However, the bids do not consider the right-of-way cost and the results show that the winner is of the desired type. Table VI shows that the winner is always investor n5. In this case the

reference value is M$ 4.64 and the optimal cost of the winner auction, n5, is M$ 4.15. TABLE VI OPTIMAL BID IN THE IEEE 24-RTS OPT.1 PLAN WITHOUT ROW Number of bidders 2N 3N 4N 5N Winner n5 n5 n5 n5 Optimal Bid (M$) 2.075 2.767 3.113 3.32

This indicates that when the right-of-way cost is either not considered or not significant (less than 50% of the project cost) the model produces efficient bids from desired investors. E. Bid Incentives An important aspect to carry out the expansion plan is to encourage the desired investors. However, they could not participate in the bidding considering the competition and that the reference values could be below their own valuation. One way of addressing this problem is to generate incentives, basically higher payments, either through higher references value, interest rates or productivity factors. This paper assumes an incentive contract, subject to a costshare factor j (1). It is intended that although the optimal bid is subject to competition and the number of bidders, the final value of the project is a linear combination between the value of the bid and the optimal cost determined for each project (1). For example, consider that the design of the contract defines j = 1, then the optimal cost is the optimal value of the project. Instead, if we consider j = 0, the expected optimal value depends on the number of bidders and the optimal bids are described in Table VI. Fig. 2 shows the optimal value vs. the cost-share factor j. For a factor j = 0.5, the optimal value of the contract is expected to be approximately M$3.14, M$3.48, M$3.62, M$3.68 and M$3.75 for 2N, 3N, 4N and 5N, respectively. 5 Project Value (M$)

The expected bid values, bjn (3), are compared with the expected optimal cost, cjn* (3), of the winning bidder). The winner and best value of the optimal bid depends on whether or not the bid includes the xjn,RoW cost (4). If the bid includes the xjn,RoW cost the results show that most of the times the winner is n5, the desired type of investor. However, there are cases in which this does not happen and it is even possible to choose a non-desired investor. For example, we see that the Rubinstein solutions present different types of winners. In the case of the R1 solution, the winner is n4 for the rules M1 and M4, and n2 for the rules M2 and M3, which shows that these rules are not efficient. For the R3 solution, the winner in all cases is n2, a non-desired type of investor. If we consider the R2 solution, the winner is n5 for rules M1 and M4. For the M2 rule the winner is n4 and for the M3 rule the winner is n3. Finally, n5 always wins using the R4, BN, KS, ES, BS, PN, and SS solutions and does not depend on the M rule (Table V).

4 2N 3N 4N 5N Cj

3 2 1 0

0.2

0.4 0.6 0.8 1.0 Cost-share factor Fig. 2. Optimal contract value of the IEEE 24-RTS without right-of-way cost.

Now if we consider that the bids include the right-of-way cost, the optimal factor is a function of the type of solution (4). For example, if we consider the case of investor n3 and the M4 allocation rule, the factor j for the R4 solution is the maximum possible. For the BN, KS, ES, ES, PN and SS solutions the factor j that is equal to the optimal cost is 0.67. For other solutions, cost-share factors lower than 0.25 are defined. This implies that the latter solutions require a bigger cost-share factor because the optimal cost is bigger than the reference value (Fig. 3).

6

5 4 3 2 1 0

0.2

R1 R2 R3 R4 BN KS ES BS PN SS Vref

0.4 0.6 0.8 1.0 Cost-share factor Fig. 3. Optimal contract value of the IEEE 24-RTS with right-of-way cost. The optimal contract considers the n3 investor and the M4 mandatory rule.

F. Transmission Expansion Plan Value The previous section shows that the impact of the cost-share factor depends on the number of bidders (1). Then, this implies that the optimal contract will be higher as more investors participate. This is not always possible, the number of investors can be reduced in the bidding process, maybe to two or even none. For example, Fig. 4 illustrates the linear ratio between (1) and (2). That is, it shows how we can increase the reference value, Vj, with respect to the expected value of the bid, bj, and the variation of j. According to (2), the decrease in the number of investors reduces the value of the expected bid, therefore, the expected value described by (1) is also reduced. One way to limit the impact of a low participation of investors is to recognize a bigger factor as to increase the number of participants of the auction. For example, assume a scenario in which the cost-share factor j varies depending on the expected value determined by (1). That is, a linear ratio is used to limit the drop in the expected project value. Figure 4a illustrates the cases with 2N, 3N, 4N and 5N auction participants (linearly increasing functions). A value of j = 0 means that the expected optimal value of the project is the value determined by the bid bj , and j = 1 means that the expected optimal value of the project is the optimal cost, cj. If we want to encourage investment and participation in the auction we must establish a value range or a limit to the expected value, Vj. To do that we can establish two strategies. The first one, ESC1, considers a contract where, if there are more bidders, they are awarded with bigger factors to recognize the optimal cost of the project. This type of incentive can be seen as an incentive to participate. The second strategy, ESC2, seeks for the desired-type bidders (although there may be bidders that desist from participating), encouraging them to continue with the bidding process and rewarding them with bigger cost-share factors to recognize their optimal costs, even in cases where only two bidders are competing. This type of incentive can be seen as an incentive to stay in the auction. The ESC1strategy results in an increased expected value and the ESC2 strategy limits the drop in the expected value as the number of bidders decreases. Fig. 4b shows the behavior of the expected project value of the Opt. 1 project. If we consider an ideal scenario in which 5 investors participate, the expected value of the 5N bid is approximately M$3.5, according to Fig. 2. In the case of ESC1, the cost-share factors increase and exceed the reference value of M$3.5 for 5N and 4N, respectively. For example, considering j = 0.5 in the ESC1 strategy, the optimal value of the proposed contract will be approximately M$3.5 with

respect to the initial solution, M$3.0. For the ESC2 strategy, the expected value of the contract is no less than M$3.5, approximately. Now, if we take this reference value and the investors quit the auction, the minimum cost-share factor is increased in order to maintain, at least, the expected value of the bid (Fig. 4b, ESC2). That is, the minimum cost-share factor is 0, 0.25, 0.5 and 0.75 for 5N, 4N, 3N and 2N, respectively. 1.0 [Vj - bj ] /bj (p.u.)

6

0.8 0.6 0.4

2N 3N 4N 5N ESC1 ESC2

0.2 0 0

0.2

0.4 0.6 Cost-share factor (a)

0.8

1.0

5 Project value (M$)

Project Value (M$)

7

4 3 2 1 0

ESC1 ESC2 0.2

0.4 0.6 Cost-share factor

0.8

1.0

(b) Fig. 4. Optimal contract value of the IEEE 24-RTS without right-of-way cost.

These incentives are intended to encourage efforts to implement the project efficiently and on time. Such an approach determines the optimal cost of the project, where the bidders have a reward according to their effort, encouraging bidders to make competing offers. Having an offer with a very small value may mean a re-negotiation of contracts or unwanted delays to the overall efficiency of the electrical system, even if they pay fines for the delay. It often occurs that the valuation of a project ignores or undervalues of the expected costs. This is part of the planning fallacy, which tends to overestimate benefits and underestimate risks. In either case a social planner should limit this type of behavior and have the tools to encourage efficient investment. V. CHILEAN SIC CASE STUDY The second case study describes a predefined regulatory plan for the SIC system in Chile. We use the investment plan implemented by the central grid operator, CDEC-SIC. The features of the expansion plan are described in Table VII. Additional features are described in [6]. Each project is defined by the following parameters: the number of bidders representing the number of owners, the reference valuation determined by the regulator ranges between the maximum, 1.1, and the minimum, 0.9, with respect to the reference value, the minimum and maximum RoW value of one hectare (Ha), the total length of the project, the Cost of Operation, Maintenance and Administration (COMA). All data is shown in Table VII.

7 TABLE VII SIC PREDEFINED EXPANSION PLAN DATA #

Project

Owners (#)

1 2 3 4 5 6

Cardones – Diego Almagro 2x220 kV (1C) Ciruelos – Pichirropulli 2x220 kV (1C) Cardones – Maintencillo 2x500 kV Maintencillo - Pan de Azúcar 2x500 kV Charrua – Ancoa 2x500 kV (1C) Pan de Azúcar – Polpaico 2x500 kV

7 9 11 17 17 33

Project valueref (M$) 37.00 45.49 79.32 130.11 140.40 280.00

A. Right-of-way Cost For each bargaining solution and investor there is a right-ofway cost shown in Table VIII (4). The least cost is obtained with the solution R4. This solution has a mandatory value that restricts bilateral negotiation. The least-desired investor wins when using the rules M1 and M2, this is because the bidder has more patience. Otherwise occurs with n5, who gets the highest values. Additionally, we observe that the altruistic solution, BS, n5 has the highest values for the right-of-way cost. B. Valuation of the Investor’s Effort Table IX shows the optimal cost of each bidder. It shows the cost overruns (14%) that the n1 bidder has and the effort (5) or "productivity" in terms of cost that the bidders n2, n3, n4, and n5 can achieve. In the case of n5 it is 11%.

RoW costmin ($) 25000 25000 25000 25000 50000 35000

RoW costmax ($) 150000 150000 150000 150000 200000 175000

Land (Ha)

Length (km)

COMA (%)

36 50 71 112 105 214

152.0 83.0 132.4 209.2 196.5 401.8

2.1 2.07 1.4 1.57 1.4 1.5

M$731.02, for n2 is M$ 633.64, for n3 is M$605.80, for n4 is M$578.22, and for n5 is of M$572.96. Considering the rightof-way costs (4), we get the results shown in Table X. This case shows again that in the projects where the right-of-way cost is not significant compared to the cost of the project, regardless of the rule or bargaining solution, the optimal cost is obtained by the desired bidder. We consider that the project value (Table VII) is a cap value and the optimal cost (that can be smaller or bigger than the cap) depends on the bidder’s type and the bargaining cost.

C. Optimal Cost The optimal cost (3) of the project is determined by the cost of the asset, the effort factor and the negotiation cost of the right of way. Disregarding right-of-way costs, we obtain the global optimal cost expansion plan. The cost for n1 is

TABLE IX EFFORT VALUE FOR THE SIC PREDEFINED EXPANSION PLAN (M$) n2 n3 n4 n5 Cj * Project n1 1 38.0 32.9 32.2 30.0 29.8 33.3 2 46.7 40.5 39.6 36.9 36.6 40.9 3 81.4 70.6 69.0 64.4 63.8 71.4 4 133.5 115.7 113.2 105.6 104.7 117.1 5 144.1 124.9 122.1 114.0 112.9 126.4 6 287.3 249.1 243.6 227.3 225.2 252.0

TABLE VIII BARGAINING SOLUTIONS OF THE SIC PREDEFINED EXPANSION PLAN (M$) M 0 R1 R2 R3 R4 NB KS ES BS PN SS

M 0 R1 R2 R3 R4 NB KS ES BS PN SS

n1 62.5 59.5 52.0 16.7 55.1 55.1 55.1 59.0 55.1 55.1

n1 793.5 790.5 783.0 747.7 786.1 786.1 786.1 790.0 786.1 786.1

n2 71.2 62.6 58.1 29.8 56.7 56.7 56.7 60.5 56.7 56.7

n2 704.8 696.2 691.8 663.4 690.3 690.3 690.3 694.1 690.3 690.3

M1 n3 78.7 64.6 65.9 39.8 55.4 55.4 55.4 58.9 55.4 55.4

M1 n3 684.5 670.4 671.7 645.6 661.2 661.2 661.2 664.7 661.2 661.2

n4 83.0 66.5 70.6 47.3 56.3 56.3 56.3 61.5 56.3 56.3

n5 87.0 68.2 76.0 49.5 56.2 56.2 56.2 61.7 56.2 56.2

n1 68.6 66.1 52.0 16.7 61.6 61.6 61.6 59.0 61.6 61.6

n2 76.0 68.9 58.1 29.8 62.9 62.9 62.9 60.5 60.8 60.8

M2 n3 82.2 70.5 65.9 39.8 62.0 62.0 62.0 58.9 60.5 60.5

n4 85.4 71.5 70.6 47.3 62.2 62.2 62.2 61.5 61.0 61.0

n5 88.9 73.2 76.0 49.5 62.1 62.1 62.1 61.7 60.7 60.7

n1 62.5 59.5 52.0 16.7 54.8 54.8 54.8 59.0 54.8 54.8

n2 71.4 62.9 58.1 29.8 56.5 56.5 56.5 60.5 55.2 55.2

M3 n3 78.5 64.2 65.9 39.8 55.1 55.1 55.1 58.9 53.8 53.8

n4 81.6 63.7 70.6 47.3 53.7 53.7 53.7 61.5 52.5 52.5

n5 87.2 68.8 76.0 49.5 55.9 55.9 55.9 61.7 54.1 54.1

n1 65.7 63.0 52.0 16.7 55.9 55.9 55.9 59.0 55.9 55.9

TABLE X OPTIMAL COST OF THE SIC PREDEFINED EXPANSION PLAN WITH RIGHT-OF-WAY COSTS (M$) M2 M3 n4 n5 n1 n2 n3 n4 n5 n1 n2 n3 n4 n5 n1 661.2 659.9 799.6 709.6 688.0 663.6 661.8 793.5 705.1 684.3 659.8 660.1 796.7 644.7 641.2 797.1 702.5 676.2 649.7 646.1 790.5 696.5 670.0 641.9 641.8 794.0 648.8 649.0 783.0 691.8 671.7 648.8 649.0 783.0 691.8 671.7 648.8 649.0 783.0 625.5 622.5 747.7 663.4 645.6 625.5 622.5 747.7 663.4 645.6 625.5 622.5 747.7 634.6 629.2 792.6 696.5 667.8 640.4 635.1 785.8 690.2 660.9 631.9 628.9 786.9 634.6 629.2 792.6 696.5 667.8 640.4 635.1 785.8 690.2 660.9 631.9 628.9 786.9 634.6 629.2 792.6 696.5 667.8 640.4 635.1 785.8 690.2 660.9 631.9 628.9 786.9 639.7 634.7 790.0 694.1 664.7 639.7 634.7 790.0 694.1 664.7 639.7 634.7 790.0 634.6 629.2 792.6 694.5 666.3 639.2 633.6 785.8 688.9 659.6 630.8 627.1 786.9 634.6 629.2 792.6 694.5 666.3 639.2 633.6 785.8 688.9 659.6 630.8 627.1 786.9

n2 68.0 58.3 58.1 29.8 52.9 52.9 52.9 60.5 52.9 52.9

n2 701.6 692.0 691.8 663.4 686.6 686.6 686.6 694.1 686.6 686.6

M4 n3 80.0 66.7 65.9 39.8 58.1 58.1 58.1 58.9 58.1 58.1

M4 n3 685.8 672.5 671.7 645.6 663.9 663.9 663.9 664.7 663.9 663.9

n4 81.6 63.8 70.6 47.3 53.2 53.2 53.2 61.5 53.2 53.2

n4 659.9 642.0 648.8 625.5 631.5 631.5 631.5 639.7 631.5 631.5

n5 87.2 68.9 76.0 49.5 57.9 57.9 57.9 61.7 57.9 57.9

n5 660.2 641.9 649.0 622.5 630.8 630.8 630.8 634.7 630.8 630.8

8

E. Bid Incentives Now, considering the competition that is generated in the auction and a linear contract to obtain the equivalent competitive value to a tender with 5 bidders we assign a costshare factor j of 0.2 for the case of 4 bidders (4N), 0.4 for the case of 3 bidders (3N), and 0.6 for the case of 2 bidders (2N). TABLE XI OPTIMAL PROJECT BID OF THE SIC EXPANSION PLAN WITHOUT ROW

400 300 200 0

ESC1 ESC2 0.2

0.4 0.6 Cost-share factor

0.8

1.0

Fig. 6. Optimal contract value of the SIC predefined expansion plan without right-of-way costs.

The methodology described is implemented in MATLAB ® 7.3. We use an Intel ® Core ™ 2 Duo [email protected] with 2 GB RAM. The MATPOWER 4.0b3 tool was used to calculate the optimal power flow [7].

2N

3N

4N

5N

cjn*

VI. CONCLUSIONS

1 2 3 4 5 6

14.9 18.3 31.9 52.3 56.5 112.6

19.8 24.4 42.5 69.8 75.3 150.1

22.3 27.4 47.9 78.5 84.7 168.9

23.8 29.3 51.0 83.7 90.3 180.2

29.8 36.6 63.8 104.7 112.9 225.2

We have presented the application of a general transmission investment model with three main components: a bargaining solution for the right-of-way cost, a principal-agent model to evaluate the optimal effort of a bidder, and a linear contract model to calculate the final value of a transmission project. All these features produce a realistic framework to analyze the bargaining process of a project, to create incentives for the desired bidders to be rewarded, and to maximize the social value of the project. The proposed methodology offers a special perspective to deal with the performance of a transmission expansion plan. This methodology provides a framework for the over-run cost and bargaining problems observed in many transmission projects. Thus, the centralized planning framework should be adjusted to the increasing number of problems that affect investments in transmission. Our method provides evidence for incentive design schemes to encourage investment and efficiently run expansion plans.

600 Project Value (M$)

500

Proj.

Fig. 5 shows the behavior of the optimal contract for the SIC case study. 500 2N 3N 4N 5N Cj

400 300 200 0

600 Project value (M$)

D. Bidding for a Contract Table XI shows the valuation of the project based on the number of participants in the auction. The results show that the winning is always bidder n5. Basically, due to the level of effort that determines a lower optimal cost. This is what the social planner seeks efficient projects with desired bidders.

0.2

0.4 0.6 0.8 1.0 Cost-share factor Fig. 5. Optimal contract of the SIC predefined expansion plan without rightof-way costs.

F. Transmission Expansion Plan Value Fig. 6 shows the behavior of the value of the project regarding the optimal cost, the number of participants of the tender and the cost-share factor, j (1). The strategies described -ESC- show how to increase the value of the optimal contract when the number of investors increases (Fig. 6 - ESC1) and remains approximately constant even if the number of investors decreases (Fig. 6 - ESC2). This type of incentive is applied to the SIC predefined expansion plan. For example, consider j = 0.5, the optimal value of the proposed contract using the ESC1 strategy is approximately M$485 with respect to the initial solution, M$425. Using the ESC2 strategy, the expected value of the contract is not less than M$464 (competitive offer with 5N), approximately. Thus, greater recognition will encourage efforts to implement the project efficiently and on time. If the number of desired bidders is low, it is possible that a single desired bidder fails to carry out all the projects, whether for reasons of budget constraint, risk and business strategy. It is therefore important to assess the increased risk on those projects where the desired bidders do not participate due to the aforementioned constraints.

VII. REFERENCES [1] [2]

[3]

[4] [5]

[6] [7]

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. 261-290, Nov 2006. E. Sauma, F. Traub, and J. Vera. (2010, May, 2010.). Effect of Delays in the Connection-to-the-Grid Time of New Generation Power Plants over Transmission Planning. Jun. Available: http://www.gsb.stanford.edu/facseminars/conferences/orpiconf/doc uments/Sauma_Enzoetal.pdf V. Rious, J.-M. Glachant, and P. Dessante. (2010, Transmission Network Investment as an Anticipation Problem. RSCAS Working Papers 4(Jan). Available: http://ideas.repec.org/p/rsc/rsceui/201004.html J. D. Molina and H. Rudnick, "Transmission Expansion Plan Ordinal and Metaheuristic Multiobjective Optimization," in Proc. of PowerTech, Trondheim, Norway, 19-23 Jun, 2011. L. P. Garces, A. J. Conejo, R. Garcia-Bertrand, and R. Romero, "A Bilevel Approach to Transmission Expansion Planning Within a Market Environment," Power Systems, IEEE Transactions on, vol. 24, pp. 1513-1522, 2009. CDEC-SIC. (2011, May, 23). Trunk Transmission System Auctions. [online] Available: https://www.cdecsic.cl/licitaciones_detalle_es.php?licitaciones_id=13 H. Wang and R. Zimmerman. TSPOPF - High Performance AC OPF Solvers for MATPOWER [Online]. Available: http://www.pserc.cornell.edu/tspopf/