impact of introducing flexibility in the colombian
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Energy 157 (2018) 131e140
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Energy
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Impact of introducing flexibility in the Colombian transmissionexpansion planning
Alvin Henao a, *, Enzo Sauma b, Angel Gonzalez a
a Department of Industrial Engineering, Universidad del Norte, km 5, Barranquilla, Colombiab Department of Industrial and Systems Engineering and UC Energy Research Center, Pontificia Universidad Cat�olica de Chile, Av. Vicu~na Mackenna 4860,Macul, Santiago, Chile
a r t i c l e i n f o
Article history:Received 4 February 2016Accepted 22 May 2018Available online 24 May 2018
Keywords:Binomial treeColombiaFlexibilityOption to deferReal optionsTransmission expansion planning
* Corresponding author. Department of IndustrialNorte, km 5, Barranquilla, Colombia.
E-mail addresses: [email protected] (A.(E. Sauma), [email protected] (A. Gonzalez).
https://doi.org/10.1016/j.energy.2018.05.1430360-5442/© 2018 Elsevier Ltd. All rights reserved.
a b s t r a c t
Power transmission expansion planning (TEP) is a process affected by uncertainty that requires a long-term vision to anticipate requirements of interconnections among generators and load centers. Flexibilityis an alternative to cope with uncertainty that increases social welfare. This article seeks for estimatingthe value of this flexibility and interpreting its meaning in the Colombian power system. In order to keepthis focus clear, we implement a well-known methodology of Real Options (RO) in a simplified version ofthe Colombian power system. Accordingly, we introduce flexibility using a simple strategy of adding newtransmission lines and make a simple comparison with a robust approach, as the one currently used inColombia. Our results show that introducing this flexibility in the Colombian TEP increases social welfarein around $35 million, which represents an upper bound for the additional investment costs incurred toprovide this adapting ability to the power system. These results suggest the need for revising the currentparadigm of robust approach to prepare future Colombian transmission plans.
© 2018 Elsevier Ltd. All rights reserved.
1. Introduction
Transmission Expansion Planning (TEP) plays a key role in allpower systems around the world because of the requirement ofinterconnecting load centers and generation units, especially as amechanism for integrating renewable energy [1]. The increasingpenetration of renewable energy poses some challenges on TEP byadding more sources of uncertainty, such as the intermittency ofsome renewables, and by dealing with shorter construction timesof new generation facilities compared to conventional plants [2e5].As a result, the network planner makes long-term investment de-cisions under deep uncertainty. Robustness and flexibility areimportant approaches to address uncertainties in this field [6e8].
A robust transmission plan is designed to be immune to nega-tive effects, considering all proposed future scenarios. One robustapproach is the use of the min-max regret method. This applies aregret function to compare the value of a selected attribute of aparticular plan under different scenarios against its value at the
Engineering, Universidad del
Henao), [email protected]
optimal solutions of the same scenarios [9]. If regret is zero, theplan is optimal, but if the regret is not zero, hedging strategies canmake a robust plan. Commonly, robust approaches assume in-vestment decisions are made today without the possibility ofdeferring them until more information is available. In this sense,robustness is a rigid approach to cope with uncertainties.
In contrast, from a managerial perspective, a transmission planis said flexible if it may be easily (i.e., not too costly) adapted [10].This approach better reflects the decision maker's behavior whenshe deals with situations that differ from planned so she is forced toadjust, adapt, or change her decision as more information becomesavailable.
A simple method to produce a transmission plan is adding newcircuits between nodes at different periods of the planning horizon.In this simple framework, the network planner is able to use arobust or a flexible approach in that process. If the planner uses arobust approach, then she needs to identify the “robust” new cir-cuits at the initial stage to support the different scenariosemployed. In contrast, if the planner adds circuits using a flexibleapproach, then she acts in response to new information. Thus,selecting a flexible approach generally involves an increase of thenet present benefits, but with an extra cost [11]. Consequently,some questions arise: What are those increments on the net
A. Henao et al. / Energy 157 (2018) 131e140132
present benefits? What is the extra cost? What is the meaning ofthis? This paper aims to provide answers to these questions.
Under managerial flexibility, a flexible transmission plan can bereached by methodologies like Adaptation Costs (AC) [12], Multi-stage Stochastic Optimization (MSO) [13,14], and Real Options (RO)[15,16]. This article uses RO because it is a well-known methodol-ogy among managers and, differently than the others, it allowsdoing a direct estimation of the flexibility value.
Maghouli et al. [17] deal with uncertainty by using both therobust and the flexible approaches. They propose a multi-objectivemodel that simultaneously minimizes total social cost, maximumadjustment cost, and maximum regret. By adding new circuits tothe network, under different scenarios, they identify the best so-lution considering all objectives. From a flexible point of view, theyuse an AC method. Accordingly, the proposed method allowsselecting the network additions with the minimum cost of adap-tation to the selected scenarios (i.e., making transmission in-vestments that have the minimum cost in case they need to beadapted). However, the authors do not estimate the value of theflexibility added.
On the other hand, MSO allows adapting an initial investmentdecision on transmission expansion after the market reaction isknown [18]. In this case, the network planner split the planninghorizon using several fixed points where she observes the conse-quences of her decision in one previous time point to make ad-justments in the next point. This procedure, commonly known as“wait and see”, works with multiple scenarios. Each scenario has anassigned probability, previously determined. Although this methodmay require a high computational effort, some solutions have beenproposed in the literature [19]. In contrast with the AC approach,MSO allows estimating the value of flexibility, although in an in-direct manner [13,18,20]. In Ref. [18] this value is obtained by thecomparison with a robust approach.
It has been widely discussed that TEP is surrounded by un-certainties [21] and involves irreversible investments [22]. InRef. [23], however, the authors argue that an opportunity to investis not an obligation, but a right to buy an asset in the future. In thissense, if an investor makes an irreversible investment, he exercisesthe option to invest. However, waiting for new information maychange the initial decision. Naturally, this flexibility has value,which should be included in the project cost-benefit analysis [24].Under this paradigm, flexibility can be viewed as a way to increasethe Net Present Value of transmission plans [25]. RO theory rec-ognizes that managerial flexibility valuemay be substantial in someprojects and provides a simple method to estimate such flexibility.
The application of RO in TEP requires the construction of aproject cash flow. Additionally, it requires the identification of op-tions, the estimation of volatilities, and a solution method for theRO problem [26]. A good explanation about the framework thatsupports RO in a context of TEP can be found in Ref. [27]. A solutionin this kind of problems can be obtained by using techniques basedon continuous time stochastic models [18], including both sto-chastic partial differential equations [28] and dynamic program-ming [27] that bears discrete time stochastic models, like thebinomial tree [29]. Techniques like Least Square Monte Carlo anddynamic programming, proposed for valuing American Options inRef. [30], allow handling complex investment projects with mul-tiple uncertain variables considered as a portfolio of options. Thesetechniques have been applied on TEP [15,16,31], but from a private-investment viewpoint. Other applications of RO on TEP can befound on the literature; some of them focus on determining theoptimal time for investing in transmission lines [27,32], or the valueadded by flexibility [33], or on proposing improvements in solvingthe stochastic optimization problem [34].
The previous works recognize the value that flexibility may add
to the investment decision in TEP, but they ignore the fact thatflexibility involves an extra cost in order to make a flexible plan. Inaddition, they generally focus on private profit, not observing theproblem from a social-welfare viewpoint. This gap is partially filledout in Ref. [11], where the authors make a RO analysis in a simpletwo-node network from a social-welfare perspective. In Ref. [11],the authors consider a network planner that optimizes socialwelfare by managing energy production from private generators.Based on that work, this paper splits the transmission expansionprocess into two parts, as a simple strategy to introduce flexibilityas a hedging mechanism against uncertainty. Accordingly, thenetwork planner will invest in the second-part component only ifthe market conditions are favorable to this investment. In conse-quence, this paper applies RO considering the second-part processas an option to defer. Although RO allows the direct estimation ofthe value of flexibility, we also include a simple comparison of theresults with a robust approach, to make explicit the meaning offlexibility.
In the case of Colombia, although power-sector regulationsrequire medium- and long-term flexible plans [35], studiesapplying flexibility approaches are scarce. This article seeks forestimating the value of flexibility in the Colombian TEP, using amethodology based on RO. In particular, a binomial tree techniqueis applied because of its simplicity, intuition, and convenience todetermine the value of the flexibility [36]. We estimate this flexi-bility using a simplification of the National Interconnected System(SIN) of Colombia.
Accordingly, the main contributions of this paper are thefollowing:
� Analyzing the impact of introducing flexibility in the ColombianTEP process, estimating and interpreting the value added by theflexibility on a simplified version of the Colombian powersystem.
� Showing that a flexible approach, based on an option to defer atransmission investment, may significantly increase socialwelfare in the Colombian TEP process.
� Showing that the simple and well-known methodology usedhere can be employed to extend our conclusions to other powersystems.
The rest of this article is organized as follows. Section 2 de-scribes the Colombian power system and the simplifiedmodel usedin this paper. Section 3 explains themethodology proposed. Section4 shows our results. Finally, Section 5 concludes the paper.
2. Methodology proposed
Fig. 1 shows a flow diagram summarizing the general method-ology followed in this paper. This methodology is similar to the onedescribed in Ref. [11] in determining the option value. However,differently than the methodology in Ref. [11], this methodologyincorporates some complexities that were previously ignored byadding transmission lines as a flexible strategy, introducingKirchhoff's current and voltage Laws, and modeling competitivegeneration firms (i.e., price-taker generators). In addition, a com-parison between flexible and robust approaches is performed inorder to better understand the implications of estimating the valueof deferring an option to invest.
The first step in the proposed methodology (first box followingStart in Fig. 1) is to define the “base case”, which, in our case, cor-responds to the no transmission expansions decision by the systemoperation. In this base case, values are estimated by solving, foreach year and each scenario, the Optimal Power Flow (OPF) modeldescribed in Appendix A.
End
Start
Define the base case and initial
conditions of the system
Define fixed expansion
Define flexible expansion
Use Monte Carlo simulation to
obtain multiple paths of the
underlying asset
Estimate volatility
Build binomial trees
Estimate the value of the option of
deferring
Fig. 1. Methodology proposed.
A. Henao et al. / Energy 157 (2018) 131e140 133
The transmission expansion process is split in two parts toincorporate flexibility. The first one is called “fixed expansion”,which corresponds to determining a transmission expansion that isoptimal under the expected evolution of the system conditions. Thesecond one is an expansion, additional to the first one, which servesas an adapting-to-contingency mechanism. The latter is called“flexible expansion”.
The following simplifying assumptions aremade in our analysis:
� Each scenario has the same probability of occurrence.� Demand is perfectly inelastic at every node. This is in agreementwith the fact that electricity demand in Colombia is highly
inelastic (average elasticity in Colombia was 0.22 from 1999 to2014) [37].
� Power supply completely covers demand, the system is reliable,and transmission losses are negligible.
� Generators' marginal costs remain constant over time. In addi-tion, generators do not have the ability to exercise marketpower.
� The analysis includes future Colombian generation expansionprojects [37].
� A regression model estimates the properties of new lines addedto the system, by using data in Table 2 [38].
Several alternatives of transmission lines for fixed expansion areevaluated in terms of their contributions on the total surplus of thesystem,DTSt . This contribution (in $/h) during the planning horizonin each scenario is computed as in (1).
DTSt ¼XnDCSn;t þ
XnDPSn;t þ
"����� X
nGlnPn;t
!*fl
�����2
������ X
nGlnPn;t
!*fl
�����1
#cl ε L; c t εf1;…; Tg (1)
The first term of the right-hand side is the incremental con-sumer surplus, DCSn;t produced by the fixed expansion on the basecase at node n and at time t. The second term is the incrementalproducer surplus DPSn;t at node n and at time t. Finally, the thirdterm, in square brackets, represents the contribution of the incre-mental congestion rent produced by the fixed expansion (sub-in-dex 2) over the base case (sub-index 1). This term uses nodal pricesPn;t at node n and at time t. The incremental consumer surplus andthe incremental producer surplus are determined in (2) and (3),respectively.
DCSn;t ¼ ��Pn;t � Pn;t�*dn (2)
DPSn;t ¼Xi2In
��Pn;t � Ci
�*qi2 �
�Pn;t � Ci
�*qi1
�(3)
The sub-indexes 2 and 1 in (2) and (3) represent the fixedexpansion and the base case, respectively. In addition, the sum-mation symbol includes all generators in node n.
For each transmission expansion alternative, incremental totalsurplus ðDTStÞ is estimated at each year of the planning horizon andits present value is obtained using the same discount rate in eachscenario. Then, the alternative with the largest expected presentvalue is selected.
2.1. Flexibility
Managerial decisions may vary during the planning horizon dueto the presence of uncertainties. In this case, having flexibility mayallow increasing the capacity of the fixed expansion when futureconditions suggest an improvement in social welfare. Naturally, thefollowing situations may occur:
� The additional or flexible expansion is never needed.� At a given time, it is better investing on the flexible expansionthan waiting for better conditions.
� At a given time, it is better waiting than investing on the flexibleexpansion at that time.
Evidently, this flexibility in the future expansion, additional tothe initial network capacity, has a cost. The value of the option to
A. Henao et al. / Energy 157 (2018) 131e140134
defer the investment may be used as an upper bound of the will-ingness to pay of the network planner for this extra cost of havingflexibility.
For simplicity purposes, we made the following additionalassumptions:
� Only uncertainty in demand growth is considered.� The time to exercise the option to defer is equivalent to theplanning horizon. This might eventually not be the case due tothe nature of the real assets and investment projects [39].
� A single transmission expansion project is analyzed to supportthe introduction of flexibility. The new transmission asset isavailable from the next period when the expansion decisionwasmade.
Following Fig. 1, our methodology requires to determine a sizefor the flexible expansion and to estimate its contribution on thetotal surplus, after the fixed expansion is already made.
RO methodology requires defining an underlying asset value(S0), which is the expected present value of the incremental totalsurplus of the flexible expansion in this case. The evolution of S0over time serves as a signal to decide if it is convenient either toinvest on the flexible expansion at a specific time or to wait.Obviously, this decision depends on the value of the option to defer.
The estimation of S0 follows the same steps done in determiningthe fixed expansion described before. A binomial tree, like the oneshown in Fig. 2, represents the evolution of S0 through the planninghorizon [36]. The construction of the binomial tree requires theestimation of the volatility of the underlying asset [39]. This vola-tility is the variability of the cash flow return estimated from thevalues of the incremental total surplus of the flexible expansion.The volatility is obtained, in this case, by Monte Carlo simulation[40], according to (4)e(6).
Zit ¼ ln
PVi
tPV0
!(4)
st ¼ std dev:�Zit�
(5)
s1 ¼ st. ffiffi
tp
(6)
where Zit is the logarithmic cash flow return for t years estimated at
Monte Carlo simulation i; PVit is the present value of DTSt at period t
at Monte Carlo simulation i; t is any time in the future; st is thevolatility through time step t; and s1 is the volatility for one timestep.
Fig. 2. Binomial tree appearance with st ¼ 1 year and a planning horizon of 8 years.
The forecast of the incremental total surplus in the system overthe planning horizon allows estimating the variable PVi
t . Thisforecast considers demand growth rate as a random variable thatfollows a triangular distribution. Under these conditions, the OPFdescribed in the appendix A is executed at each period of theplanning horizon, allowing the estimation of DTSt using (1)e(3),but with sub-index 2 representing the systemwith both the flexibleexpansion and the fixed expansion and sub-index 1 representingthe systemwith only the fixed expansion, in this case. Future valuesof DTSt are brought to present t by using a discount rate of 10%.
The process described above is repeated 1000 times by MonteCarlo simulation, so (4)e(6) can be applied at the end.
In Fig. 2, the evolution of S0 can go up or down due to therealization of the demand uncertainty in themodel. To represent anupwardmovement of S0 this value is multiplied by a factorU that isa number larger than 1. On the other hand, when the value goesdown, S0 is multiplied by a factor D smaller than 1. U and D arecalculated as follows:
U ¼ exp�s1
ffiffiffiffiffidt
p �(7)
D ¼ 1 =U (8)
As mentioned earlier, a key assumption in this approach is fixingthe life of the option as the planning horizon (T). This allows us toestimate the value of the option at the end of the planning horizon.
The last nodes of the binomial tree (i.e., the nodes at the end ofthe life of the option) contain the range of possible values that theunderlying asset may take when the option expires (these valuesrange from SoU8 to SoD8, if T ¼ 8 as in Fig. 2). The maximizationcriterion in (9) determines the value of the option to defer theflexible expansion when the option expires. This criterion suggeststhat the option has value if the net value reached by S0 at the end ofthe life of the option ( PVT ;j) is positive:
OVT ;j ¼ maxPVT ;j � XT � PenaltyT ;0
�(9)
where XT is the exercise price or strike price (i.e., investmentneeded to obtain the present value PV of the project at period T).For simplicity, this investment cost does not change through theplanning horizon; PVT ;j is the value of the underlying asset at time Tat node j (the upper node is j ¼ 1); and PenaltyT is the opportunitycost lost due to the fact that the flexible expansion is not executedat time T (it would be zero if there is not congestion in the system).
The penalty term is included because network congestion re-duces social welfare. Penalty magnitude at time T is estimated asthe incremental total surplus produced by the flexible expansion atthat time if such expansion were made.
By applying backward induction, the option value is obtained ateach stage of the binomial tree, until the initial node is reached att ¼ 0.
The option value (OVt;j) for prior periods to the last one areestimated applying the following maximization criterion(although, other approaches are possible [16].
OVt;j ¼ maxPVt;j � Xt � Penaltyt ; DVt;j
�(10)
where DVt;j is the expected discounted value of continuing with theoption open (i.e., it represents the value of not exercising the optionin period t and waiting until next period to decide). The followingequation calculates DVt;j:
A. Henao et al. / Energy 157 (2018) 131e140 135
DVt;j ¼hp � OVtþdt;j þ ð1� pÞ�OVtþdt;jþ1
ie�Rf �dt (11)
where p is a probability adjusted by risk [41] and it is estimated by(12).
p ¼ eRf � DU� D
(12)
where Rf is the risk free rate of return.Then, it is possible to invest at a stage t if PVt;j � Xt > DVt;j.When t ¼ 0 is reached, the Present Value with Options (PVO) is
obtained, and it contains the option value to defer the flexibleexpansion. To estimate the option value at time 0 (OV0;1), the valueof the project without flexibilities, PVWF, is subtracted from PVO(PVWF ¼ S0 � X0).
OV0;1 ¼ PVO� PVWF (13)
2.2. Robust approach
Next, we explain the robust approach we use to compare theintroduction of flexibility into the TEP process.
Using (1)e(3) and the equations presented in Appendix A, wecan evaluate the performance of the eligible transmission lines ineach scenario of demand growth. In particular, the comparison ofincremental total surplus provided by each candidate line allows usthe identification of the optimal value for each scenario. Deviationsfrom this optimum yield to a regret value for each eligible trans-mission line, as describes in (14).
Regreti;j ¼��TSi;j � TS*
�� (14)
where Regreti;j is the regret function for candidate line i at scenario j
and TS* is the incremental total surplus for optimum at scenario j.Thus, the robust solution is given by mini½maxjðRegreti;jÞ�.
Fig. 3. Reduced version of the Colombian SIN.
Table 1Generation units' characteristics.
Generator Type Node Marginal Cost ($/MWh) Max Power (MW)
Unit 1 Coal 1 98.6 290Unit 2 Gas 1 399.4 2127Unit 3 Gas 1 465.6 187Unit 4 Hydro 2 232.5 550Unit 5 Coal 3 83.5 315Unit 6 Hydro 4 180.7 2949.8Unit 7 Gas 4 636.9 485Unit 8 Hydro 4 190.9 1806Unit 9 Gas 4 392.5 278Unit 10 Hydro 5 144.8 1894.6Unit 11 Coal 5 85.1 225Unit 12 Hydro 5 207.7 1000Unit 13 Gas 5 28.6 50.7Unit 14 Hydro 6 1230.4 881.9Unit 15 Gas 6 142 205Unit 16 Gas 6 621.4 230.9Unit 17 Hydro 6 239.2 945Unit 18 Coal 2 89.1 414.0Unit 19 Hydro 3 369.7 800.0Total 15,635
3. Colombian power sector
Before applying the proposed methodology to the Colombiacase, it is worth to describe the general characteristics of theColombian power system. The Colombian National InterconnectedSystem (SIN) is an electric system composed of generation units,interconnection networks, regional and interregional transmissionnetworks, distribution networks, and electric loads. The SIN hasbeen divided into six regions, defined by considering the network
congestion (that forces generationwithin such areas or limit tradeswith the rest of the SIN regions) [35]. These regions are the base forthe system simplification employed in this paper.
In 2015, the SIN had a total generation capacity of 15 GW. Twothirds of them come from hydroelectric power plants [42]. The SINhas approximately 24,000 kms of transmission lines, 57% of whichcorresponds to the National Transmission System (STN). The STN isa set of transmission lines with at least a voltage of 220 kV [35]. Thetransmission lines below 220 kV (that do not belong to the LocalDistribution System, SDL) represent the Regional TransmissionSystem (STR).
The Colombian network planner performs the TEP using robustcriteria over certain scenarios to identify and evaluate solutionsfrom a technical and economic point of view [37].
We model the SIN network with six nodes (or buses), eachcontaining load centers and main generation units. Historical data
of energy generation and prices were taken from Ref. [37]. Fig. 3shows a diagram of this reduced system.
The localization, technology, marginal costs, and capacity ofgeneration units are shown in Table 1. Table 2 contains informationabout transmission lines. Nodal and total loads at the beginning ofthe horizon are presented in Table 3.
In the TEP formulation of the system described in Fig. 3, we
Table 2Transmission lines' characteristics.
Line name Max flow (MW) Reactance X (p.u.) Starting Node Final Node
L1 1200 0.0423 1 3L2 1300 0.0209 1 2L3 1300 0.0316 2 4L4 500 0.1381 3 5L5 400 0.0602 4 3L6 200 0.1604 4 3L7 800 0.033 4 3L8 700 0.0261 4 5L9 300 0.0452 4 4L10 250 0.0648 4 6L11 250 0.0994 4 6L12 550 0.027 4 6L13 215 0.1018 5 6L14 215 0.2182 5 6
Table 3Loads at the beginning of the horizon.
Node Load (MW)
Node 1 1990.3Node 2 141.8Node 3 772.4Node 4 1773.9Node 5 2821Node 6 1940.2Total (MW) 9439.6
A. Henao et al. / Energy 157 (2018) 131e140136
consider a planning horizon of 14 years. Within this horizon, powerload grows annually. We consider three scenarios for the annualload growth rate (2%, 2.5%, and 2.7%), the same at every node of theSIN. We assume that, although load at nodes changes annually, itremains constant within each year.
4. Results
Results from the base case (i.e., OPF with no transmission ex-pansions) in the system described in Fig. 3 suggest that there iscongestion between nodes 4 and 6. Accordingly, the inclusion of anew transmission line in this path seems to be a good candidate forexpansion. Table 4 shows the candidate lines selected, their in-vestment costs, and their contributions to the total surplus if theyare built. Reactance of candidate lines are estimated from Table 2 byadjusting a regression equation to the corresponding data [38].
In addition, Table 4 presents the results under three scenarios ofthe demand growth rate (probabilities are in parenthesis). The lastcolumn shows the maximum values of total surplus under eachscenario. Finally, the last row shows the net value of the incre-mental total surplus that results when the investment costs aresubtracted from the expected value. Consequently, a fixed expan-sion of 2000MW is the best solution because it presents the largestcontribution to total surplus.
Table 4Results of valuation of candidate lines (values in millions of dollars).
Demand Growth Rate Incremental Total Surplus in millions
350MW - Investment $89.4 2000MW - Inv
2.0% (0.3) $27,000 $41,1022.5% (0.4) $27,356 $59,7522.7% (0.3) $25,512 $69,862Expected Value $26,696 $57,190Net value $26,607 $56,966
4.1. Flexible approach
A transmission line of 350MW is proposed as the flexibleexpansion. Estimations of the incremental total surplus producedby this flexible expansion (in addition to the fixed expansion) areshown in Table 5. In consequence, the underlying asset value (S0)was estimated in $ 2361 million (probabilities of each scenario are
the same as shown in Table 4).Using Monte Carlo simulation, we estimate the value of the
parameters s1, U, D, and p. Results are shown in Table 6 (risk freerate Rf ¼ 0:03).
These values are needed in order to build the binomial treeshowing the evolution of the underlying asset over time. Subtrac-tion of investment ($89.4 millions for this 350MW line) and pen-alties from this tree produces the binomial treewith net values thatis shown in Fig. 4. Appendix B presents the penalties used in thiscase.
From Fig. 4, we can observe that there are conditions underwhich investment on flexible expansion produces negativecontribution on values of total surplus, so it is not convenientinvesting in these cases. However, delaying the investment mayrepresent a good alternative in some cases. Fig. 4 does not allowseeing this. The binomial tree of the option value, shown in Fig. 5, ismore useful for that purpose.
If the network planner executes the flexible expansion at period14 (see Fig. 5), she will obtain an increment of social welfareexpressed by those numbers inside the boxes. Positive numbers inthe last period indicate that the planner should invest. Positivenumber in the rest of periods may indicate investing immediatelyor waiting until the next period (the decisionwill depend onwhichvalue is larger). Finally, boxes marked with a zero indicate that theoption to defer until next period has no value, and the investment isnot convenient because it reduces social welfare. The application ofthe maximization criterion represented in (9) produces thosevalues in Fig. 5.
The values presented in columns 1 through 13 in Fig. 5 are theresult of comparing the execution of the flexible expansion at thattime against the delay of that investment until the next periodusing (10) and (11). As a result, Fig. 5 shows the maximumvalues ofthis comparison. The option to defer expires when the plannerdecides to make flexible investment.
Fig. 6 illustrates the best decision in every case. This figureshows the cases where the planner invest (boxes marked with lightgray color), where she does not invest in flexible expansion (boxesmarked with dark color), and where she waits for new information(boxes with no color) for making decisions. Fig. 6 shows a clearzone (after period 6), during planning horizon, where the optionhas expired.
According to these figures, it is better to wait in period 0 than toinvest in flexible expansion because this decision increases thevalue of the investment. In other words, delaying the decision tomake the flexible investment increases the incremental total sur-plus in the system. The value of this flexibility is $35millions. This is
Optimal value
estment $223.5 2350MW - Investment $312.9
$40,948 $41,102$60,651 $60,651$68,495 $69,862$57,093$56,780
Table 5Results of S0 in the flexible expansion case.
Demand Growth rate Expected Value S0 (Millions of dollars/y)
2% 2.50% 2.70%
Total surplus added by the flexible expansion $1075.41 $2567.76 $3370.10 $2360.76
Table 6Results of Monte Carlo simulations for theparameter estimation.
Parameters Values
s1 0.91U 2.5D 0.4p 0.3
Fig. 4. Binomial tree with the net values of the underlying asset for the Colombian case(amounts in millions of dollars).
Fig. 5. Binomial tree of the option values (amounts in millions of dollars).
Fig. 6. Decision tree.
A. Henao et al. / Energy 157 (2018) 131e140 137
the option value that results from the difference between of $2306minus $2271 (see Figs. 5 and 4 at period 0).
Table 7 shows a summary of these results.The additional expansion (in addition to the original system) can
be built only if the original system is able to accept it. Allowing thisgenerally involves some extra costs, corresponding to taller orwider transmission towers in the original system, among otheractions. The network will find flexibility useful if the option value islarger than the necessary extra cost. That means that the optionvalue to delay a flexible expansion becomes an upper bound of thewillingness to pay for providing the system with this adaptationability.
4.2. Sensitivity analysis
As a sensitivity analysis, we explore the impact on the value ofthe option to defer of changes in the generation costs. We select toperform a sensitivity analysis with respect to this parameterbecause it is the one with the largest influence on the option value.
Recall that Table 1 shows the types of generators located in eachnode of the reduced version of the Colombian power system shownin Fig. 3. Each type of generator has its own cost of generation thatmight lead to congestion depending on its localization. The mostcommon generation technologies in Colombia are hydro and gas, sothis sensitivity analysis explores the effects on the value of theoption to defer if hydro generators reduce their cost in 50% (keepingthe rest unchanged), and if gas generators reduce their cost in 50%(keeping the rest unchanged). The results of these sensitivity ana-lyses are shown in Table 8.
Table 8 shows that a reduction in the generation cost of hydropower plants increases the value of the option to defer, but thereduction in the cost of gas power plants reduces the option value.The explanation of this phenomenon is detailed next.
The proposed transmission expansion is located between nodes4 and 6. Hydro generators at node 4 have a capacity 6 times largerthan gas generators in the same node. In node 6, this capacity is 4times larger. Therefore, changes in cost of hydro generators have alarger effect on the option value than changes in the cost of gasgenerators. Moreover, the total capacity of hydro generators dou-bles the capacity of gas generators.
Under the context of cost minimization, the OPF gives priority tohydro generators so the possibility of delaying the transmissioninvestment increases its value because the needs of the system canbe coveredwith nearby generators. This is not true in the case of gasgenerators because gas generators with larger capacity are far fromthe location of the transmission expansion, justifying theexpansion.
4.3. Robust approach
Using (14), we determine the regret of each transmissionexpansion alternative. Table 9 shows these results.
Results from Table 9 suggest that a unique expansion of2000MW (robust expansion) is required to support demandgrowth scenarios. In order to explain this, it is convenient toremember that total surplus is obtained by the sum of consumer
Table 7Summary of results.
Description Values
Flexible expansion 350MW in addition to 2000MWProject Value Without Flexibility e PVWF
(Millions of dollars)$2271
Project Value With Options e PVO(Millions of dollars)
$2306
Option Value, OV0,1 (Millions of dollars) $35
Table 8Sensitivity analysis when changing generation costs.
Description Originalsystem
Hydrocase
Gascase
Project Value Without Flexibility e PVWF(Millions of dollars)
$2271 $2966 $4181
Project Value With Options e PVO (Millionsof dollars)
$2306 $3011 $4211
Option Value, OV0,1 (Millions of dollars) $35 $45 $30
Table 9Results using the regret function.
Demand Growth Rate Regret |TS -TS*|
350MW 2000MW 2350MW
2.0% $ 14,101 $ - $ 153.52.5% $ 33,294 $ 898.5 $ -2.7% $ 44,350 $ - $ 1366Maximum Regret $ 44,350 $ 898.5 $ 1366Min-Max $ 898.5
A. Henao et al. / Energy 157 (2018) 131e140138
surplus, producer surplus and congestion rents so, in this case, acomparison among these terms shows that an expansion from2000 to 2350 reduces incremental consumer surplus and incre-mental congestion rent, but increases incremental producer sur-plus. However, this increment does not compensate the reductionof both incremental consumer surplus and incremental congestionrents.
It is interesting to observe that the robust expansion coincideswith the fixed expansion, although there is an important differencein the approach. A robust expansion does not allow the possibilityof an expansion over the initial infrastructure. In contrast, theflexible expansion conceives from the beginning that possibility. Asa result, it anticipates possible adaptations and creates options forthat (height of tower, space for additional connectors, etc.). Obvi-ously, creating options produces extra costs. As mentioned previ-ously, the value of the option gives an upper limit of the extra costthat is rational to pay, which should not be exceeded.
In the case study analyzed here, it was evident that the flexibleexpansion should not always be built (recall Fig. 6). That is, therewill be the possibility of losing the extra cost invested if the plannerdoes not invest in the flexible expansion. That situation is similar tothe loss of the premium paid by the option buyer in financial op-tions, when the underlying stock does not close above the strikeprice at the expiration date.
5. Conclusions
The Colombian TEP is performed based on a robust approach tocope with uncertainty. However, this paradigm is contrasted withthe possibility of adding flexibility to the TEP. We showed that theincorporation of flexibility increases social welfare while managinguncertainty. In this paper, we use a reduced version of the
Colombian power system, a simple strategy to introduce flexibilityby adding lines, a well-knownmethodology among managers (RO),and a very simple comparison with a robust approach to keep thefocus of the paper clear. Although there are more elaboratedmethods, techniques and models to support flexibility, we prefer tokeep the methodology simple to maintain the focus of the paperclear.
The incorporation of flexibility allows the investors to adapttheir decisions according to new information. Planning with flexi-bility may be a smart decision because it may increase the value ofthe project. Planners are able to use AC, MSO or RO to build aflexible plan. RO allows estimating an upper limit for the extra costincurred to make the system able to support future adaptations.Thus, network planners should calculate how much additional in-vestment is required to make a flexible system and compare it tothe value of the flexibility added.
Under a flexible system, it may occur that the flexible invest-ment is needed, in which case the extra cost will accomplish itspurpose. However, it is also possible that the system does notrequire future adaptations so the extra-cost would be lost. Thissituation suggests that the extra-cost is like a premium paid by thenetwork planner in order to have the right to exercise the option toadapt the system to uncertain scenarios.
Based on the assumptions made on this paper, the followingtopics are left for future research work: effects of adding flexibilityon a real version of the Colombian interconnection system, analysisof regulatory changes in the TEP process to introduce flexibility,inclusion of realistic lead-time for transmission projects, andapplication of other methodologies to introduce flexibility andestimation of volatility.
Acknowledgments
This work was partially supported by CONICYT, FONDECYT/Regular 1161112 grant and by CONICYT, FONDAP 15110019 grant(SERC-CHILE). A. Henao would like to thank the financial supportfrom Universidad del Norte. We specially thanks Dr. David Pozo forproviding the data of the reduced version of the Colombian powersystem.
Appendix A. Optimal Power Flow (OPF) model
The following equations model a typical OPF formulation, usinga DC approximation of Kirchhoff's Laws.
minqi; fl;dn
Xi2I
Ci*qi (A1)
s.t.Xi2In
qi þXl2L
Glnfl ¼ dn; cn εN (A2)
fl ¼Sbxl
Xn2N
Glndn; cl εL (A3)
�Fmaxl � fl � Fmax
l ; cl εL (A4)
0 � qi � Qmaxi ; ci (A5)
�p
2� dn � p
2; cn (A6)
A. Henao et al. / Energy 157 (2018) 131e140 139
d1 ¼ 0 (A7)
where:
i: Generator index, i2f1; 2; …; 19gn: Node index, n2N ¼ f1;2;…;6gl : Line index.N : Set of nodes.In: Set of producers in node n.I : Set of producers.L : Set of transmission lines.Ci : Marginal cost of generator i.dn : Load in node n.Fmaxl : Maximum capacity of line l.Qmaxi : Maximum power given by generator i.
Gln: Line to node Incidence matrix.qi : Power produced by generator i (decision variable).fl: Flow through line l (decision variable).dn: voltage angle in node n (decision variable).
The objective function, defined in (A1) is the minimization ofthe operation cost. Thus, the OPF selects the least-cost generationthatmeet demandwhile taking into account theirmaximumpowerand marginal costs. A DC approximation of Kirchhoff's Laws arerepresented in (A2) and (A3). Constraints (A4), (A5), (A6), and (A7)define operational limits and the range of the decision variables.
Appendix B. Penalties
The table below shows the values of the incremental total sur-plus if flexible expansion is added at the specified time in eachscenario. We consider these values as proxy for the opportunitycost of the society due to postponing the flexible investment de-cision. Thus, these values are taken as the penalties for postponingthe investment decision.
Penalties (millions of dollars)
Time 0 1 2 3 4
2.0% $0.00 $0.00 $0.00 $0.00 $0.002.50% $0.00 $0.00 $0.00 $0.00 $0.002.70% $0.00 $0.00 $0.00 $0.00 $0.00Weighted average $0.00 $0.00 $0.00 $0.00 $0.00Time 5 6 7 8 9
2.0% $0.31 $0.25 $357.00 $357.00 $357.002.50% $0.24 $2.18 $357.00 $357.00 $357.002.70% $2.10 $2.25 $357.00 $357.00 $357.00Weighted average $0.82 $1.62 $357.00 $357.00 $357.00Time 10 11 12 13 14
2.0% $357.00 $357.00 $357.00 $357.00 $357.002.50% $357.00 $357.00 $357.00 $3518.49 $2541.802.70% $357.00 $357.00 $3521.17 $2544.33 $2827.12Weighted average $357.00 $357.00 $1306.25 $2277.79 $1971.95
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