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Research

Cite this article: Moreno R, Street A, Arroyo

JM, Mancarella P. 2017 Planning low-carbon

electricity systems under uncertainty

considering operational exibility and smart

grid technologies. Phil.Trans.R.Soc.A375:

20160305.

http://dx.doi.org/10.1098/rsta.2016.0305

Accepted: 12 May 2017

One contribution of 13 to a theme issue

‘Energy management: exibility, risk and

optimization’.

Subject Areas:

power and energy systems, electrical

engineering, energy

Keywords:

low-carbon power system planning, smart

grid, exibility, stochastic optimization, robust

optimization, power system economics

Author for correspondence:

Rodrigo Moreno

e-mail: rmorenovieyra@ing.uchile.cl

Planning low-carbon

electricity systems under

uncertainty considering

operational exibility and

smart grid technologies

Rodrigo Moreno1,2, Alexandre Street3,

José M. Arroyo4and Pierluigi Mancarella5,6

1Department of Electrical Engineering (Energy Center), University of

Chile, Santiago, Chile

2Department of Electrical and Electronic Engineering, Imperial

College London, London SW7 2AZ, UK

3Department of Electrical Engineering, Pontical Catholic University

of Rio de Janeiro, Rio de Janeiro, Brazil

4Departamento de Ingeniería Eléctrica, Electrónica, Automática y

Comunicaciones, Universidad de Castilla-La Mancha, Ciudad Real,

Spain

5Department of Electrical and Electronic Engineering, University of

Melbourne, Parkville, VIC 3010, Australia

6School of Electrical and Electronic Engineering, The University of

Manchester, Sackville Street, Manchester M13 9PL, UK

RM , 0000-0001-5538-445X;PM,0000-0002-9247-1402

Electricity grid operators and planners need to

deal with both the rapidly increasing integration of

renewables and an unprecedented level of uncertainty

that originates from unknown generation outputs,

changing commercial and regulatory frameworks

aimed to foster low-carbon technologies, the evolving

availability of market information on feasibility and

costs of various technologies, etc. In this context,

there is a signiﬁcant risk of locking-in to inefﬁcient

investment planning solutions determined by

current deterministic engineering practices that

neither capture uncertainty nor represent the actual

operation of the planned infrastructure under high

penetration of renewables. We therefore present

an alternative optimization framework to plan

electricity grids that deals with uncertain scenarios

2017 The Author(s) Published by the Royal Society. Allrights reser ved.

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and represents increased operational details. The presented framework is able to model the

effects of an array of ﬂexible, smart grid technologies that can efﬁciently displace the need

for conventional solutions. We then argue, and demonstrate via the proposed framework and

an illustrative example, that proper modelling of uncertainty and operational constraints in

planning is key to valuing operationally ﬂexible solutions leading to optimal investment in a

smart grid context. Finally, we review the most used practices in power system planning under

uncertainty, highlight the challenges of incorporating operational aspects and advocate the

need for new and computationally effective optimization tools to properly value the beneﬁts of

ﬂexible, smart grid solutions in planning. Such tools are essential to accelerate the development

of a low-carbon energy system and investment in the most appropriate portfolio of renewable

energy sources and complementary enabling smart technologies.

This article is part of the themed issue ‘Energy management: ﬂexibility, risk and

optimization’.

1. Introduction

(a) The evolving landscape: from conventional electricity systems to low-carbon smart

grids

Current electricity grids comprise large generating units that produce power generally far from

load centres, high-voltage transmission networks that ship power from production centres to load

centres (up to the so-called primary substations) and lower-voltage distribution networks that

transfer power from primary substations to commercial, industrial and residential consumers.

In this electricity system, referred to as conventional: (i) generation is typically carbon-intensive,

powered by fossil fuels, and in some cases supported by hydro plants that are usually large so as

to take advantage of economies of scale; (ii) transfer capability of networks (that are passive and

inﬂexible) is mainly delivered through more investment in asset-heavy infrastructure (e.g. lines

and transformers); and (iii) demand is mostly dominated by consumers’ end-use requirements

only and thus unresponsive to system/market conditions. In this context, operators aim to run

electricity grids in an economically efﬁcient and reliable manner by mainly controlling generation

outputs. Such a control allows operators to maintain the instantaneous production–demand

balance and transfers within network capacity limits. In addition, sufﬁcient security margins in

electricity infrastructure (generation and network) are retained to withstand credible outages (e.g.

sudden failure of a generating unit or network circuit). In planning time scales, network design

decisions attempt to eliminate congestions through investment in asset-heavy infrastructure up

to the point where the marginal beneﬁt of more network capacity (i.e. savings in congestion

costs and reliability costs driven by the installation of further lines and transformers1)isequalto

the marginal cost of network investment [1]. While network infrastructure is centrally planned,

generation investment is generally market-driven and hence decentralized; historically, network

planners have thus responded through more network investment to new connection requirements

from generation, which engenders system expansion.

In the coming years, the above-mentioned conventional electricity network will evolve

towards the so-called smart grid. Firstly, generation will increasingly be low-carbon and

distributed across the electricity system, even being located at the point of power consumption

(e.g. rooftop photovoltaic (PV) panels). Low-carbon generation from renewables such as wind

and solar power will create the need to counteract their variable and partly unpredictable power

injections through operational measures so as to maintain the instantaneous production–demand

balance (i.e. system frequency), generation outputs and network transfers within capacity limits

1Congestion cost is driven by network transfer limits and is equal to the extra cost of operation due to network capacity

constraints, while reliability cost is driven by the demand curtailment associated with network capacity and is equal to the

expected unsupplied demand times the value of lost load, VoLL.

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and the required security margins in generation and transfer capability (i.e. reserves and network

redundancy). Secondly, distribution networks will transition from mostly passive networks,where

the control problem is resolved at the planning stage by designing for the worst-case peak-

demand scenarios, to active systems, where new information and communications technologies

(ICT) and controllable distributed energy resources, including storage and ﬂexible demand,

will provide real-time supply–demand balance to efﬁciently integrate local renewables while

also interacting with the transmission system [2,3]. Thirdly, the transmission system itself will

become much more active through: (i) new controllable and ﬂexible technologies, such as

ﬂexible AC transmission systems (FACTS) [4] and high-voltage DC (HVDC) systems [5], that can

control power ﬂows through the network without changing power injections/withdrawals; (ii)

system integrity protection schemes (SIPS) that can enforce rapid increase/reduction of power in

importing/exporting areas after a network outage occurs by, for instance, curtailing generation

and/or demand [6]; and (iii) various wide-area monitoring and control equipment supported

by ICT technologies that will increase the capability of system operators to monitor and control

electricity assets in real time and throughout wider areas through increased communication

[7,8]. Last and more importantly, demand will be controllable and end-users will become active

participants in system and market operation, thereby opening up opportunities for aggregating

and coordinating consumers and system needs (e.g. for system balancing and congestion

management), taking advantage of ﬂexibility from smart appliances (e.g. dishwashers and tumble

dryers), electric heaters, batteries, etc. [9,10]. Also, electriﬁcation of other energy demands from

the heating/cooling (e.g. electric heat pumps and heating ventilation and air conditioning

equipment) and transport (e.g. electric vehicles) sectors can further expand the opportunities to

control electricity loads and coordinate them with system needs, owing to their intrinsic virtual

storage capabilities [11,12].

In the low-carbon smart grid context outlined above, and focusing on the system as a whole

and the transmission level, a change in system frequency due to system level supply–demand

imbalance can be resolved through the combined action of various fast generating units and loads

(e.g. disconnection of non-critical load from dishwashers and/or refrigerators [10], contribution

from distributed battery systems, including those from parked electric vehicles, connection of

distributed back-up generation, etc.), while network congestions can be resolved through changes

in topology and/or impedances (e.g. changing FACTS set points) rather than through costly

changes in the output of generation. This increase in operational ﬂexibility can be used to address

both real-time operation and time-ahead scheduling, where strategic decisions are made in

advance (e.g. a few hours ahead relative to real time) to deal with the uncertain evolution of wind

and solar power generation, and so adapt to different realizations that may happen in real time.

In this context, generation reserves can be coordinated with demand response, charge/discharge

actions from storage and even network topology reconﬁguration so as to deliver the needed

balancing services requested in real time due to wind/solar forecast errors and their potentially

high variability [9,13]. In particular, such coordination of multiple, ﬂexible operational measures

can increase network utilization and decrease holding levels of generation reserves that may

be signiﬁcantly costly due to the high levels of wind and solar power expected in the near

future [14,15].

In smart grid planning, it remains unclear whether new asset-heavy investment (e.g. a new

line and/or transformer) will be needed even in the presence of increased ICT infrastructure and

fast control of network devices, storage and demand, which can rapidly instigate reduction of

transfers and even eliminate network congestion [14]. The availability of distributed storage,

for instance, could locally provide power to consumers while battery charging and transfer

actions take place during hours where the network is not congested. Likewise, fast control of

network devices could eliminate post-contingency, real-time congestion by changing network

topology (e.g. through line switching), impedances (e.g. through the adjustment of FACTS set

points and soft normally open points [16], i.e. relying on power electronics-based technology)

and power ﬂows (e.g. through dispatchable HVDC links), thereby reducing the need for

network redundancy. Furthermore, the reliability role of redundant transmission and generation

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infrastructure could be substituted by the deployment of ﬂexible devices that can carry out

advanced operational measures, creating a strong relation between planning and operation

that should be carefully studied when expanding the system [17]. In addition, economics

and reliability will not be the only drivers of network planning, and thus new investment

solutions should also facilitate the achievement of environmentally driven energy policy targets.

Furthermore, owing to the increased levels of requests for new renewable generation connections

featuring extremely shorter construction times (shorter than those needed to build network

infrastructure and deﬁnitely shorter than those needed to build large conventional power plants),

network planners will need to undertake proactive investment, in anticipation of connection

requirements from generation, taking real advantage of the economies of scale of transmission

investment.2In the smart grid paradigm, network planners should thus proactively drive system

expansion rather than react to generation proposals, and support network decisions through the

deployment of ﬂexible, smart network technologies in order to more effectively adapt to the

unfolding future scenarios.

(b) Opportunities for advanced optimization models to plan electricity grids

The increase in automation and ﬂexibility due to new generation, storage, network, demand and

ICT technologies is paramount to face intermittent and uncertain power outputs from renewable

generation and defer the need for conventional network infrastructure. In fact, any attempt to

face the increase in renewables through current practices and without the adequate upgrade

in emerging network technologies could be extremely costly and even infeasible [14]. To truly

unleash ﬂexibility, however, operators will need to control a much wider array of set points

to ensure that economics and reliability are maintained at acceptable levels. For example, it is

expected that in the UK generating units providing fast frequency control (i.e. fast balancing)

could increase from about 10 (large generating units providing frequency control) to some 600 000

(if 10% of small PV and wind units support frequency control); network automatic controls

such as voltage regulation devices could increase from 10 000 to 900 000; and automatic controls

in homes (energy management systems) could increase from virtually zero to 15 million (if

half of the installed smart meters were to link with energy management devices) [18]. Along

with the increase in control set points (and thus optimization variables), the amounts of data

available in operational time scales will escalate at all levels due to technologies such as smart

meters—at the consumer level—and phasor measurement units [19]—at both transmission and

distribution network levels—which will allow operators to signiﬁcantly improve their visibility

and thus state estimation of the system. With a clearer picture of system conditions in real

time through increased volumes of data and with a higher level of controllability of network

equipment, operators will thus be able to run the system in a more secure fashion, using network

infrastructure at higher levels (i.e. with less redundancy and less congestion) and therefore

improving the overall economic performance of the system operation activity as well as deferring

the need for conventional electricity infrastructure investment.

In addition, there is an increasing need to capture coordination actions across various energy

vectors, including all electricity sectors (from local energy management systems, distribution and

transmission networks to generation), gas, heat/cooling and transport [12]. In fact, ﬂexibility from

electricity demand can be coupled with heat and/or mobility needs from consumers, and this may

have important impacts on electricity network operation and investment beyond distribution

networks, affecting also the transmission and generation infrastructure. For example, ﬂexible

demand from electric vehicles (i.e. battery charging/discharging actions) or heat pumps [11]can

be controlled so as to reduce electricity demand during peak hours, provide frequency control

services, provide congestion management services (in distribution and transmission networks),

2Building a 1000 MW line in anticipation of connection requirements is likely to be more efﬁcient than building 10 lines of

100 MW as a result of 10 different connection requirements at different points in time, albeit there is a risk associated with the

potential stranded network infrastructure if generation projects do not materialize as anticipated.

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etc., ultimately driving both less costly operation and less investment in network and generation

assets [9,20]. Coupling with other pieces of infrastructure, for example the gas network, can

then further increase the power system’s ﬂexibility and support integration of larger volumes

of renewables [21], as well as even provide new forms of storage [22].

Recognition of power system dynamics and stability phenomena (that are usually in the

time scale of a few seconds) is also becoming increasingly important in both the operational

and planning decision-making processes [23,24], and this is also crucially linked with the

penetration of advanced ICT infrastructure in power networks. Indeed, it is envisaged that

current redundancy levels (that are used to provide security under the current operational

doctrine) will be replaced with increased automation, monitoring, communication, control, etc.

(i.e. via corrective control actions) [24]. However, these might fail to properly operate in real time,

and such malfunctions, for example in the form of delays in communication, could originate

network stability problems and even a cascading load disconnection [25,26]. Hence, adequate

balance between network redundancy and advanced corrective control actions will need to be

recognized so as to properly determine (i) transfer limits and congestion levels in operational

time scales and (ii) the need for new network reinforcements in the long term, considering both

power and ICT network reliability [27,28].

Finally, an adequate treatment of uncertainty in both operational and planning time scales

will be essential in the presence of renewables in order to minimize the risk of locking in to

inefﬁcient solutions which ignore the multiple possible evolutions that networks may experience.

In operational time scales, prediction of renewable generation production from wind and solar

power plants will give rise to certain levels of forecast error against which operators hedge

through balancing services [29,30]. Thus, if wind power outputs, for example, are less than

expected, there will be a mix of generation and demand-based services that can counteract

the lack of wind energy production. It will also be important that network congestion be

efﬁciently managed in real time, even under those conditions where large forecast errors

are realized. In this context, the location of reserve services and the possibility to undertake

corrective actions over ﬂexible network components will be paramount. In planning time

scales, network design will need to anticipate renewables connection and thus face uncertainty

levels associated with the location and volume of new generation. Hence, planners will

require a strategic approach and determine a balanced portfolio of present and future network

investments in such a way that present, ‘here-and-now’ decisions (using stochastic programming

terminology) are sufﬁciently ﬂexible to be adapted by further investment in the future when

information regarding new generation capacity becomes more certain. Under this paradigm,

ﬂexible investment options and ‘wait-and-see’ investment strategies may be valuable to avoid

the implementation of network solutions that cannot be efﬁciently adapted to the unfolding

future [31–33].

Overall, the comparison among alternative solutions when planning electricity grids will

become increasingly complex since an extremely large set of options need to be objectively

assessed before implementation, including calculation of their performance in operational time

scales. In fact, as we will also demonstrate in this work, neglecting operational aspects, as

customarily done in traditional planning tools, will provide inefﬁcient solutions in a low-carbon

smart grid context. Moreover, there will be a large volume of information that will become

available due to new monitoring technology and a large number of control variables across

voltage levels, electricity sectors/markets and energy vectors that need to be coordinated even

across borders (in the so-called whole-system approach). In this regard, time resolution will also

be critical to capture the effect of variable generation, the need for ancillary services (e.g. reserves)

and even the occurrence of complex stability phenomena that can jeopardize system security. In

addition, uncertainty will need to be recognized at various time scales in order to minimize the

risk of locking in to inefﬁcient solutions [34]. Furthermore, there are also multiple objectives,

beyond economics, that need to be balanced, including security of supply, carbon emissions and

energy policy targets that should be met in the long term. Clearly, this creates a challenging

environment for modellers who attempt to identify optimal decisions in planning.

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(c) Paper scope and contributions

On the above premises, the goals of this paper are threefold:

—- to explain the new level of complexity associated with the low-carbon, smart grid context

(especially when compared with that of conventional electricity systems), highlighting

how this affects infrastructure planning concepts and practices;

—- to propose an alternative optimization framework to plan infrastructure in smart grids,

exposing the modelling and computational challenges related to its implementation in

actual power systems; and

—- to demonstrate the importance of considering uncertainty along with operational

ﬂexibility when planning investment in innovative, smart grid technologies.

Note that there are a range of studies in the context of power system expansion planning under

uncertainty [17,31,33–64] and power system expansion planning with representation of increased

operational details, including unit commitment constraints [58,60,65–68]. Expanding on this, we

propose a uniﬁed framework that couples infrastructure investment with market clearing/unit

commitment and real-time decisions, while considering the uncertainties in both planning and

operational time scales. We argue that the proposed framework is of utmost importance in the

smart grid context, where the value of new, innovative technologies is fundamentally based on the

provision of ﬂexibility services that allow planners and operators to efﬁciently deal with various

sources of uncertainty and constraints affecting both investment decisions and system operation.

The remainder of this paper is organized as follows. Section 2 presents the proposed modelling

framework to value smart grid technologies. Section 3 describes the formulation of an instance

of the proposed framework and its application to a small, textbook-like case study. The main

notation used throughout the paper is described here. Section 4 reviews the state of the art in

power system planning under uncertainty and discusses the challenges, at the computational

and algorithmic levels, of embedding operational aspects and scaling the optimization models to

real-size power networks. Finally, section 5 concludes.

2. The modelling framework to value smart grid technologies

As argued in the previous section, the true value of smart grid technologies lies in their

ﬂexibility and thus ability to cope with uncertainty and variability,3and this will need to be

properly recognized when planning electricity system infrastructure. In this context, we propose

a modelling framework based on mathematical programming concepts, which can capture the

effects of relevant operational constraints (so as to recognize the necessary ﬂexibility levels to

deal with a number of variable conditions in operational time scales), as well as the importance

of uncertainty when planning new investment. To do so, the proposed framework considers

decision variables in two different time scales (investment and operation), and the presence of

long-term uncertainty to reﬂect the changing landscape faced by system planners, especially in

terms of available technologies, costs and market conditions, energy policy and incentives, etc.

(which may affect, for example, the future development of generation capacity that will in turn

affect network expansion). In this framework, optimal investment decisions are made before the

realization of uncertain scenarios (e.g. investment cost of a new technology), whereas operational

decisions aim to run the resulting infrastructure after each possible uncertain scenario is realized

and for a number of operating conditions (e.g. demand levels, wind power outputs, etc.). The

optimal investment decisions minimize a given metric or objective function (e.g. expected total

cost, maximum cost, maximum regret among all scenarios) and comply with a set of constraints

that ensure feasibility under an array of physical laws and market rules, as well as certain

reliability levels (e.g. system reserves requirements). The mathematical framework is shown

3Uncertainty refers to the set of alternative scenarios that may potentially occur (in the long or short term) following a known

or unknown probability distribution, while variability is certain and refers to the time-varying operating conditions that the

electricity system will face in operational time scales (i.e. combination of demand levels and renewable generation outputs).

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in equations (2.1)–(2.3), where functional fε{·} represents the chosen utility function to select

the optimal investment vector x(·), CI(·,·)andCO(·,·) represent the investment and operational

cost functions, respectively, y(·) is the matrix of operational decisions (per component and per

operating condition or time period), εrepresents a given scenario of the uncertainty set Eand X(·)

and Y(·,·) are the feasibility sets of investment and operational decisions, respectively, where the

latter depends on the optimal investment vector x(·):

min

x(·),y(·)fε{CI(x(ε), ε)+CO(y(ε), ε)}(2.1)

subject to x(ε)∈X(ε), ∀ε∈E(2.2)

and y(ε)∈Y(x(ε), ε), ∀ε∈E. (2.3)

The proposed framework is general enough to accommodate both two- and multi-stage

optimization programs, x(·) not being dependent on εin the former case. Functional fε{·} deﬁnes

whether the model corresponds to a stochastic [69,70] or robust optimization program [71]. Set

Y(·,·), which aims to capture how the resulting infrastructure is operated and its corresponding

costs, contains the physical (and market) laws associated with power system operation and may

include ﬁrst and second Kirchhoff’s laws and power ﬂow equations, generation and network

capacity constraints (including minimum and maximum limits), inter-temporal constraints

including minimum up and down times and ramping limitations, system reserves requirements,

etc. [72]. Note that operational constraints characterize the system ﬂexibility levels from existing

and prospective generation and network infrastructure. Therefore, these constraints are critical

to determine the need for ﬂexible, smart grid technologies when increasing levels of renewable

generation are integrated into the electricity system.

Furthermore, we can add another level of complexity to our framework if we consider that

system operation has to be planned ahead of real time (as well as investment has to be planned

ahead of system operation). Under this new consideration, network infrastructure has to be

also ﬂexible to deal with the forecast errors (or short-term uncertainty) related to actual system

conditions that will realize in the short-term future (e.g. next hours, next day) with respect to the

expected conditions forecasted in advance by system operators. Clearly, poor forecast methods

used, for example, to predict future wind and solar power outputs will need higher ﬂexibility

levels from the electricity system infrastructure so as to deal with unforeseen events. In this

context, new investments should be able to cope not only with various long-term uncertainties

(e.g. trends in the investment cost of a new technology that may affect the development of

generation capacity in the future), but also uncertain changes that may occur more rapidly

in real time (e.g. due to wind forecast errors). To that end, the above-mentioned framework

can be extended as shown in equations (2.4)–(2.7), where fε,ξ{·} models the objective function

being minimized, whereas y(·)andz(·,·) represent the operational decisions undertaken in the

scheduling or operations planning time scale (e.g. day ahead) and in real time, respectively, ξ

represents a given scenario of the uncertainty set Ξthat contains a number of realizations of

operational parameters such as outputs of wind and solar power generation, generation and

network outages, etc., and set Z(·,·,·,·) represents the operational decisions that can be made in

real time (e.g. actual deployment of generation reserves), which clearly are constrained by prior

decisions made during the investment time scale (vector x(·)) and operations planning time scale

(matrix y(·)):

min

x(·),y(·),z(·,·)fε,ξ{CI(x(ε), ε)+CO(y(ε), z(ε,ξ), ε,ξ)}(2.4)

subject to x(ε)∈X(ε), ∀ε∈E, (2.5)

y(ε)∈Y(x(ε), ε,ξ), ∀ε∈E,∀ξ∈Ξ(2.6)

and z(ε,ξ)∈Z(x(ε), y(ε), ε,ξ), ∀ε∈E,∀ξ∈Ξ. (2.7)

As discussed in subsequent sections, both frameworks pose important challenges at the

computational and algorithmic levels that will need to be carefully addressed in the future to

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work with scalable optimization models suitable for large-scale power networks. For illustration

purposes, the next section is devoted to the description of an instance of equations (2.1)–(2.3) and

its application to a small, textbook-like case study.

3. Eects of long-term uncertainty and short-term operational constraints on

system investment portfolios

(a) Overview

In order to illustrate the effects of relevant system operational constraints, as well as

the importance of uncertainty when planning new investment, we formulate an integrated

generation and network planning model jointly capturing both operational aspects and long-term

uncertainty. The resulting optimization program is a particular instance of equations (2.1)–

(2.3) that is subsequently applied to a three-node test system. The proposed model allows the

system operator/planner to manage (i) short-term constraints associated with the (in)ﬂexibility

of conventional plants to cope with uncertainty and fast variability of renewables’ outputs and

(ii) long-term uncertainty associated with the evolution of investment costs of new technologies.

In this context, the proposed framework provides the system operator/planner with efﬁcient

investment solutions (which we consider here as portfolios of generation, transmission, storage

and ﬂexible transmission equipment) in order to facilitate the operational management of the

system (especially under high penetration of renewables) and permit a cost-effective adaptation

of the infrastructure even under the realization of unfavourable scenarios in the long term.

Importantly, the model also recognizes the prolonged construction times of investments such as

conventional plants and long transmission lines (that create a lag between the investment decision

and commissioning time4), as well as the faster installation process of infrastructure such as wind

and solar power units, batteries and FACTS devices (that are assumed to be installed right after

investment decisions have been made).

(b) Problem formulation

Using the notation presented in table 1, we formulate our integrated generation and network

planning problem as a stochastic, mixed-integer linear program. Note that, for unit consistency,

some variables and parameters are expressed in per unit (pu), and hourly time periods are

considered in the operation.

(i) Long-term uncertainty

Long-term uncertainty is modelled through a multi-stage scenario tree (ﬁgure 1), which describes

potential evolutions of future investment costs (especially in renewable technologies such as

solar power generation), albeit further uncertainties can be included in a straightforward manner.

Hence, as formulated in equation (3.1), the model minimizes the expected value of the total cost,

which includes the costs of investments and system operation as shown in equations (3.2) and

(3.3), respectively:

min

m∈M

ρmrm(τIIm+τOOm), (3.1)

Im=

g∈ˆ

G

πIG

g¯

PG

g,m+

l∈ˆ

L

πIL

lμL

l,m+

b∈ˆ

B

πIB

b¯

PB

b,m+

l∈ˆ

LQ

πIQ

lμQ

l,m,∀m∈M(3.2)

and Om=

t∈T

g∈G

πOG

gPG

g,m,t,∀m∈M. (3.3)

4Commissioning time refers to when an asset starts its operation (e.g. when a line starts transferring power). The commissioning

time minus the decision time is equal to the construction time of an asset.

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Table 1. Nomenclature (pu =per unit).

symbol meaning unit

variables

..........................................................................................................................................................................................................

¯

Cexpected total cost across all scenarios $

..........................................................................................................................................................................................................

Cmaximum total cost across all scenarios $

..........................................................................................................................................................................................................

fl,m,tpower ow of line lin scenario tree node min time period (hour) tMW

fL

l,m,tpower ow of line lin scenario tree node min time period (hour) t(without the

contribution from the FACTS device)

MW

fQ

l,m,tpower ow contribution of the FACTS device in line lin scenario tree node min time period

(hour) t

MW

Iminvestment cost of scenario tree node m$

Omoperational cost of scenario tree node m$

¯

PB

b,mcapacity of battery storage plant bin scenario tree node mMW

PB

b,m,tpower output of battery storage plant bin scenario tree node min time period (hour) tMW

PB+

b,m,tdischarge power of battery storage plant bin scenario tree node min time period (hour) tMW

PB−

b,m,tcharge power of battery storage plant bin scenario tree node min time period (hour) tMW

PBE

b,m,tenergy stored in battery storage plant bin scenario tree node min time period (hour) tMWh

¯

PG

g,mcapacity of generator gin scenario tree node mMW

PG

g,m,tpower output of generator gin scenario tree node min time period (hour) tMW

θFrom/To

l,m,tvoltage angle in From/To node of line lin scenario tree node min time period (hour) trad

μL

l,minstallation of line lin scenario tree node m{0,1}

μQ

l,minstallation of the FACTS device in line lin scenario tree node m{0,1}

parameters

AG

g,tavailability factor of generator gin time period (hour) tpu

Dn,m,tdemand in node nin scenario tree node min time period (hour) tMW

¯

flcapacity of line lMW

¯

fQ

lmaximum capability to shift power of the FACTS device in line lMW

rmdiscount factor of scenario tree node mpu

RDW

gdownward ramp rate limit of generator gasapercentageofitscapacityperhour %h

−1

RUP

gupward ramp rate limit of generator gasapercentageofitscapacityperhour %h

−1

Xlreactance of line lΩ

ηbround-trip eciency of battery storage plant bpu

λrisk-aversion factor pu

πIB

bunitary investment cost of battery storage plant b(annuitized) $ MW−1yr−1

πIG

gunitary investment cost of generator g(annuitized) $ MW−1yr−1

πIL

linvestment cost of line l(annuitized) $ yr−1

πIQ

linvestment cost of the FACTS device in line l(annuitized) $ yr−1

πOG

gvariable cost of generator g$MWh

−1

(Continued.)

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Table 1. (Continued.)

symbol meaning unit

ρmprobability of scenario tree node mpu

τB

benergy capacity of battery storage plant bmeasured in hours at full power capacity h

τIscaling factor to transform annuitized investment cost into investment cost in an epoch pu

τOscaling factor to transform operational cost in representative days into operational cost in

an epoch

pu

Ψsuciently large positive constant MW

set-related symbols

Bset of battery storage plants

ˆ

Bset of candidate battery storage plants

Bnset of battery storage plants in node n

Fromnset of lines that start from node n

Gset of generators

ˆ

Gset of candidate generators

GCset of conventional generators

ˆ

GCset of candidate conventional generators

Gnset of generators in node n

Lset of lines

ˆ

Lset of candidate lines

ˆ

LQset of candidate FACTS devices

Mset of scenario tree nodes

MSset of the scenario tree nodes that contains one child node of each parent node in the

scenario tree

Nset of nodes

p(m) parent node of scenario tree node m

Smset of sibling nodes of scenario tree node m

SC set of uncertain scenarios

Scjset of scenario tree nodes in scenario j

Tset of time periods (hours)

Tonset of lines that go to node n

As formulated in equation (3.2), investment costs include terms related to new conventional

and renewable generation, new lines, new storage plants and new FACTS devices (shown in this

order in equation (3.2)). Equation (3.3) models the fuel costs of thermal power units across all time

periods.

As per equations (3.4) and (3.5), we assume that investments are irreversible and that some of

them feature a lag of one epoch (i.e. several years) between decision and commissioning times,

respectively. Note that equation (3.5) constrains some investment decisions to be equal among

sibling nodes, which is caused by the above-mentioned one-epoch lag. Owing to their prolonged

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2 is

parent

of 4

1 is

parent

of 2

2 and 3

are

siblings

epoch

1

epoch

2

epoch

3

time

m=2

m=4

m=5

m=1

m=6

m=7

m=3

r4

r2

r1

r5

r6

r3

r7

r

24

r

25

r

12

r

36

r13

r

37

Figure 1. Multi-stage scenario tree indicating epochs (i.e. discretization of the time horizon; equivalent to a group of years),

parent nodes, sibling nodes, transition probabilities (ρij) and node probabilities (ρm).

construction times, investment in conventional assets cannot be commissioned in epoch 1 as

imposed in equation (3.6):

¯

PG

g,m,μL

l,m,¯

PB

b,m,μQ

l,m≥¯

PG

g,p(m),μL

l,p(m),¯

PB

b,p(m),μQ

l,p(m),∀g∈ˆ

G,∀l∈ˆ

L,∀b∈ˆ

B,∀m∈{M\m=1},

(3.4)

¯

PG

g,m=¯

PG

g,k,∀g∈ˆ

GC,∀m∈MS,∀k∈Sm,

μL

l,m=μL

l,k,∀l∈ˆ

L,∀m∈MS,∀k∈Sm,⎫

⎬

⎭

(3.5)

¯

PG

g,1 =0, ∀g∈ˆ

GC,

and μL

l,1 =0, ∀l∈ˆ

L.⎫

⎬

⎭

(3.6)

(ii) System operation

The effect of the transmission network is characterized by a linearized power ﬂow model.

Equation (3.7) models the ﬁrst Kirchhoff’s law for nodal injections and withdrawals, whereby the

sum of power inputs is equal to the sum of power outputs in a node. Power ﬂows through a line

can be decomposed into a transfer contribution from the line itself and another from its FACTS

device (if installed) as shown in equation (3.8) (this is known as the power injection model of a

FACTS device [73]), and the total transfer is limited by the capacity of the line (see equation (3.9)).

Equation (3.10) shows that FACTS devices (in this case a quad-booster phase-shifting transformer

[73]) also have limited capability to shift power. In addition, equation (3.11) represents the second

Kirchhoff’s law, where a disjunctive, big-M-based model [74] is used for new lines, Ψbeing

a sufﬁciently large positive constant. For existing network infrastructure, we can assume that

μL

l,m=μQ

l,m=1 in all equations. Similarly, for those lines where there are no candidate FACTS

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devices (i.e. l/∈ˆ

LQ), we assume μQ

l,m=0 and, therefore, fQ

l,m,t=0:

g∈Gn

PG

g,m,t+

b∈Bn

PB

b,m,t−Dn,m,t=

l∈Fromn

fl,m,t−

l∈Ton

fl,m,t,∀n∈N,∀m∈M,∀t∈T, (3.7)

fl,m,t=fL

l,m,t+fQ

l,m,t,∀l∈L,∀m∈M,∀t∈T, (3.8)

−¯

flμL

l,m≤fl,m,t≤¯

flμL

l,m,∀l∈L,∀m∈M,∀t∈T, (3.9)

−¯

fQ

lμQ

l,m≤fQ

l,m,t≤¯

fQ

lμQ

l,m,∀l∈L,∀m∈M,∀t∈T, (3.10)

and −Ψ(1 −μL

l,m)+θFrom

l,m,t−θTo

l,m,t

Xl

≤fL

l,m,t≤θFrom

l,m,t−θTo

l,m,t

Xl

+Ψ(1 −μL

l,m),

∀l∈L,∀m∈M,∀t∈T. (3.11)

Generating units are also constrained by both their available power capacity (driven by the

availability of renewable resources, the maximum generation capacity for all units and outages

for conventional units) and their ramp rate limits (equations (3.12) and (3.13), respectively, where

we assume that only conventional generation features ramping-related limitations). Furthermore,

energy storage plants, which are modelled through equations (3.14)–(3.17), can provide services

associated with energy arbitrage (i.e. charge/discharge actions to improve the system load

factor, charging during off-peak periods and discharging during peak periods following the

minimization of the total cost in equation (3.1)) and ﬂexibility (since storage plants are not affected

by ramp rate constraints such as those speciﬁed in equation (3.13) for generating units):

PG

g,m,t≤¯

PG

g,mAG

g,t,∀g∈G,∀m∈M,∀t∈T, (3.12)

−RDW

g¯

PG

g,m≤PG

g,m,t−PG

g,m,t−1≤RUP

g¯

PG

g,m,∀g∈GC,∀m∈M,∀t∈T, (3.13)

−¯

PB

b,m≤PB

b,m,t≤¯

PB

b,m,∀b∈B,∀m∈M,∀t∈T, (3.14)

PB

b,m,t=PB+

b,m,t−PB−

b,m,t,∀b∈B,∀m∈M,∀t∈T, (3.15)

PBE

b,m,t=PBE

b,m,t−1−PB+

b,m,t+PB−

b,m,tηb,∀b∈B,∀m∈M,∀t∈T(3.16)

and PBE

b,m,t≤¯

PB

b,mτB

b,∀b∈B,∀m∈M,∀t∈T. (3.17)

Finally, all decision variables are continuous and non-negative, except for (i) power transfers,

voltage angles and net outputs of storage plants, which may be negative, and (ii) investment

variables associated with new network infrastructure, which are binary, indicating whether a

candidate asset has been chosen for investment (binary variable equal to 1) or not (binary variable

equalto0).

Note that further details can be included in a straightforward manner in the operational

model by adding more variables, constraints and cost functions related to units’ minimum stable

generations, start-up and shut-down costs, minimum up and down times, etc. (see, for instance,

[65,67,68,75]), albeit the current level of detail (equations (3.1)–(3.17)) sufﬁces to illustrate the

importance of operational constraints (through ramp rate and time-coupling constraints, besides

power ﬂow and capacity constraints) when planning future infrastructure. Likewise, further

details associated with the necessary security margins (e.g. generation and network reserves)

to face random faults of system components can also be critical in investment planning models

[28]. In the context of stochastic programming, these considerations will signiﬁcantly increase the

number of variables and constraints and thus the computational burden, as discussed in §4.

(iii) Risk aversion

While the previous objective function in equation (3.1) corresponds to a risk-neutral planner, the

decision-making process of a risk-averse, conservative planner can also be accommodated in the

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G1

G3 N3 S

D

N1 N2

L23

L13

L¢23

G2

W

B

QB

L12

Figure 2. Three-node system topology, where candidate assets for investment are shown in dashed lines.

proposed optimization framework. To that end, equation (3.1) is replaced with equations (3.18)–

(3.20) where a weighted average of the expected total cost and the maximum total cost across

scenarios is minimized:

min{λ¯

C+(1 −λ)C}, (3.18)

¯

C=

m∈M

ρmrm(τIIm+τOOm) (3.19)

and C≥

m∈Scj

rm(τIIm+τOOm), ∀j∈SC. (3.20)

(iv) Planning under perfect (deterministic) information

Note that we can also minimize each scenario’s total cost as shown in equation (3.21). Here, the

planner assumes perfect information of the future and thus the associated (deterministic) solution

may represent an investment option that cannot be efﬁciently adapted if a different scenario

(which was, in fact, ignored) is realized. This case can also be considered as a base, reference

case:

min ⎧

⎨

⎩

m∈Scj

rm(τIIm+τOOm)⎫

⎬

⎭

,∀j∈SC. (3.21)

(c) Illustrative example

(i) Input data

We study the electricity network depicted in ﬁgure 2 with three nodes (N1, N2 and N3),

three existing conventional generators (G1, G2 and G3), three existing lines (L12, L13 and L23)

and one load (D). In order to face the forecasted demand growth along a planning horizon

comprising three epochs spanning ﬁve years each, the system can be expanded through wind

power generation (W), solar power generation (S), a line (L’23), a FACTS device, namely a quad-

booster phase-shifting transformer (QB) [73], and a battery storage system (B). The input data

used to run our stochastic program are shown in table 2, where base values for power and voltage

are 100 MVA and 66kV, respectively.

For illustration purposes, uncertainty is solely associated with the future evolution of

investment costs of solar power generation. The problem’s scenario tree is illustrated in ﬁgure 3,

which describes both the transition probabilities between nodes and PV investment costs in each

node (relative to that listed in table 2). Note that investment costs of solar power generation might

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Table 2. Relevant input data.

investment cost

wind power generation ($ MW−1yr−1) 150 000

solar power generation ($ MW−1yr−1) 250 000

battery ($ MW−1yr−1) 130 000a

QB ($ yr−1) 8000

L’23 ($ yr−1) 10 000

operational cost

G1 ($ MWh−1)50

G2 ($ MWh−1)30

G3 ($ MWh−1)100

demand

yearly growth rate (%) 1

peak (MW) 100

capacity

G1 (MW) 60

G2 (MW) 60

G3 (MW) 100

L12 (MW) ∞

L13 (MW) ∞

L23 (MW) 57

L’23 (MW ) 57

QB (MW) 9

storage

eciency (%) 90

energy capacity (in hours at maximum power output) (h) 2

further data

yearly discount rate (%) 5

lines’ reactance (pu) 0.01

G1 ramp rate (% h−1)10

G2 ramp rate (% h−1)30

aInvestment cost in the rst epoch. Weassume that the investment cost of the battery storage system decreases down to 50% and 20% of its

original value in epochs 2 and 3, respectively.

fall signiﬁcantly in the future in scenario 1. Under such a scenario, solar power plants might be

installed at the consumers’ location, leaving conventional generation and network assets stranded

(at least partially). On the other hand, if future investment costs remain at today’s levels, as

modelled in scenario 4, wind power generation (which is assumed to be signiﬁcantly less costly

at present) is likely to be installed. Given that the location of candidate wind power generation

is far from the load centre, investment in this technology would also require further transmission

investment that needs to be planned signiﬁcantly ahead of actual commissioning. In fact, we

assume that while the transmission line features a lag of 1 epoch (e.g. 5 years), the remaining

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m=4

m=2

m=5

m=6

m=3

m=7

epoch

1

epoch

2

epoch

3time

m=1

100%

scenario 4: high investment

cost of solar power generation

scenario 3: mid-high investmen

t

cost of solar power generation

scenario 2: mid-low investment

cost of solar power generation

scenario 1: low investment

cost of solar power generation

90%

90%

80%

50%

50%

40%

0.5

0.5

0.5

0.5

0.5

0.5

Figure 3. Three-stage scenario tree indicating investment cost evolutionof solar power generation (as a percentage of the cost

listed in table 2) and transition probabilities. We dene set MS={2,4,6}, which contains one child node per parent node.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

power (pu)

time (h)

demand

wind power

solar power

Figure 4. Demand, wind and solar power proles (per unit, pu) in the representative day.

assets can be commissioned faster (i.e. with negligible lag). Further to the need for dealing with

investment uncertainty, the planner may also face important challenges at the operational time

scale when managing increased amounts of renewable generation. In this outlook, demand, wind

and solar power proﬁles, as shown in ﬁgure 4, are used to clearly illustrate the need for ramping,

which will be used here as an example of an operational constraint to be considered in planning.

Owing to the illustrative purpose of this example, days are assumed equal across a year and thus

weekly and seasonal variations are ignored. While this simpliﬁcation does not affect our general

conclusions, further details like those neglected in this example may need to be captured for real

applications.

The proposed model has been implemented using FICO®Xpress [76]. For the sake of

reproducibility, the code and associated data ﬁles can be downloaded from [77].

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Table 3. Results from the deterministic model: expansion plan per epoch and scenario, and costs per scenario (W, S, L’23, B

and QB mean investments in wind power generation, solar power generation, line, battery and quad-booster transformer,

respectively, and the additional capacity in MW is indicated within ).

scenario 1 low scenario 2 mid-low scenario 3 mid-high scenario 4 high

expansion plan per epoch and scenario

epoch 1 W16W16W19W19

epoch 2 S39S39W7, L’23 W7, L’23

epoch 3 S18S5,W2W1W1

costs per scenario (k$)

investment 59 896 60 047 38 665 38 665

operation 240 974 244 321 270 624 270 624

total 300 870 304 368 309 289 309 289

(ii) Results

Eects of long-term uncertainty on system planning

Table 3 shows the results per epoch and scenario attained by the deterministic model (equation

(3.21)). While in scenarios 1 and 2 the signiﬁcant drop in investment cost of solar power generation

encourages the adoption of this technology in the future (producing power right at the point of

consumption), the still relatively high investment cost of solar power generation in later epochs

of scenarios 3 and 4 encourages investment in wind power, far from the load centre. Expectedly,

conventional network reinforcement in the form of a new transmission line is needed in scenarios

3 and 4 (unlike in scenarios 1 and 2) in order to transfer power from remote areas to the load

centre. In the ﬁrst epoch when the investment cost of solar power generation remains relatively

high, wind power plants are built in all scenarios.

Table 4 presents the results per epoch and scenario determined by the risk-neutral stochastic

model. In contrast to the above deterministic case, this table demonstrates that in no scenario is

there any conventional network investment needed to cope with the uncertainty levels faced by

the planner in the ﬁrst epoch. Indeed, the model determines that epoch 1 is too early to decide

whether to invest in a long power line (commissioned at the beginning of epoch 2) that may

end up stranded under the realization of scenarios 1 and 2 where solar power emerges right at

the location of consumption. Rather, the stochastic approach chooses to ‘wait and see’ and thus

potential network congestions (under scenarios 3 and 4) are managed in the operational time scale

through smart technologies such as FACTS/QB devices that can be chosen and quickly installed

right in epoch 2. This solution is complemented later on with a battery storage plant to face larger

amounts of wind power that materialize towards the end of the analysed planning horizon, also

considering the drop in cost of batteries in the meantime.

It is important to point out that table 3 shows the costs associated with the best expansion plan

for each scenario considered by the planner to characterize the uncertain future. In contrast, the

solution summarized in table 4 corresponds to the investment plan performing best on average

for the scenario set under consideration. In other words, the costs reported in table 3 will only

be incurred if the corresponding scenario actually materializes. Otherwise, the costs associated

with such expansion plans may signiﬁcantly differ should other scenarios occur. Moreover, the

expected total costs for the deterministic solutions summarized in table 3 are greater than or equal

to that of the optimal stochastic solution. Table 5 presents the total costs per scenario and the

expected total costs for the deterministic solutions reported in table 3 and the stochastic solution

given in table 4. As can be observed, the stochastic solution is identical to the best expansion plans

for scenarios 1 and 2, whereas, for scenarios 3 and 4, a slight 0.03% cost increase is incurred over

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Tabl e 4. Results from the risk-neutral stochastic model: expansion plan per epoch and scenario,costs per scenario and expected

costs (W, S, L’23, B and QB mean investments in wind power generation, solar powergeneration, line, battery and quad-booster

transformer, respectively, and the additional capacity in MW is indicated within ). Note that the results follow the scenario

tree structure depicted in gure 3.

scenario 1 low scenario 2 mid-low scenario 3 mid-high scenario 4 high

expansion plan per epoch and scenario

epoch 1 W16

epoch 2 S39W10,QB

epoch 3 S18S5,W2B1B1

costs per scenario (k$)

investment 59 896 60 047 35 853 35 853

operation 240 974 244 321 273 517 273 517

total 300 870 304 368 309 370 309 370

expected costs (k$)

investment 47 912

operation 258 082

total 305 994

Table 5. Total costs per scenario and expec ted total costs for the deterministic solutions and the stochastic solution.

deterministic

solution for

scenario 1

deterministic

solution for

scenario 2

deterministic

solution for

scenario 3

deterministic

solution for

scenario 4

stochastic

solution

total cost per scenar io (k$)

scenario 1 low 300 870 301 287 309 289 309 289 300 870

scenario 2 mid-low 304 853 304 368 309 289 309 289 304 368

scenario 3 mid-high 330 777 327 587 309 289 309 289 309 370

scenario 4 high 334 760 330 668 309 289 309 289 309 370

expected total cost (k$) 317 815 315 977 309 289 309 289 305 994

the best expansion plans for such scenarios. On the other hand, cost increases as high as 8.2%

can be experienced if the planner adopts a deterministic solution and a different from forecasted

scenario is realized. Moreover, table 5 also shows that the stochastic solution outperforms all

deterministic solutions in terms of the expected total cost, with considerable improvement factors

ranging between 3.7% and 1.1%.

Interestingly, although the stochastic model constrains the ﬁnal volume of wind capacity

connected in scenarios 3 and 4, it is possible to demonstrate that, overall, this is better than

building a line that ends up stranded in scenarios 1 and 2. In fact, a sensitivity analysis where we

force the stochastic model to commission a new transmission line in epoch 2 (shown in table 6)

reveals that costs under scenarios 1 and 2 escalate with respect to those observed under the truly

stochastic solution and this cannot be compensated for by the cost reduction under scenarios 3

and 4. Furthermore, the early investment in a transmission line is not only economically inefﬁcient

but also prevents the utilization of smart grid technologies (FACTS/QB and battery plant) that

have been displaced by the line.

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Table 6. Results from the risk-neutral stochastic model with forced commissioning of conventional network infrastructure:

expansion plan per epoch and scenario, costs per scenario and expected costs (W, S, L’23, B and QB mean investments in wind

power generation, solar powergeneration, line, battery and quad-booster transformer, respectively,and the additional capacity

in MW is indicated within ). Note that the results follow the scenario tree struc ture depicted in gure 3.

scenario 1 low scenario 2 mid-low scenario 3 mid-high scenario 4 high

expansion plan per epoch and scenario

epoch 1 W16

epoch 2 S39, L’23 W10, L’23

epoch 3 S18S5,W2W1W1

costs per scenario (k$)

investment 59 959 60 110 36 487 36 487

operation 240 975 244 321 272 859 272 859

total 300 934 304 431 309 346 309 346

expected costs (k$)

investment 48 261

operation 257 753

total 306 014

Note that the stochastic model builds minimum amounts of wind power infrastructure in

epoch 1 so as to ‘wait and see’ whether the investment cost of solar power decreases; if it

does, then solar power investment is preferred (scenarios 1 and 2), more wind power being

commissioned otherwise. Table 7, where the maximum amount of wind power has been forcedly

commissioned in epoch 1, shows that installing higher levels of wind power generation in

epoch 1 (as the deterministic model suggests) will limit the ability to integrate solar power

in the future when investment costs fall, thereby decreasing the overall cost efﬁciency of the

stochastic solution.

It is important to point out that the forced commissioning of network and generation assets

yields solutions with expected total costs that are slightly higher than that incurred by the optimal

stochastic solution and well below those featured by the deterministic solutions. Thus, tables 4,

6and 7show the inefﬁciencies (advantages) of implementing deterministic (stochastic) solutions

when planning under uncertainty. Note that, in this illustrative example (which demonstrates

the need to make investment plans more ﬂexible through smart grid technologies and thus face

uncertain scenarios more efﬁciently), uncertainty is associated with investment costs of solar

power generation only. Hence, the consequences of following deterministic planning can be much

more signiﬁcant in real systems subject to additional sources of uncertainty such as energy policy

incentives, development of new technologies, demand growth, various decentralized decisions

from market participants, etc.

Finally, risk aversion is analysed through the minimization of a weighted average of the

expected and the maximum total costs across scenarios (table 8), with parameter λin equation

(3.18) taken as equal to 0.5 for the low-aversion case, and equal to zero in the high-aversion

case (corresponding to the min–max cost optimization). As compared with the risk-neutral

solution shown in table 4, the optimal risk-averse investment plans feature larger volumes of

wind power generation in epoch 1. Although both expansion plans are less efﬁcient solutions in

terms of the expected total cost, increasing the levels of wind power generation commissioned in

epoch 1 is an effective measure to reduce the costs in the more costly scenarios, i.e. scenarios 3

and 4. Interestingly, for low levels of risk aversion, the planner prefers to install larger amounts

of wind power generation capacity in epoch 1 rather than investing in conventional network

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Table 7. Results from the risk-neutral stochastic model with forced commissioning of higher levels of wind power generation

in the rst epoch: expansion plan per epoch and scenario, costs per scenario and expected costs (W, S, L’23, B and QB mean

investments in wind power generation, solar power generation, line, battery and quad-booster transformer, respectively, and

the additional capacity in MW is indicated within ). Note that the results followthe scenario tree structure depicted in gure 3.

scenario 1 low scenario 2 mid-low scenario 3 mid-high scenario 4 high

expansion plan per epoch and scenario

epoch 1 W19

epoch 2 S38W7,QB

epoch 3 S19S5B1B1

costs per scenario (k$)

investment 64 533 63 856 38 030 38 030

operation 236 500 240 593 271 283 271 283

total 301 033 304 449 309 313 309 313

expected costs (k$)

investment 51 113

operation 254 914

total 306 027

infrastructure that can allow the planner to install further wind power generation capacity

towards the last epoch. This is so because conventional network investment would unnecessarily

increase the total costs in scenarios 1 and 2 (and thus the expected total cost). For higher levels of

risk aversion and, in the extreme case, for the fully averse min–max cost solution, conventional

network investment is necessary to minimize the total costs under the worst conditions, which

also eliminates the need for smart grid solutions. It is important to mention that this extreme

solution focuses on total cost minimization under the worst conditions (in this case, scenarios 3

and 4) irrespective of the inefﬁciencies caused in terms of the expected total cost and, in particular,

under those scenarios where solar power generation is realized in the future (when the new

transmission line becomes unnecessary and stranded).

Eects of short-term operational constraints on system planning

In the previous examples, the results discussed in tables 3–8account for ramp rate constraints in

all scenarios, as indicated in table 2.Table 9 shows the resulting infrastructure investment and the

total costs for the solutions to the following three different deterministic cases for scenario 1, which

corresponds to a low investment cost of solar power generation.

(i) When the infrastructure is optimized by ignoring ramp rate constraints: although the

total cost is the lowest, it underestimates the actual cost of operating large volumes of

renewables.

(ii) When the infrastructure is optimized by ignoring ramp rate constraints and the actual

cost of operation is adequately quantiﬁed in a second stage when considering actual

ramp rate limitations: the total cost is the highest since the planner ignores important

operational constraints (when the system infrastructure was determined).

(iii) When the infrastructure is adequately planned and optimized by accounting for ramp

rate constraints (G1 and G2’s ramp rates equal to 10% h−1): this corresponds to the true

cost of the optimal solution.

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Tabl e 8. Results from therisk-aversemodel: expansion planper epoch andscenario,costs per scenarioand expected costs (W,S,

L’23, Band QB meaninvestmentsin wind powergeneration,solar power generation,line, batteryand quad-booster transformer,

respectively, and the additional capacity in MW is indicated within ). Note that the results follow the scenario tree structure

depicted in gure 3.

scenario 1 low scenario 2 mid-low scenario 3 mid-high scenario 4 high

risk-averse solution 1 (low aversion)

expansion plan per epoch and scenario

epoch 1 W19

epoch 2 S38W7,QB

epoch 3 S19S6B1B1

costs per scenario (k$)

investment 63 485 62 888 37 603 37 603

operation 237 501 241 525 271 721 271 721

total 300 986 304 413 309 324 309 324

expected costs (k$)

investment 50 395

operation 255 617

total 306 012

risk-averse solution 2 (high aversion, min–max cost)

expansion plan per epoch and scenario

epoch 1 W19

epoch 2 S38, L’23 W7, L’23

epoch 3 S19S5W1W1

costs per scenario (k$)

investment 64 597 63 919 38 665 38 665

operation 236 499 240 593 270 624 270 624

total 301 096 304 512 309 289 309 289

expected costs (k$)

investment 51 461

operation 254 586

total 306 047

The results shown in table 9 demonstrate that modelling detailed operational constraints can

be critical when planning electrical infrastructure. For example, if this electricity system were

planned with fully ﬂexible ramp rates from conventional plants (in other words, ignoring ramp

rate constraints as in solution (i), which is a widespread implicit assumption in most planning

exercises and available planning tools), larger volumes of renewables would be installed with

no need for storage plants. This would clearly increase the cost of operation in reality, when

considering the actual system’s ﬂexibility limitations (see solution (ii)), and would probably result

in investment into suboptimal technologies. In contrast, if ramp rates are adequately considered

in the planning model as in solution (iii), then less renewable energy capacity is built because of

the diminished ability of the system to cope with intermittency. Further, storage capacity becomes

necessary to provide operational ﬂexibility.

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Table 9. Results from the deterministic model under three dierent assumptions on system’s exibility for scenario 1

corresponding to a low investmentcost of solar power generation: expansion plan per epoch and solution, and costs per solution

(W, S, L’23, B and QB mean investments in wind power generation, solar power generation, line, battery and quad-booster

transformer,respectively,and the additional capacityin MW isindicatedwithin ). Roman numeralsof the solutions correspond

to the cases described in the main text.

solution (i) solution (ii) solution (iii)

expansion plan per epoch and solution

epoch 1 W16W16W14

epoch 2 S39S39S40

epoch 3 S18S18S11,B5

costs per solution (k$)

investment 59 896 59 896 55 507

operation 240 974 242 437 246 467

total 300 870 302 333 301 974

Tabl e 10. Results from the risk-neutral stochastic model without ramp rate constraints on G1 and G2: expansion plan per epoch

and scenario, costs per scenario and expected costs (W, S, L’23, B and QB mean investments in wind power generation, solar

power generation, line, battery and quad-booster transformer, respectively, the additional capacity in MW is indicated within

and N/I refers to ‘no investment’). Note that the results follow the scenario tree structure depicted in gure 3.

scenario 1 low scenario 2 mid-low scenario 3 mid-high scenario 4 high

expansion plan per epoch and scenario

epoch 1 W16

epoch 2 S39W10,QB

epoch 3 S18S5,W2N/I N/I

costs per scenario (k$)

investment 59 896 60 047 35 827 35 827

operation 240 974 244 321 273 525 273 525

total 300 870 304 368 309 352 309 352

expected costs (k$)

investment 47 899

operation 258 087

total 305 986

Furthermore, table 10 shows that if the system is planned through the risk-neutral stochastic

model while ignoring ramp rate constraints, storage infrastructure (originally installed in scenarios

3 and 4 as shown in table 4) is not needed. This illustrates the importance of considering both the

system’s ﬂexibility limitations (table 9) and uncertainty (table 10) in order to properly capture the

value of smart grid technologies and invest in the most appropriate portfolio in the ﬁrst place.

Discussion

As mentioned earlier, it is important to emphasize that there are a range of studies in the context of

power system expansion planning under uncertainty (see, for instance, [17,31,33–64]) and power

system expansion planning with representation of increased operational details, including unit

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commitment constraints (see, for instance, [58,60,65–68]). Expanding on this, we argue that the

proposed uniﬁed framework evaluated above is of utmost importance in the smart grid context,

where the value of new, innovative technologies is fundamentally based on the provision of

ﬂexibility services that allow planners and operators to efﬁciently deal with various sources of

uncertainty and constraints affecting both investment decisions and system operation.

Furthermore, notice that the above results correspond to an instance of equations (2.1)–(2.3),

where operational uncertainty has been neglected and, instead, replaced with the modelling of

variability in operational parameters such as wind and solar power outputs. Another level of

complexity will be to run an instance of equations (2.4)–(2.7) where uncertainty is considered

in both planning and operational time scales. Under this consideration, generation and network

infrastructure have to be also ﬂexible to deal with equipment outages and forecast errors that

may happen in real time, and thus new investments should be able to cope with both long-term

uncertainties and also uncertain changes that may occur rapidly in real time. There are various

studies that demonstrate the importance of short-term uncertainty and security for power system

planning (see, for instance, [28]), and we believe that, with the advent of new technology, such

importance will be exacerbated since innovative technologies can successfully provide (at least

partially) the ﬂexibility and security needed in real time and hence displace (or at least defer)

part of the new conventional infrastructure. Therefore, modelling both long- and short-term

uncertainty will be key to properly quantify the beneﬁts of smart grid technologies.

4. Modelling and computational challenges

Following up on our multi-stage stochastic programming modelling framework presented above

and demonstrated on a small test system, and with reference to the state of the art in power

system planning under uncertainty, in this section we aim: (i) to present in more general

terms the main families of problems dealing with power system planning under uncertainty

(of which our framework is a subset); (ii) to discuss the main challenges, at the computational

and algorithmic levels, of embedding detailed operational aspects in planning under uncertainty

problems, especially when dealing with real-size networks; and (iii) to advocate the need for new

optimization tools that could address these computational challenges in order to properly value

the beneﬁts of ﬂexible, smart grid solutions in planning.

Current models for power system planning under uncertainty lie within one of the four

classes of problems resulting from the combination of: (i) the temporal framework adopted for

decisions with uncertainty about future evolution, namely, dynamic (or multi-stage), whereby in

each period decisions are made under uncertainty of successive ones, versus static (or single-stage),

whereby decisions are made in a single period under the same information level for all future

periods [48]; and (ii) the modelling of uncertainty characterization,namely,stochastic programming,

which uses probabilistic models to characterize uncertainty [70], versus robust optimization,which

ensures feasibility for a user-deﬁned set of uncertainty realizations or scenarios and is particularly

useful when the deﬁnition of a probability distribution is not an easy task, e.g. occurrence of

contingencies, strategic actions of market participants and evolution of fuel prices [71].

(a) Multi-stage stochastic planning

The class of multi-stage stochastic optimization models [70] is of great importance in power

system planning and encompasses many applications in many subareas (e.g. hydrothermal

planning [35], transmission expansion planning [42,49] and generation expansion planning

[41,50,51]) and aims to emulate the way decisions are made in the real world while uncertainty

is revealed over time. For computational tractability, this logic is applied only to investment

decisions and long-term uncertainties, while short-term operational uncertainties and decisions

are generally simpliﬁed or even disregarded, on the basis of the (traditionally) weak time

dependence between short-term and investment-related decisions. However, as we argued

through our multi-stage optimization textbook-like examples, it is key that operational aspects

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that come with smart grid technologies and ﬂexible devices are suitably taken into account in

planning too.

The requirement for more detailed modelling of operational ﬂexibility introduces manifold

computational and methodological challenges in multi-stage stochastic planning problems, which

may be intractable for real-life benchmarks. Decomposition techniques can, therefore, be most

useful. In this outlook, a natural decomposition scheme arises from the temporal and conditional

nature of the uncertainty revelation structure, whereby the problem can be cast as a discrete-time

control problem that follows Bellman’s principle, thus being suitable for dynamic programming.

Benders-type decomposition procedures and dynamic programming [70], as well as sampling

techniques such as Monte Carlo simulation, are well-known approaches used to ﬁnd high-quality

solutions [35,78,79] due to their scalability and computational efﬁciency to solve realistic problems

from industry. For instance, the stochastic dual dynamic programming (SDDP) approach [35]is

used in most hydrothermal-based power systems worldwide. However, the discrete nature of

state variables in planning problems introduces loss of convexity that prevents the direct use of

the SDDP approach. This is also the case when considering nonlinear relations or discrete actions

at the market or operational levels, such as those arising when modelling network ﬂexibility. In

this sense, recent advances in decomposition methods provide algorithms capable of dealing with

some of these non-convexities in different ways [80–82]. In general, developing computationally

efﬁcient methods to address non-convexities is an important research avenue to be able to fully

capture the beneﬁts of smart grid technologies in realistic planning problems.

(b) Static stochastic and robust planning

The class of static stochastic and robust planning models arises when, roughly speaking, all

the investment decisions x(·) in the model comprising equations (2.4)–(2.7) are in the ﬁrst

stage, and the uncertainty realizations are in the second stage (two-stage stochastic or two-

stage robust problems). Static planning is largely used in industry applications, particularly for

grid reinforcements, whereby the decision and uncertainty hierarchy is typically the following:

‘investment decisions for the whole time horizon (x)’ ‘long-term uncertainty revelation (ε)’

‘unit commitment or market scheduling decisions (y(ε))’ ‘short-term uncertainty revelation

(ξ)’ ‘short-term operational decisions (z(ε,ξ))’. Depending on the application, one of the three

decision levels is simpliﬁed and, in general, two-stage decision models are used to address static

problems. Such two-stage models use a simpliﬁed (or ‘myopic’) and in general more conservative

approach for investment dynamics. Thus, much more detailed operational aspects and short-

term uncertainties can be considered relative to the multi-stage approach (which uses simpliﬁed

operational models to avoid tractability issues). As a result, the two-stage framework is largely

adopted by industry. However, two-stage models require the consideration of many epochs

and snapshots of the system operation. Therefore, again, Benders decomposition [80], Dantzig–

Wolfe [41], progressive hedging [54] and column-and-constraint generation methods [17,83]are

common approaches used to decompose the original problems into two stages: the planning

stage, where the set of decision variables comprises variables x(·), and the operational stage,

where decision variables y(·) are scenario-dependent.

While in the stochastic framework, with a probabilistic model being used to characterize

uncertainty, the decision-maker risk attitude is expressed through utility functions and risk

measures [64], in robust optimization the decision-maker imposes feasibility for all scenarios

within a given set, denoted by uncertainty set, thereby giving rise to a worst-case setting. In

this case, the risk level (or ‘conservativeness’ level, as commonly referred to in the literature on

robust optimization [71,84,85]) is accounted for by means of the comprehensiveness (or ‘broadness’,

in the sense of cardinality) of the set of scenarios. Available approaches to uncertainty modelling

(e.g. ellipsoidal uncertainty sets [85,86] and polyhedral uncertainty sets [84]) deﬁne convex sets

relying on nominal values for the uncertain coefﬁcients of linear inequality constraints and on

distance norms limiting the size of the uncertainty sets. Therefore, differently from the stochastic

optimization framework, where the explicit evaluation of the short-term operational cost for all

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the scenarios is needed, in the robust optimization framework the worst-case scenario is found

through parametrized optimization problems. Hence, robust optimization implicitly accounts

for all the scenarios within the uncertainty set through efﬁcient search algorithms, thereby

addressing some of the tractability issues associated with the consideration of many scenarios

in the stochastic optimization approach. This is a salient virtue of the robust approach that

has recently triggered an increasing number of publications in robust models for power system

operation and planning (e.g. [17,39,46,48,53,62,63,87,88]). This virtue notwithstanding, there are

still several challenges that should be addressed in the ﬁeld of robust optimization models for

power system planning. In particular, two relevant avenues of research are devoted to (i) the

characterization of uncertainty sets that preserve relevant statistical properties of the uncertainties

and (ii) addressing the tractability issues associated with the presence of non-convexities in

the second-stage problem. Recently, promising results propose new perspectives for merging

stochastic and robust optimization, thereby giving rise to the notion of distributionally robust

optimization [71,89].

5. Conclusion

In the context of the transition from conventional power systems to low-carbon energy systems,

whereby a control-based paradigm relying on ﬂexible smart grid technologies replaces the

classic asset-based paradigm, this paper has discussed some of the main challenges associated

with planning future energy systems to securely and cost-effectively integrate renewable energy

sources. In particular, we have presented an optimization framework to plan electricity grids that

deals with uncertainty and properly accounts for relevant operational constraints, so that truly

optimal solutions, based on ﬂexible technologies, can be determined (although this is limited, in

practice, by computational burden and the ability to deal with very large optimization problems

through suitable algorithms). In fact, the resulting investment options contain an array of smart

grid technologies that can effectively provide the needed ﬂexibility in operation to cope with

renewables’ ﬂuctuations and the necessary adaptation capacity in the long term to adjust system

expansion more cost-effectively even under the occurrence of unfavourable scenarios. We thus

demonstrated that and how smart grid technologies can displace the need for conventional

network assets that are instead preferable when uncertainty is ignored and operational constraints

are neglected.

Two key conceptual results of our work, with important policy implications, are (i) investment

in speciﬁc low-carbon generation technologies over time may strongly depend on the

(deterministic, stochastic or robust) approach and risk aversion of the planner, and (ii) optimal

generation investment decisions need to be complemented by decisions on investment in smart

grid technologies and/or network assets. In these regards, the proposed model allows one to

fully capture synergies and competition among different generation, smart and transmission

technologies so that the truly optimal solution is attained. In contrast, the deterministic planning

approaches (often in the form of ‘roadmaps’) and relevant planning tools that are currently used

are bound to deliver solutions that are not only more expensive, but also misleading in terms

of optimal technology mix (generation, storage, etc.). This is particularly important in the light

of, for example, setting policy incentives for certain technologies, for which enabling smart grid

technologies such as FACTS and batteries should also be considered to be optimal complements

to renewable generation.

Following up on the ﬁndings highlighted above, we then also presented the state of the art in

power system planning under uncertainty and discussed the need for new optimization tools

that can properly value the beneﬁts of ﬂexible, smart grid solutions by embedding detailed

operational aspects in the planning problem. In particular, important progress is necessary in

terms of sampling techniques, decomposition methods, parallel and cloud computing, among

others, to fully unlock the application of mathematical programming to plan investments in

electricity grids when considering a large number of potential future scenarios in the long term,

and a detailed representation of system operation in the short term.

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Data accessibility. The code and associated data ﬁles can be downloaded from [77].

Authors’ contributions. R.M. designed and conducted the numerical studies. The manuscript was drafted by R.M.

and A.S. All authors edited and approved the manuscript.

Competing interests. We declare we have no competing interests.

Funding. R.M. gratefully acknowledges the ﬁnancial support of Conicyt-Chile (through grants Fondecyt/

Iniciacion/11130612, Newton-Picarte/MR/N026721/1, Fondef/ID15I10592, SERC Fondap/15110019 and the

Complex Engineering Systems Institute (CONICYT–PIA–FB0816; ICM P-05-004-F)). A.S. acknowledges the

National Council for Research and Development (CNPq), Brazil. J.M.A. acknowledges the ﬁnancial support of

the Ministry of Economy and Competitiveness of Spain under Project ENE2015-63879-R (MINECO/FEDER,

UE) and the Junta de Comunidades de Castilla-La Mancha under Project POII-2014-012-P. P.M. acknowledges

the partial support of the UK EPSRC through the ‘MY-STORE’ research project (EP/N001974/1).

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