# A Prosumer-Centric Framework for Concurrent Generation and Transmission
Planning — Part II

Ni Wang, Remco Verzijlbergh, Petra Heijnen, and Paulien Herder N. Wang, R.
Verzijlbergh and P. Heijnen is with the Faculty of Technology, Policy and
Management, Delft University of Technology, Delft, the Netherlands (e-mails:
{n.wang, r.a.verzijlbergh, p.w.heijnen}@tudelft.nl). P. Herder is with the
Faculty of Applied Sciences, Delft University of Technology, Delft, the
Netherlands (e-mail: p.m.herder@tudelft.nl). This research received funding
from the Netherlands Organisation for Scientific Research (NWO) [project
number: 647.002.007].

###### Abstract

We propose a framework where generation and transmission capacities are
planned concurrently in market environments with a focus on the prosumers.
This paper is a continuation of Part I and presents numerical results from
three archetypal case studies. Following the proposed framework, optimal
planning decisions are shown in all the cases. Furthermore, in case study I,
no-regret planning decisions considering the uncertainties in future
electricity market designs are discussed. In case study II, we look at a
situation where the social resistance of wind energy prevails and the
prosumers choose to not invest in wind energy. This preference results in an
increased system cost, and thus may harm other prosumers. The framework is
used as a negotiation simulator to deal with this problem. Case study III
presents numerical results for a mixed bilateral/pool market. The case studies
utilize realistic data from the Dutch power system and the European power
system to provide policy-relevant results that aid their decarbonization in
various market environments.

###### Index Terms:

Generation and transmission planning, Peer-to-peer electricity markets, Pool
electricity markets, Mixed bilateral/pool electricity markets, Distributed
optimization.

##  I Introduction

This paper is a continuation of its Part I. There, we provide a prosumer-
centric approach for concurrent generation and transmission planning, where 1)
the costs and benefits for the prosumers along the planning horizon are
explicitly modeled, 2) the investment preferences of the prosumers are
included, 3) distributed algorithms are provided to solve the planning problem
in a decentralized manner. The formulations of the investment problems
integrating three different electricity markets, i.e. the P2P market, the pool
market and the mixed bilateral/pool market, respectively, are given.

We position this work as a general modeling framework to systematically assess
the influences of various electricity market designs on the planning
decisions. Therefore, the basic usage of our framework is to investigate the
optimal investment decisions and the associated costs and benefits given the
different market environments. By this, the framework is able to provide
recommendations on issues that pertain to social, political and policy
aspects. To further illustrate its applicability, in this paper, three
archetypal case studies will be presented. These case studies focus on
different aspects of the framework and will be done in various realistic
contexts. Now they will be briefly introduced and more details will be given
in the corresponding sections.

Case study I discusses the no-regret planning decisions taking into account
the uncertainty in electricity market designs of the future. Case study II
focuses on the effects of the investment preferences of the prosumers. There,
the framework is used as a simulator for the negotiation that is incurred by
the conflicting interests of the prosumers. Case study III presents numerical
results for a mixed bilateral/pool market. The first two archetypal case
studies are based on the national RES planning process in the Netherlands for
2030, while the last one takes a European perspective and models a highly
renewable Europe in 2050.

In Section II \- IV, the three case studies are described and the numerical
results are presented, respectively. Section V concludes the paper.

##  II Case Study I

To combat climate change, the Dutch government aims to reduce the Netherlands’
greenhouse gas emissions by 49% by 2030 compared to 1990 levels. In the
electricity sector, a major focus is to increase the share of wind and solar
energy, i.e. from 14.25 TWh (12% of the total production) in 2018 to 84 TWh
(72% of the total production) in 2030. Accordingly, massive investments in
wind and solar energy are needed. The investments are facilitated through a
national program Regionale Energiestrategie (Regional Energy Strategies) [1].
In this program, the country is divided into 30 energy regions, where each
region proposes its own investments in onshore wind and solar energy and
meanwhile, the TSO proposes the transmission network expansion plan.

This problem is dealt with using our framework. The generation and
transmission planning problem is considered as optimization problems, where
each agent minimizes their own cost concurrently. The cost for a region
includes not only the costs that are considered in energy system planning
models (i.e. CapEx, FOM and VOM), but also the expected trading costs in the
electricity markets. In this case study, the influences of two market designs
on the planning decisions will be investigated to account for the uncertainty
in future market designs. In addition to a pool market, a P2P market as a
promising next-generation electricity market will be discussed as the
alternative future market design.

We start by describing the model set-up in Section II-A. Then, the optimal
investment decisions and the relevant costs under different markets are
presented in Section II-B. At last, in Section II-C, how the uncertainty is
taken into account is described and the no-regret investment decisions are
shown.

###  II-A Model set-up

The Netherlands is divided into 30 nodes which are connected by transmission
lines. The considered time horizon is one year with hourly resolution. The
input data includes, among others, the spatio-temporal variations of wind and
solar capacity factors and their maximum potentials which is obtained from
[2]. The considered technologies are onshore wind turbines, solar PV, OCGT,
hydrogen storage and flow battery storage. Due to the differences in their
efficiencies and cost parameters, hydrogen storage is best suited for long-
term usage, whereas battery storage provides an option for short-term storage.
Furthermore, in this national planning process, the regions are not asked to
propose capacity in offshore wind and thus offshore wind is not considered.
Since quite a significant generation expansion is needed, for the ease of
presenting results, the existing generation and storage capacities are not
considered. The existing transmission network is considered and the capacity
can be expanded. Direct current (DC) power flow calculations are performed
based on the existing capacities. 2030 estimations for the techno-economic
parameters are used and are taken from [3] except for the network cost
parameter which is taken from [2].

Regarding the product differentiation term in P2P markets, two points of view
may be used [4]. The first one is the system view. The product differentiation
can be regarded as an externality cost such as transaction cost (TC) or
exogenous grid cost. The second view is the prosumer view. It represents the
willingness to pay of the prosumers in relation to the trades. In this case
study, we employ the first one, it will be used to represent the TC in the P2P
markets. Later, in the second case study, the perspective of the latter will
be investigated.

###  II-B Optimal investment decisions under different market designs

We first look at the optimal investment decisions in terms of the total
installed capacities for the system. Here, two different markets are
considered, i.e. the pool market and the P2P market. We start with the pool
market. Then, the P2P markets are analyzed with increasing TCs, starting from
10 % of the average electricity price to 30 %, 50 %, 70 % and 90 % to
investigate the influences of TC on the planning decisions.

Figure 1: Optimal total capacities for the Netherlands under pool market and
P2P market with different TC (% of the average electricity price).

Figure 2: Optimal capacities over the Netherlands. Left to right: pool market,
P2P market with 10 % TC.

From left to right in Figure 1, the following general trends can be derived.
First of all, the capacity of onshore wind declines while the size of solar
and storage capacity increases. Secondly, hydrogen storage and battery storage
play an equally important role and both have a significant surge in capacity.
Thirdly, the expansion of transmission network capacity is marginal for all
the cases. To be more specific, the capacity of onshore wind needs to climb to
around 35 GW. Even though offshore wind energy is not considered in this
model, this indicates how much offshore wind capacity is needed. Given that
the capacity factors for offshore wind turbines are generally higher than
those of the onshore, the obtained result is more conservative compared to the
case when offshore wind is considered. Moreover, to reach the emission goals,
the needed capacity for solar PV ranges from 41 GW to 55 GW. Due to the
increase in solar capacity, the system levelized cost of energy (LCOE)
increases for each case, which are 88 €/MWh, 90 €/MWh, 93 €/MWh, 96 €/MWh, 97
€/MWh and 99 €/MWh, respectively.

Figure 2 shows the optimal capacities over the country for the two markets,
respectively. We start by looking at the planning decisions under the pool
market (left figure). Because no preferences are considered in the pool
market, cost-optimal results are obtained. Wind capacity is mostly placed on
the coastal regions where the wind resources are good, i.e. the northwestern
borders. Compared to wind, the capacity factors of solar PV are more evenly
spread over the country. The right figure shows the planning decisions under
the P2P market design with 10 % TC. The results are significantly different
from those in the left figure. Compared to the cost-optimal results, trading
barriers are introduced by the use of TC, and thus the resulted capacities are
more local, where they are in line with the energy demands of the regions.

TABLE I: Costs for Friesland region. Scenarios | Planning phase | Operational phase | Total cost (M€)
---|---|---|---
Alternatives | Cost (M€) | Uncertain future markets | Cost (M€)
Scenario 1 | Pool market (alternative 1) | 852 | Pool | 108 | 960
Scenario 2 | P2P | 65 | 917
Scenario 3 | P2P market (alternative 2) | 589 | Pool | 102 | 691
Scenario 4 | P2P | 65 | 654

Figure 3: Geographical distribution over the Netherlands of selected market in
the planning phase, and the saved costs compared to the other alternative in
%.

Figure 4: Geographical distribution over the Netherlands of preferred market
in the future, and the degree of preferences in terms of the relative cost
changes in %.

###  II-C No-regret planning considering uncertainty in future market designs

While it is important to include the trading costs when planning, there is
uncertainty in what the future market design will be, especially with the
rapid development of the P2P markets. Therefore, it may well happen that
planning decisions are made based on a certain type of markets, however, in
the future, the market form changes, resulting in cost increases in the
operational phase and thus, non-optimal planning decisions. In practice,
reserve capacities are often used to deal with various uncertainties such as
variable demand and intermittent RES supply. In the literature, these
uncertainties are often addressed using stochastic or robust optimization
methods, whereas the uncertainty in market designs are not studied. In this
subsection, we will show how our approach will help to deal with this problem.

Here, we cope with this issue by using the idea of robust planning. In robust
planning, the key idea is that different decisions would lead to different
scenarios pertaining to the uncertainties. To assess the scenarios, a min-max
problem is solved, meaning that given all the worst-case scenarios (the
maximization problem) that may happen in the future, the best decision (the
minimization problem) is selected as the solution. In our case, the decision
has to be made between two alternatives, i.e. whether to consider the pool
market or the P2P market, the uncertainty is the future market design.

We will now explain the idea further by looking at an exemplary region. The
costs of this region is given in Table I. The discussed region is Friesland,
which has good wind resources and low energy demand. As explained, one has to
do the planning while anticipating a certain market, in this case, either the
pool market or the P2P market. The choice of the markets will result in
different planning decisions and the anticipated costs.

As listed in Table I, there will be four scenarios. We start with scenario 1
as an example. In this scenario, in the planning phase, the planning decisions
are made by assuming a pool market in the future. Then in the future
(operational phase), i.e. when the capacities have already been built, it will
be the pool market in practice. For scenario 1 and scenario 2, the planning
decision is made based on the anticipation of a pool market in the future
(alternative 1). As a result, many wind capacities have to be built and thus
the investment cost is high (852 M€). For scenario 3 and scenario 4, the other
alternative is taken, which is to anticipate a P2P market in the future,
resulting in fewer investments for the region (589 M€).

In the operational phase, the electricity market design is uncertain between
the pool and the P2P markets. Compared to the P2P market (in scenario 2 and
scenario 4), the pool market (in scenario 1 and scenario 3) results in more
exports for Friesland, however with lower prices, leading to higher
operational costs. If these uncertain market designs are not considered, one
would only get results of scenario 1 and scenario 4, as we analyzed in the
previous sections. However, taking into account the uncertainty, we notice
that the costs are different for the four scenarios. This finding creates
difficulties for planning, as it is unclear, which market to consider in order
to achieve the lowest total costs.

By listing all the possible scenarios, we are able to find the worst-case
scenario when a certain alternative is chosen. If the pool market is
anticipated (in scenario 1 and scenario 2), the worst-case scenario is
scenario 1 resulting in a total cost of 960 M€. If the other alternative is
chosen, i.e. when the P2P market is considered in the planning phase (in
scenario 3 and scenario 4), the worst-case scenario is scenario 3 leading to a
total cost of 691 M€. The solution with the lower cost between the two (691
M€) is considered as a no-regret solution. It means that for this region, no
matter what the future market design turns out to be, opting for the second
alternative (i.e. to plan considering the P2P market) would save 39 % of the
costs compared to the first alternative in the worst case.

Moreover, even when the second alternative is chosen, there is a cost
difference between scenario 3 and scenario 4. This indicates a preference from
this region regarding what the future market designs should be. Scenario 4
reveals lower cost compared to that of scenario 3, and thus the P2P market
will be preferred, also in the long run, by the region. In addition, the
absolute value of the cost differences shows the degree of preference.

At last, we present the results for all the regions. Figure 3 shows the
markets that the regions would consider in the planning phase, and the
corresponding saved costs in terms of percentages. It turns out that a third
of the regions would do the planning based on a P2P market, and most of these
regions are close to the load centers. The differences in costs display a wide
range from 0 to 240 %. In particular, the region with the highest cost
difference would do the planning based on the P2P market, which, the same as
Friesland, is a net export region. Figure 4 illustrates the regions’ preferred
markets and their degrees of preferences. Although most regions would prefer a
pool market in the future, the cost differences are rather small compared to
the potential saved costs when choosing alternatives, indicating milder
degrees of preferences.

##  III Case Study II

Figure 5: Geographical distributions of wind energy capacity in benchmark and
scenario 1, and net increase in total costs in %.

In this case study, we again look at the RES planning of the Netherlands, but
with a focus on the investment preferences of the regions, in particular
related to the wind energy investment.

Due to social acceptance issues, wind energy in the Netherlands is
contentious, and thus all the regions are careful with proposing wind energy
in their regions. The initial investment proposals show that solar energy is
more popular than wind, despite the higher cost. Abandoning wind energy will
surely make the system LCOE increase which would affect all the regions. Our
framework is capable of taking into account the bilateral relations between
the regions, and could be used as a simulator to study the effects of such
decisions of not investing in wind energy (results to be shown in Section
III-A) and helps with the negotiation process (results to be shown in Section
III-B).

Figure 6: Costs of Friesland in various scenarios.

###  III-A No wind for a region

We first look at the benchmark situation where there is no preferences towards
certain RES, i.e. wind energy resistance is not taken into account. Next, the
investment preferences against wind energy are considered. These two
situations will be referred to as benchmark and scenario 1, respectively. For
illustration purposes, we consider only one region that acts against wind
energy. The influences of this choice will be shown.

Figure 5 depicts the wind capacity distribution for the benchmark situation.
It shows a concentration in capacity in the coastal regions (later referred to
as capacity centers) , i.e. the northwestern border of the country. Due to the
discrepancy of load centers and capacity centers, most of the energy that is
produced in capacity centers will be transmitted to the load centers which are
in the west and middle.

Among the capacity centers, we look at the Friesland region. It is a farm
region and is located in the north. There, energy demands are low (3 % of the
total energy demand) whereas the optimal wind capacity is the largest in the
country (20 % of the total wind capacity). For this reason, as a hypothetical
case, we assume that due to high social resistance, this region proposes no
investments in wind energy. Other regions or other groups of regions may also
be chosen, where the gist of the case study still applies. Taking this
preference into account, Figure 5 displays the new capacity distribution.
Because of the decrease in wind capacity in Friesland, almost all other
regions will have to build more generation capacities. In particular, Drenthe
has to build 4680 MW more in wind energy.

Furthermore, we look at the cost changes in percentages. Friesland, as a
result of fewer costs in investment yet more costs for importing energy, ends
up with a total cost that is 16 % higher. Although a few of its neighboring
regions benefit by gaining more revenues from exporting energy, most of the
regions are incurred with more costs due to the large increases in CapEx. This
demonstrates that the deliberate choice of one region influences the planning
decisions of all other regions. More specifically, most of the regions suffer,
though unwillingly, both in terms of increased total costs and forced wind
energy investments locally.

###  III-B Other regions’ negotiation strategies

The RES planning is a negotiation process between the regions. When other
regions are unwilling to comply with Friesland, which would be the case in our
hypothetical example, their bilateral relationships deteriorate. This, in
turn, have an effect on Friesland as well. Here, our framework is used to
simulate the negotiations between the regions. This will be done by
considering the preference costs associated with the energy trades which is
used as a means to represent the willingness to pay for the prosumers. In this
exemplary case, as a result of Friesland’s choice, the willingness to pay for
other regions with regard to trades with Friesland becomes low. In other
words, a high cost is imposed on the trades from other regions as their
negotiation strategy to Friesland’s proposal.

Figure 6 shows the costs of Friesland in various scenarios. We start with the
benchmark. Due to the large investment needs in wind energy, the CapEx and
OpEx costs (together referred to as investment costs later) are high. However,
since most of the energy that is generated there will be transmitted and sold
to other regions, Friesland will gain a significant revenue from selling
energy as well. Overall, its total cost in the benchmark case is 393 M€ . The
benchmark case provides the least cost solution for this region. If
preferences against wind energy are taken into account, its cost will
increase. In that case, its investment costs decline to 14 % of those in the
benchmark case. Accordingly, due to the lack of local generation capacities,
it has to import energy, with the net export percentage dropping from 230 % to
-50 %.

Figure 7: Costs of all the regions compared to benchmark in scenario 1 to
scenario 5.

Then, we look at the results of the following scenarios (2-4) when other
regions start to negotiate with Friesland, with increasing levels of
preference cost (10 € /MWh, 50 € /MWh, 100 € /MWh). These indicate degrees of
the willingness to pay with trades that involve Friesland. The results show
that as the trading barriers between other regions and Friesland become
larger, the energy trades shrink and thus, Friesland will be forced to rely
more on its own energy production which drives up its total costs. In the
extreme case (scenario 5), the region will be isolated by others and has no
other choice, but to be energy self-sufficient. All these scenarios are not
desirable for Friesland, and therefore, it has to reconsider its decision of
not investing in wind energy.

Now we turn to the cost changes for other regions. A key question to answer
here is, by imposing trade barriers with Friesland, what are the consequences
for other regions? Figure 7 shows the cost changes relative to the benchmark
cost for five cases. There are mainly two groups of regions to be discussed.
One group is Friesland’s neighboring regions with similar wind conditions and
low energy demands. Among all, Drenthe builds more wind capacity and is
incurred with more costs compared to the benchmark. Flevoland and Groningen
have fewer costs, since they benefit from more energy sales. The other group
consists of the load centers, Noord-Holland Zuid (Amsterdam region) and
Rotterdam-Den Haag, which rely heavily on imports. Due to the choice of
Friesland, the energy prices go up leading to higher costs for the load
centers as well. In particular, when Friesland does not invest in wind and
others take no actions, Rotterdam-Den Haag has a higher cost increase than
Friesland. Nevertheless, in all other cases, Friesland bears the most cost
increase, especially when the counteraction is strong in negotiations.

###  III-C Discussions

We would like to emphasize that the topic of this case study is not to propose
a national P2P market and study the effects of the P2P markets on the planning
decisions nor to argue for the best values to quantify the willingness to pay,
but rather to show how the product differentiation term can help to express
the preferences of the prosumers and to simulate the inter-regional
negotiation process. To this end, we have focused on Friesland as an
illustrative region, but we would like to take the discussion away from
Friesland into more general inter-regional negotiations. Moreover, we assume
all regions have the same willingness to pay, and they all counter one
particular region’s choices. The exact values for willingness to pay depend on
the bilateral relationships between other regions and the region under study,
which can relate to economic aspects such as how much influence they perceive
for their own regions or socio-political aspects such as the political
tensions between them. Some regions may even benefit, as shown already. In
addition, with various values for willingness to pay from the regions,
depending on the results, they may again choose to change their perceptions.
Therefore, the actual results highly depend on the case-specific situation
when the framework is used in practice. Our case study highlights how our
framework can be used to investigate these kinds of policy-relevant
challenges.

##  IV Case Study III

In this case study, we focus on the other part of the framework, i.e. the
mixed bilateral and pool market. We expand the geographical context to Europe.
Here, the ENTSO-E perspective is taken, where a pan-European electricity
market is considered. We follow the model assumptions and techno-economics
parameters in [3], where a highly renewable power system was modeled to meet
the carbon target in 2050. In that model, 30 countries are connected using
HVDC links and a cost-optimal system portfolio was obtained. Compared to the
previous case studies, this case study includes more technologies such as
offshore wind, hydropower and run-of-river. On top of the cost-optimal
solution, to further explore the outcomes of the European power system
decarbonization, in this case study, we consider both bilateral and pool
markets. In the bilateral market, the countries may choose to trade with
others based on their own wills, whereas in the pool market, the energy is
traded homogeneously in a pool. Here, bilateral contracts are assumed to
account for 70 % of the energy exchanges of each country [5].

Figure 8: Capacity distribution over Europe for generation, storage and
transmission for benchmark, scenario 1 (with non-green index) and scenario 2
(with cross-regional TC). The thickness of the lines represent the
transmission capacities.

Figure 9: System total capacities for benchmark, scenario 1 (with non-green
index) and scenario 2 (with cross-regional TC).

Figure 10: Average costs for the 30 European countries in benchmark, scenario
1 (with non-green index) and scenario 2 (with cross-regional TC).

###  IV-A Scenario definition

Three scenarios are investigated in this case study. In addition to a
benchmark situation where countries have no preferences in bilateral markets,
two other scenarios named scenario 1 and scenario 2 with different usages of
the preference terms will be discussed.

In scenario 1, the preference terms represent bilateral attitudes (or
willingness to pay as discussed in Section II-A) between the countries. This
is to show the effects of differentiated preferences on the investment
decisions. In general, the values of the preference terms are dependent on the
geopolitical relationships. We choose to quantify the preferences based on the
”non-green index” of each country. The idea is that the preference terms are
interpreted as a popularity index among the European countries. To be more
specific, the countries with a higher percentage of RES in their national
generation mix are more popular in bilateral trading for other countries. To
translate the popularity in costs, we look at the percentage of non-RES
generation. For example, Norway is highly dependent on hydropower and only has
a non-RES percentage of 25.4 % in the generation mix. Switzerland ranks second
in national RES penetration and has a non-RES share of 37.6 %. The percentages
are translated directly to costs, i.e. 25.4 € /MWh and 37.6 € /MWh for Norway
and Switzerland, respectively. As these costs are derived merely to
differentiate the bilateral attitudes, relative values matter more than actual
values. We further calibrate all the costs using the lowest one (25.4 € /MWh),
which ends up with 0 for Norway and 12.2 € /MWh for Switzerland. Figure 11
shows the non-green index of European countries. Luxemburg has the highest
preference costs (67.6 € /MWh) when trading with others.

In scenario 2, we use the preference terms to represent the TCs between
regions. The TCs could be considered as actual costs or deemed as a measure
for cross-regional trading barriers. Different divisions of regions in Europe
could be deployed, based on geographical locations or relationships between
various multi-national European organizations and agreements. Here, we refer
to the TSO regional coordination scheme from ENTSO-E [6]. In particular, five
regional security coordinators are considered (Figure 12 shows the
geographical locations of the five regions).

###  IV-B Investment capacities and costs

In this subsection, we show the results obtained from the different scenarios.
First, the total investment capacity of the system will be given. Then, the
geographical distribution of the capacities will be shown. Finally, the
average costs of countries are discussed.

Figure 9 illustrates the total investment capacities for the three cases.
Several trends could be observed. Firstly, onshore wind energy is the most
predominant technology, whose capacity is 42 %, 33 % and 40 % of the
generation mix, respectively in the three scenarios. Next, solar energy is the
second most needed technology and its capacity evens the onshore wind capacity
in scenario 1. Then, scenario 3 that encourages intra-regional trades shows a
similar generation portfolio with that of the benchmark scenario. Lastly, all
other technologies such as offshore wind and storage only contribute a small
part to the energy mix.

Figure 11: Non-green index, LMP and annual energy trading volumes for each
country in scenario 1.

Figure 12: Annual bilateral trading volumes represented by thickness of the
lines in benchmark, scenario 1 (with non-green index) and scenario 2 (with
cross-regional TC).

We now look at the geographical distribution of the capacities over Europe in
Figure 8. The meteorological conditions of wind and solar energy are
drastically different in various parts of Europe. Therefore, in general, wind
capacities are built in the middle and the north, whereas solar capacities are
concentrated towards the south. Transmission networks are needed to connect
the different locations. Most of the generation capacities are concentrated in
load centers of Europe, such as France and Germany. Moreover, in the benchmark
scenario, it can be found that Norway has built lots of wind energy and thus
being a major net export country. In scenario 1 and scenario 2, fewer
capacities are built but still many revenues are gained due to the increase in
energy price.

In our framework, the average costs are calculated differently from the
traditional LCOE. Normally, when calculating LCOE, only the incurred
investment costs are included. However, here, we also include the trading
costs, which means that the average costs for a country can even be negative
provided that lots of revenues are gained from energy trading. Figure 10
depicts the average costs per country for the three scenarios. Overall, the
means of the average costs are 73 € /MWh, 80 € /MWh, 72 € /MWh, respectively
for the three scenarios. The general trend is also reflected in the costs for
most countries, with a few exceptions. For example, Norway is the most popular
country to trade with in scenario 1, resulting in the lowest average cost (2 €
/MWh) therein. Denmark, in contrast, sees a surge in costs in this scenario.
This is because it changes from a net exporter to a net importer. It is also
found that Luxembourg becomes the only country with a negative average cost in
scenario 2. It benefits from the TC-free intra-regional trades and turns its
position into a net exporter.

###  IV-C Deeper look on the influences of the differentiated non-green index

After showing the capacities and costs under the three scenarios, we now take
a deeper look at how the non-green index influences the results in scenario 1.
Figure 11 illustrates the relationships of the non-green index, the locational
marginal price (LMP) and the annual trading volumes for each country. The bar
plots show the non-green index representing the preference terms for each
bilateral trades. Naturally, one tends to think that the ranking of the non-
green index will result in the same ranking for the volumes of the bilateral
energy exchanges. This is true for Norway for being the largest exporter,
Switzerland’s exports are outnumbered by countries such as France, Germany,
Sweden and Finland. In fact, the volumes of the bilateral energy exchanges are
joint effects of several factors, such as the volumes of local energy
production. This is the case with countries where the productions are high,
such as France and Germany. Therefore, in terms of export percentages, these
countries have lower values than Switzerland. In addition to the national
energy production, the other contributing factor is the price of the bilateral
trades. The price not only consists of the preference cost, but also the LMP.
For the case of Switzerland, its final price might be higher than other
countries, which is the other reason to explain its final energy exchange.

###  IV-D Further analysis on bilateral trades

Figure 12 shows the annual bilateral trades for the three scenarios. In the
benchmark scenario, the bilateral exchanges happen actively across Europe,
making use of the different wind and solar resources at different locations.
In scenario 1, with various preference costs imposed on the trades, the
trading volumes drastically decrease with more locally produced energy.
Norway, taking advantage of its lowest non-green index, becomes the most
active trader across Europe and is selling energy to all other countries. In
scenario 2, trading barriers are created by introducing a TC across regions,
which notably change the trading trends. Here, bilateral trades are no longer
active across regions, but most of them are kept within regions, showing
strong intra-regional preferences.

##  V Conclusion

With the rapid development of distributed energy sources, prosumers play an
increasingly crucial role in modern power systems. Although peer-to-peer
electricity markets have been studied extensively, a largely unrealized value
of such markets is to consider their implications on investments. In this
paper, we integrate the peer-to-peer markets in planning models, in order to
help the prosumers make optimal investment decisions while accounting for
their true wishes. Moreover, to systematically assess the influences of market
designs on the planning decisions, we consider pool markets and mixed
bilateral/pool markets in separate planning models as well. In Part I of the
paper, we develop and elaborate the problem formulations for prosumers, and
its coordination with various other actors are also investigated to solve a
concurrent generation and transmission planning problem. Here, the roles of
the TSO, the energy market operator and the government are revealed. These
different yet interconnected optimization problems of the agents are
formulated, based on which a centralized problem and a distributed
optimization model are derived for each market.

This framework has been applied to three archetypal case studies. They aim at
answering different but all technically, socially and politically relevant
questions. For all the cases, the investment decisions and the associated
costs are presented. The first case studies the institutional uncertainty in
future market designs and helps to obtain no-regret planning decisions for the
prosumers. The second case focuses on a situation where a group of prosumers
have to decide on their own investments in order to meet a joint carbon
target. There, we consider the technology preferences of the prosumers, in
particular, an assumed unwillingness to invest in wind energy due to social
acceptance issues. The framework acts as a negotiation simulator to inform the
group about the consequences of such a preference. Furthermore, by using the
bilateral preference terms, it is able to simulate the negotiation strategies
that are to the benefit of the prosumers. The third case investigates a
European highly renewable power system in 2050. The bilateral relationships
between the countries that are shaped by e.g., geopolitical reasons are shown
to greatly influence the national planning decisions, and thus the results
give practical insights on the decarbonization of the European power system.

In order to focus the contribution on integrating electricity markets into
planning models, some simplifications are made. In order to apply the
framework, adjustments could be made from both planning and operational
aspects depending on the purpose of the potential applications. These include,
e.g., adding more actors such as the distributions system operator, using AC
power flow calculation including losses and considering imperfect competition.
Moreover, the complexity of the distributed optimization algorithms
drastically increases, as the number of agents becomes large. It is less of an
issue in planning problems, as there are no urgent operational needs to obtain
the results within a limited amount of time. Nevertheless, future research
could investigate more in this direction to optimize for speed as well as
limit information exchange between the agents. In fact, our framework has been
constructed in different modules that allow for these modifications.

## References

  * [1] Unie van Waterschappen, “Regionale Energie Strategie,” 2019. [Online]. Available: https://www.regionale-energiestrategie.nl/default.aspx
  * [2] N. Wang, R. A. Verzijlbergh, P. W. Heijnen, and P. M. Herder, “A spatially explicit planning approach for power systems with a high share of renewable energy sources,” _Applied Energy_ , vol. 260, p. 114233, feb 2020.
  * [3] D. P. Schlachtberger, T. Brown, S. Schramm, and M. Greiner, “The benefits of cooperation in a highly renewable European electricity network,” _Energy_ , vol. 134, pp. 469–481, 2017.
  * [4] E. Sorin, L. Bobo, P. Pinson, and S. Member, “Consensus-Based Approach to Peer-to-Peer Electricity Markets with Product Differentiation,” _IEEE Transactions on Power Systems_ , vol. 34, no. 2, pp. 994–1004, 2019.
  * [5] European Commission, “Quarterly Report on European Electricity Markets,” Tech. Rep. 4, 2021.
  * [6] ENTSO-E, “Enhanced TSO Regional Coordination for Europe,” Tech. Rep., 2019.


Charged rotating black strings
in Einsteinian cubic gravity

Hamid R<EMAIL_ADDRESS>
Department of Nanotechnology, Graduate University of Advanced Technology,
Kerman, Iran

Abstract

We construct, for the first time, charged rotating black string solutions in
four-dimensional Einsteinian cubic gravity, which are asymptotically anti–de
Sitter. By assuming that the solutions are completely regular at the horizon
and studying their near-horizon behavior, we find some thermodynamic
properties, which can be accessed analytically. We compute independently the
Hawking temperature, the Wald entropy, the mass, the angular momentum, the
charge, and the electrostatic potential of the solutions, analytically. Using
these, we show that the first law of thermodynamics for rotating black strings
is exactly satisfied in both charged and uncharged cases. We also observe
that, in the absence of Maxwell field, some of the solutions have positive
specific heat, which makes them thermodynamically stable.

##  1 Introduction

One of the most relevant approaches in modification of general relativity (GR)
is to take into account the higher-curvature terms. In addition to the
Lovelock densities [1], there exists another class of theories, so-called
quasitopological gravities (QTG) [2, 3], which admit analytic static black
holes in dimensions larger than 4. The most general theory of gravity up to a
cubic order in curvature is called generalized quasitopological gravity [4, 5,
6, 7, 8], whose static and spherically symmetric vacuum solutions are fully
described by a single function which comes from second-order linearized
equations around maximally symmetric backgrounds [6, 8, 9, 10, 11, 12]. In
this theory, the Lovelock and quasitopological gravities, which are determined
by an algebraic equation, have been recovered as special cases in four
dimensions.

Recently, it was shown that up to cubic order in curvature, there is a theory
so-called Einsteinian cubic gravity (ECG) as the most general higher-curvature
modification of Einstein gravity, which only propagates the usual transverse
and traceless graviton on maximally symmetric backgrounds in general
dimensions [13]. This theory admits nonhairy single-function generalizations
of the Schwarzschild black hole similar to the Lovelock and quasi-topological
theories in dimensions larger than 4 [14, 15].

Here we review some attempts in finding the solutions in the context of ECG
theory. In [16], the authors show that static charged black holes in four-
dimensional ECG differ from their counterparts in GR in several aspects: (i)
nonuniqueness of regular solutions, (ii) noncompliance with the extremality
bound, and (iii) the absence of an inner horizon. In [17], using a continued
fraction ansatz, an analytic approximation for a spherically symmetric black
hole solution to ECG has been found. ECG theory has been regarded as a
holographic toy model of a nonsupersymmetric conformal field theory (CFT) in
three dimensions in [18]. Also, the first generalizations of the Einstein
gravity Taub-NUT/bolt solutions for any higher-curvature theory in four
dimensions have been represented in [19]. In [20], using on-shell amplitudes,
a rotating black hole solution in ECG theory is derived. Recently, slowly
rotating black hole solutions of ECG in four dimensions with flat and anti-de
Sitter (AdS) asymptotes were constructed and various physical properties of
the solutions were studied [21]. In [22], new solutions of Einsteinian cubic
gravity coupled to a Maxwell field that describe the near-horizon geometry of
charged rotating black holes are obtained.

In the present work, we will study asymptotically AdS charged rotating black
strings [23, 24] in the context of ECG theory. It has been shown that for
charged rotating toroidal black hole solutions only configurations that are
supersymmetric are: (i) the nonrotating electrically charged naked
singularities, and (ii) an extreme rotating toroidal black hole with zero
electric and magnetic charges [25]. The solutions of charged rotating black
strings, which are indeed cylindrically symmetric black holes, have also been
studied in [26, 27] and are generalized to include higher dimensions as black
strings/branes in [28]. The thermodynamic properties of solutions are also
investigated in [29, 30]. Rotating black string solutions have been also
explored in the presence of nonlinear electrodynamics [31, 32], dilaton
gravity [33, 34], holographic superfluids [35], $f(R)$ gravity [36], and
mimetic gravity [37, 38].

The structure of the paper is organized as follows. In the next section, we
will study, in detail, uncharged solutions both asymptotically and near a
black string horizon. We then obtain, using a Taylor expansion around the
horizon, exact expressions for the mass and surface gravity. We also compute
the angular momentum of solutions, as well as the Wald entropy, and show that
the first law of black string thermodynamics holds exactly. In Sec. 3, we
investigate the charged solutions. As the uncharged case, by studying the
near-horizon behavior of solutions, we get the mass, angular momentum, as well
as some thermodynamic properties. We also find the electrostatic potential and
total charge of solutions. Having had these quantities, we easily check the
first law of black string thermodynamics in the presence of Maxwell field and
find an exact agreement. Section 4 is also devoted to summary and concluding
remarks.

##  2 Asymptotically AdS uncharged solutions

The action of four-dimensional ECG theory in the presence of cosmological
constant reads,

$S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}(R-2\Lambda-2G^{2}\lambda{\cal P}),$
(2.1)

where $G$ is the Newton gravitational constant, $R$ represents the Ricci
scalar, $\Lambda=-3/{l^{2}}$ is the negative cosmological constant of AdS
space,222Note that the asymptotically de–Sitter solutions can be obtained by
simply taking $l\rightarrow il$ [28]. and the cubic-in-curvature correction to
the Einstein-Hilbert action is incorporated in

$\displaystyle{\cal
P}=12R_{a}{}^{c}{}_{b}{}^{d}R_{c}{}^{e}{}_{d}{}^{f}R_{e}{}^{a}{}_{f}{}^{b}+R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}-12R_{abcd}R^{ac}R^{bd}+8R_{a}{}^{b}R_{b}{}^{c}R_{c}{}^{a}.$
(2.2)

We will assume the dimensionless ECG coupling constant to be non-negative
throughout the paper, i.e., $\lambda\geq 0$. Einstein gravity is also
recovered by setting $\lambda=0$.

Our aim here is to construct the asymptotically AdS charged rotating black
string solutions of ECG theory and investigate their properties. Therefore, we
assume the metric of four-dimensional spacetime with cylindrical or toroidal
horizons can be written as [23, 24, 25]

$\displaystyle ds^{2}=-f(r)g^{2}(r)\left(\Xi dt-
ad\phi\right)^{2}+\frac{1}{f(r)}dr^{2}+\frac{r^{2}}{l^{4}}\left(adt-\Xi
l^{2}d\phi\right)^{2}+\frac{r^{2}}{l^{2}}dz^{2},$ (2.3)

where

$\Xi=\sqrt{1+a^{2}/l^{2}}.$ (2.4)

The constants $a$ and $l$ have dimensions of length and can be interpreted as
the rotation parameter and the AdS radius, respectively. In the following, we
are going to study the solutions of ECG with cylindrical symmetry. This
implies that the spacetime admits a commutative two-dimensional Lie group
$G_{2}$. The ranges of the time and radial coordinates are
$-\infty<t<\infty,0\leq r<\infty$, and the topology of the horizon can be
regarded as follows:

  1. (i)

the flat torus $T^{2}$ with topology $S^{1}\times S^{1}$ (i.e.,
$G_{2}=U(1)\times U(1)$) and the ranges $0\leq\phi<2\pi,0\leq z<2\pi l$, which
describes a closed black string,

  2. (ii)

the cylinder with topology $\mathbb{R}\times S^{1}$ (i.e.,
$G_{2}=\mathbb{R}\times U(1)$) and the ranges
$0\leq\phi<2\pi,-\infty<z<\infty$, which describes a stationary black string,

  3. (iii)

the infinite plane with topology $\mathbb{R}^{2}$ and the ranges
$-\infty<\phi<\infty,-\infty<z<\infty$, which does not rotate.

We consider the topology (ii) throughout the paper.333For the case of a
toroidal horizon, the entropy, mass, angular momentum, and charge of the
string are obtained from their respective densities by multiplying them by
$2\pi l$ [29]. Now, we are in a position to evaluate the field equations of
action (2.1) on the ansatz (2.3) and find the corresponding equations for the
functions $g(r)$ and $f(r)$. Here, we will use a method introduced in [14, 15]
for static and spherically symmetric spacetimes. By considering the action as
a functional of these functions, $S[g,f]$, one finds

$\frac{\delta S[g,f]}{\delta g}=\frac{\delta S[g,f]}{\delta
f}=0\Leftrightarrow{\cal E}_{tt}={\cal E}_{rr}={\cal E}_{t\phi}=0.$ (2.5)

Here ${\cal E}_{tt}$, ${\cal E}_{rr}$, and ${\cal E}_{t\phi}$ are,
respectively, the $tt$, $rr$, and $t\phi$ components of the corresponding
field equations. Notice that the metric (2.3) is obtained after a boost along
the $(t,\phi)$ coordinates from the static black brane solution. The boost is
only locally defined, so the global structure of the metric is different, but
this explains why the metric takes this particular form and why it is not
necessary to solve other components of the equations of motion. This shows
that the equations for $f(r)$ and $g(r)$ can be obtained from the action
functional $S[g,f]$ without the need to compute the full non-linear equations
explicitly. It can be seen that the action $S[g,f]$ can be written as

$\displaystyle S[g,f]=\frac{1}{8\pi G}\int drg(r).\left\\{-\frac{1}{3}\Lambda
r^{3}-rf-G^{2}\lambda\left[4f^{\prime
3}-12ff^{\prime}f^{\prime\prime}-24\frac{f^{2}\left(f^{\prime}-rf^{\prime\prime}\right)}{r^{2}}\right]\right\\}^{\prime}$
(2.6)

plus some terms involving at least two derivatives of $g$. Here, a prime
denotes a derivative with respect to $r$. By variation of the above action
with respect to $g$ and $f$, one can get the equations of them. It can be seen
from that, $g$ is multiplied by a total derivative. Therefore, $\delta_{g}S=0$
can be solved by setting

$g^{\prime}(r)=0.$ (2.7)

As a result, the ECG theory admits solutions characterized by a single
function $f(r)$. So we set $g=1$.444Without loss of generality we can set
$g(r)=1$, for simplicity. In general, one can choose $g=1/\sqrt{f_{\infty}}$,
where $f_{\infty}=\lim_{r\rightarrow\infty}f(r)$, to normalize the speed of
light on the boundary or in the dual CFT to be $c=1$. However, we can set
$g=1$ by reparametrization of time ($t$) and angular ($\phi$) coordinates of
the metric, if desired. The equation $\delta_{g}S=0$ yields, after integrating
once, the following equation for $f$:

$-\frac{1}{3}\Lambda r^{3}-rf-G^{2}\lambda\left[4f^{\prime
3}-12ff^{\prime}f^{\prime\prime}-24\frac{f^{2}\left(f^{\prime}-rf^{\prime\prime}\right)}{r^{2}}\right]=r_{0},$
(2.8)

where $r_{0}$ is an integration constant which is related to the mass of
string as $r_{0}=GM$.

###  2.1 Asymptotic solution

As can be seen from Eq. (2.8), when $\lambda=0$ we obtain

$f(r)=\frac{r^{2}}{l^{2}}-\frac{GM}{r},$ (2.9)

where we have set $\Lambda=-3/{l^{2}}$ and $r_{0}=GM$. This is, of course,
nothing but the usual AdS uncharged solution of rotating black strings [24,
23]. When $\lambda$ is turned on, the asymptotic quantities get corrected in
this case. To obtain these corrections, we first examine the large-$r$
behavior of the solution. To do so, we assume that the metric function $f$ can
be expressed as a particular solution in the form of a $1/r$ expansion, plus
the general solution of the corresponding homogeneous equation,

$f=f_{1/r}+f_{h}\quad\text{with}\quad f_{1/r}(r)=\frac{r^{2}}{l_{\rm
eff}^{2}}+\sum_{n=1}^{\infty}\frac{b_{n}}{r^{n}}.$ (2.10)

Substituting this series expansion into Eq. (2.8), one finds the large-$r$
expansion reads

$f_{1/r}(r)=\frac{r^{2}}{l_{\rm eff}^{2}}-\frac{G_{\rm eff}M}{r}+{\cal
O}\left(r^{-3}\right),$ (2.11)

where the effective radius of the AdS space, ${l_{\rm eff}}$, is a solution of
the equation

$\frac{16G^{2}\lambda}{l_{\rm eff}^{6}}-\frac{1}{l_{\rm
eff}^{2}}+\frac{1}{l^{2}}=0,$ (2.12)

and the effective gravitational constant is given by

$G_{\rm eff}=\frac{G}{1-48\frac{G^{2}\lambda}{l_{\rm eff}^{4}}}.$ (2.13)

On the other hand, the linearized homogeneous equation satisfied by $f_{h}(r)$
at the large-$r$ limit, reads

$f_{h}^{\prime\prime}(r)-\frac{4}{r}f_{h}^{\prime}(r)-\gamma^{2}rf_{h}(r)=0,$
(2.14)

where

$\gamma^{2}=\frac{l_{\rm eff}^{2}}{36GG_{\rm eff}^{2}M\lambda}.$ (2.15)

Here, we have kept only the leading terms in the large-$r$ limit. The solution
of Eq. (2.14), in the case of $\gamma^{2}>0$, is

$f_{h}^{(+)}(r)=r^{5/2}\left[AI_{5/3}\left(\frac{2\gamma
r^{3/2}}{3}\right)+BK_{5/3}\left(\frac{2\gamma r^{3/2}}{3}\right)\right],$
(2.16)

where $I$ and $K$ are the modified Bessel functions of the first and second
kinds, and $A$ and $B$ are some constants. In the limit of large $r$ we can
approximate the solution by

$f_{h}^{(+)}(r)\sim r^{5/2}\left[A\exp\left(\frac{2\gamma
r^{3/2}}{3}\right)+B\exp\left(-\frac{2\gamma r^{3/2}}{3}\right)\right],$
(2.17)

and so we must set $A=0$ to ensure the AdS boundary conditions are satisfied.
As a result no ghost excitations can propagate to infinity. We shall see
shortly that the contribution of the second term can be dismissed. If
$\gamma^{2}<0$, then the homogeneous solution asymptotically takes the
following form:

$f_{h}^{(-)}(r)=r^{5/2}\left[CJ_{5/3}\left(\frac{2\left|\gamma\right|r^{3/2}}{3}\right)+DY_{5/3}\left(\frac{2\left|\gamma\right|r^{3/2}}{3}\right)\right],$
(2.18)

where $J$ and $Y$ are the Bessel functions of the first and second kind, and
$C$ and $D$ are arbitrary constants. In this situation, the solution
oscillates rapidly, and its amplitude becomes larger than $r^{2}/l_{\rm
eff}^{2}$ at large $r$. It therefore does not approach AdS asymptotically, and
so we must set $C=D=0$ to get rid of this homogeneous part of the solution.
For the rest of our considerations, to avoid any oscillating behavior near
infinity, we restrict the solutions to the constraint $\gamma^{2}>0$. Finally,
we note that the particular solution (2.11) polynomially decreases with $1/r$
and is the dominant part of the total solution $f(r)=f_{1/r}+f_{h}^{(+)}$ for
sufficiently large $r$; we therefore neglect the term $f_{h}^{(+)}$ in Eq.
(2.17) in the sequel.

###  2.2 Near-horizon solution

Horizon is defined by surface $r=r_{h}$, at which $f(r_{h})=0$ and
$f^{\prime}(r_{h})\geq 0$. By Taylor expanding around the horizon (assuming
$f$ to be completely regular there),
$f(r)=\Sigma_{n=1}^{\infty}a_{n}(r-r_{h})^{n}$, and solving (2.8) order by
order in powers of $(r-r_{h})$, the coefficients $a_{n}$ can be determined.
Here, $a_{n}=f^{(n)}(r_{h})/n!$. The general definition of surface gravity is

$\kappa=\sqrt{-\frac{1}{2}(\nabla^{a}\chi^{b})(\nabla_{a}\chi_{b})}.$ (2.19)

Here, the null generator of the black string horizon is given by
$\chi=\partial_{t}+\Omega\partial_{\phi}$, where the angular velocity of the
event horizon is given by

$\displaystyle\Omega=\frac{a}{l^{2}\Xi}.$ (2.20)

A simple calculation for the spacetime (2.3) with $g(r)=1$ gives
$\kappa_{g}=f^{\prime}(r_{h})/2\Xi$. The two lowest-order equations form an
algebraic system that is used to fix the mass $M$ and surface gravity
$\kappa_{g}$ in terms of horizon radius $r_{h}$, are given by

$\frac{r_{h}^{3}}{l^{2}}-GM-32G^{2}\lambda\kappa_{g}^{3}\Xi^{3}=0,$ (2.21)
$\frac{3r_{h}^{2}}{l^{2}}-2\kappa_{g}\Xi r_{h}=0.$ (2.22)

Solving the above equations, we get the following quantities for surface
gravity and mass of black strings

$\kappa_{g}=\frac{3r_{h}}{2l^{2}\Xi},$ (2.23)
$M=\frac{r_{h}^{3}}{Gl^{2}}\left(1-\frac{108G^{2}\lambda}{l^{4}}\right).$
(2.24)

Note that, for higher-order equations, we can treat $a_{2}$ as a free
parameter and find the $n$th coefficient $a_{n}$, in terms of $a_{2}$, and get
a family of solutions with only one free parameter $a_{2}$. For the value of
$a_{2}$, we are able to construct numerically the solution up to a
sufficiently large $r$, for which the solution approaches to the asymptotic
expansion in (2.10) [14].

###  2.3 Thermodynamics

We begin this section with a discussion of the black string entropy. In a
higher curvature theory of gravity, the Bekenstein-Hawking entropy is modified
by additional terms which can be obtained using the Wald entropy formula [39,
40],

${\sf S}=-2\pi\int_{H}d^{2}x\sqrt{\gamma}\frac{\delta{\cal L}}{\delta
R_{abcd}}\epsilon_{ab}\epsilon_{cd},$ (2.25)

where $\frac{\delta{\cal L}}{\delta R_{abcd}}$ is the Euler-Lagrange
derivative of gravitational Lagrangian, $\gamma$ is the determinant of the
induced metric on the horizon, and $\epsilon_{ab}$ is the binormal of the
horizon, normalized to satisfy $\epsilon_{ab}\epsilon^{ab}=-2$, which can be
derived from the following relation:

$\triangledown_{a}\chi_{b}=\kappa_{g}\epsilon_{ab}.$ (2.26)

Using (2.26), one can easily find the nonzero components for the antisymmetric
binormal of the horizon, that are

$\displaystyle\epsilon_{tr}=-\epsilon_{rt}=-\Xi,$
$\displaystyle\epsilon_{r\phi}=-\epsilon_{\phi r}=-a.$ (2.27)

The result for entropy now reads [14],

$\displaystyle{\sf S}$ $\displaystyle=$
$\displaystyle\frac{1}{4G}\int_{H}d^{2}x\sqrt{\gamma}\left[1+G^{2}\lambda\left(36R_{b}{}^{e}{}_{d}{}^{f}R_{aecf}+3R_{ab}{}^{ef}R_{cdef}\right.\right.$
(2.28)
$\displaystyle\qquad\qquad\qquad\left.\left.-12R_{ac}R_{db}-24R^{ef}R_{eafc}g_{bd}+24g_{bd}R_{ce}R^{e}{}_{a}\right)\epsilon^{ab}\epsilon^{cd}\right].$

For the metric (2.3) and with a cylindrical horizon placed at $r=r_{h}$, one
finds the following value for entropy per unit length,

${\cal S}=\frac{\pi
r_{h}^{2}\Xi}{2Gl}\left(1-\frac{108G^{2}\lambda}{l^{4}}\right),$ (2.29)

where $f^{\prime}(r_{h})=2\Xi\kappa_{g}$, and Eq. (2.23) has been taken into
account to write the final result in terms of horizon radius $r_{h}$. The
Hawking temperature of our solution can easily be written in terms of the
horizon radius as

$\displaystyle T=\frac{\kappa_{g}}{2\pi}=\frac{3r_{h}}{4\pi l^{2}\Xi}.$ (2.30)

We finish this section by calculating the mass and angular momentum of black
strings by using the counterterm method inspired by AdS/CFT conjecture [41].
To do so, we add the Gibbons–Hawking boundary term, which removes the
divergences of the action (2.1). Here, the suitable boundary action is given
by [18]

$S_{b}=\frac{1+48\frac{G^{2}\lambda}{l_{\rm eff}^{4}}}{8\pi
G}\int_{\partial\cal{M}}d^{3}x\sqrt{-\gamma}\Theta,$ (2.31)

where $\gamma$ is the determinant of the induced metric on the boundary and
$\Theta$ is the trace of the extrinsic curvature $\Theta_{ab}$ of the
boundary. Note that $l_{\rm eff}$ is a scale length factor that depends on $l$
and $\lambda$ that must reduce to $l$ as $\lambda$ goes to zero. That is
indeed the root of Eq. (2.12). We use the counterterm method [42, 43] to
eliminate the divergences of action. In this approach, we add some local
surface integrals to the action to make it finite. This method also has been
applied to the case of third-order Lovelock gravity with flat boundary ${\cal
R}_{abcd}(\gamma)=0$ [44]. The counterterms for the case of ECG, which makes
the action finite up to four dimensions, are given in [18]; they are

$S_{ct}=\frac{1+48\frac{G^{2}\lambda}{l_{\rm eff}^{4}}}{8\pi
G}\int_{\partial\cal{M}}d^{3}x\sqrt{-\gamma}\left(\frac{2}{l_{\rm
eff}}-\frac{l_{\rm eff}}{2}{\cal R}\right),$ (2.32)

where ${\cal R}$ is the Ricci scalar for the boundary metric $\gamma$. Notice
that this counterterm has exactly the same form as the Einstein gravity with
zero curvature boundary, in which $G$ and $l$ are replaced by
$G/\left(1+48\frac{G^{2}\lambda}{l_{\rm eff}^{4}}\right)$ and $l_{\rm eff}$,
respectively. The total action can be written as a linear combination of the
action of bulk, boundary, and the counterterm,

$S_{total}=S+S_{b}+S_{ct}.$ (2.33)

Having had the total finite action, one can use the Brown-York definition of
stress energy-momentum tensor [45], by varying the action with respect to
boundary metric $\gamma_{ab}$, and find a divergence-free stress tensor as

$T^{ab}=\frac{1}{8\pi G_{\rm
eff}}\left(\Theta^{ab}-\Theta\gamma^{ab}+\frac{2}{l_{\rm
eff}}\gamma^{ab}-\frac{l_{\rm eff}}{2}{\cal G}^{ab}\right),$ (2.34)

where ${\cal G}_{ab}={\cal R}_{ab}-{\cal R}\gamma_{ab}/2$ is the Einstein
tensor of boundary metric $\gamma_{ab}$. For asymptotically AdS solutions with
flat horizons, the only nonvanishing counterterm, is the first term in (2.32),
which yields the stress-energy tensor (2.34) up to the third term. Using the
above stress tensor, one can define the quasilocal conserved quantities for an
asymptotically AdS spacetime. The conserved charge associated to a Killing
vector $\xi_{a}$ is given by

$Q_{\xi}=\int_{\Sigma}d^{3}x\sqrt{\sigma}u^{a}T_{ab}\xi^{b},$ (2.35)

where $u_{a}=-N\delta^{0}_{a}$, while $N$ and $\sigma$ are the lapse function
and the metric of spacelike surface $\Sigma$, which appear in the ADM–like
decomposition of the boundary metric,

$\gamma_{ij}dx^{i}dx^{j}=-N^{2}dt^{2}+\sigma_{ab}\left(dx^{a}+V^{a}dt\right)\left(dx^{b}+V^{b}dt\right).$
(2.36)

Here also $V^{a}$ is the shift vector. To obtain the total mass (energy), we
should set $\xi=\partial_{t}$, i.e., the Killing vector conjugate to time
coordinate $t$, and to obtain the angular momentum, we should set
$\xi=\partial_{\phi}$, i.e., the Killing vector conjugate to angular
coordinate $\phi$. Using the definition (2.35) for conserved charges, we find
the total mass per unit length of horizon to be given by

${\cal M}=\frac{1}{8l}\left(3\Xi^{2}-1\right)M,$ (2.37)

while the angular momentum per unit length of horizon is given by

${\cal J}=\frac{3}{8l}\Xi aM=\frac{3}{8}\Xi\sqrt{\Xi^{2}-1}M,$ (2.38)

where in writing the last equality, the Eq. (2.4) has been used. Substituting
the mass (2.24) into the above equations, one arrives at

$\displaystyle{\cal
M}=\frac{r_{h}^{3}}{8Gl^{3}}\left(3\Xi^{2}-1\right)\left(1-\frac{108G^{2}\lambda}{l^{4}}\right),$
(2.39) $\displaystyle{\cal
J}=\frac{3r_{h}^{3}}{8Gl^{2}}\Xi\sqrt{\Xi^{2}-1}\left(1-\frac{108G^{2}\lambda}{l^{4}}\right).$
(2.40)

Using the above results, one can easily approve that the first law of
thermodynamics for uncharged rotating black strings,

$d{\cal M}=Td{\cal S}+\Omega d{\cal J},$ (2.41)

also holds exactly in the context of Einsteinian cubic gravity.

###  2.4 Thermal stability

The local stability, in an arbitrary ensemble, requires that the entropy
${\cal S}$ be a convex function of their extensive variables, or its Legendre
transformation must be a concave function of their intensive variables. Thus,
the local stability can be explored by finding the determinant of the Hessian
matrix of ${\cal S}$ with respect to its extensive variables $X_{i}$, ${\bf
H}_{X_{i}X_{j}}^{\cal S}=\partial^{2}{\cal S}/\partial X_{i}\partial X_{j}$,
or the determinant of the Hessian of the Gibbs function, with respect to its
intensive variables $Y_{i}$, ${\bf
H}_{Y_{i}Y_{j}}^{G}=\partial^{2}{G}/\partial Y_{i}\partial Y_{j}$, where the
thermodynamic variables of the system depend on the ensemble, which is used
[46]. In the other words, the more $X_{i}$ or $Y_{i}$ we regard as variable
parameters, the smaller is the region of stability. In the case of uncharged
solutions, the entropy is a function of mass and angular momentum per unit
length. In canonical ensemble, the angular momentum is the fixed parameter,
and therefore, the positivity of specific heat,

$C=T\left(\frac{\partial{\cal S}}{\partial T}\right)_{\cal J},$ (2.42)

guarantees the local stability [46]. Writing the entropy (2.29) in terms of
temperature using Eq. (2.30) and then substituting it into the above
expression, one finds that the specific heat parametrized by horizon radius
$r_{h}$ is given by

$C=\frac{\pi
r_{h}^{2}\Xi\left(1-\frac{108G^{2}\lambda}{l^{4}}\right)\left(1+\Xi^{2}\right)}{2Gl\left(5\Xi^{2}-4\right)}.$
(2.43)

It can be seen that for $\lambda\leq l^{4}/108G^{2}$, the solutions are
locally stable in the canonical ensemble.

In the grand canonical ensemble, the thermodynamic variables are the
temperature and the angular velocity. So, it is more convenient to work with
thermodynamic potential $G(T,\Omega)={\cal M}-T{\cal S}-\Omega{\cal J}$, which
is indeed the Legendre transformation of the energy ${\cal M}({\cal S},{\cal
J})$ with respect to ${\cal S},{\cal J}$. Here, the determinant of Hessian
matrix is given by

$|{\bf
H}_{T,\Omega}^{G}|=\left[\frac{\pi\Xi^{2}r_{h}^{2}}{G}\left(1-\frac{108G^{2}\lambda}{l^{4}}\right)\right]^{2}\left(1-\frac{\Xi^{2}}{2}\right).$
(2.44)

It is a matter of calculation to show that the solutions are locally stable in
grand canonical ensemble if $\Xi\leq\sqrt{2}$ or using Eq. (2.4), $a\leq l$.

##  3 Asymptotically AdS charged solutions

In this section we add a Maxwell field to the action as

$S=\int d^{4}x\sqrt{-g}\left[\frac{1}{16\pi
G}\left(R-2\Lambda-2G^{2}\lambda{\cal
P}\right)-\frac{1}{4}F_{ab}F^{ab}\right],$ (3.1)

where $F_{ab}=2\partial_{[a}A_{b]}$. The vector potential and nonzero
components of electromagnetic field strength tensor are given by [24]

$\displaystyle
A_{a}=A_{0}(r)\left(\Xi\delta_{a}^{t}-a\delta_{a}^{\phi}\right);$
$\displaystyle F_{tr}=-F_{rt}=-\Xi A^{\prime}_{0}(r),F_{\phi
r}=-F_{r\phi}=aA^{\prime}_{0}(r).$ (3.2)

Having had the electromagnetic field strength tensor, it is not difficult to
show that both $t$ and $\phi$ components of the Maxwell equation,
$\triangledown_{a}F^{ab}=0$, lead to the same equation,

$rA^{\prime\prime}_{0}g-rA^{\prime}_{0}g^{\prime}+2A^{\prime}_{0}g=0.$ (3.3)

Here also, as the uncharged case, we find that $g^{\prime}=0$ and therefore
set $g=1$. This yields the following value for the gauge potential

$A_{0}(r)=\frac{q}{4\pi r},$ (3.4)

where $q$ is an integration constant, which is indeed the electric charge of
the black string. As before, the equation for $f$ can be obtained from the
variation of (3.1) with respect to $g$ and then setting $g=1$, that is

$-\frac{1}{3}\Lambda r^{2}+\frac{GQ^{2}}{r}-rf-G^{2}\lambda\left[4f^{\prime
3}-12ff^{\prime}f^{\prime\prime}-24\frac{f^{2}\left(f^{\prime}-rf^{\prime\prime}\right)}{r^{2}}\right]=r_{0},$
(3.5)

where we have defined $Q^{2}\equiv q^{2}/4\pi$, and $r_{0}$ is an integration
constant, which is related to the mass of the black hole as the uncharged
case, $r_{0}=GM$. In the next section, we will perform an asymptotic expansion
around $r\rightarrow\infty$.

###  3.1 Asymptotic solution

In the case of charged solutions, by setting $\lambda=0$ in Eq. (2.8), one
finds

$f(r)=\frac{r^{2}}{l^{2}}-\frac{GM}{r}+\frac{GQ^{2}}{r^{2}},$ (3.6)

which is the usual asymptotically AdS charged solution of rotating black
strings [24, 23]. As the uncharged case, one can examine the large-$r$
behavior of the solution. To this end, we assume that $f$ can be expressed as
a particular solution in the form of a $1/r$ expansion, plus the general
solution of the corresponding homogeneous equation. Just like the uncharged
case, this expansion is given by Eq. (2.10). Plugging this series expansion
into Eq. (3.5), one observes that the large-$r$ expansion takes the following
form:

$f_{1/r}(r)=\frac{r^{2}}{l_{\rm eff}^{2}}-\frac{G_{\rm eff}M}{r}+\frac{G_{\rm
eff}Q^{2}}{r^{2}}+{\cal O}(r^{-3}),$ (3.7)

where the effective radius of the AdS space $l_{\rm eff}$, and the effective
gravitational constant $G_{\rm eff}$ are defined by Eqs. (2.12) and (2.13),
respectively. The linearized homogeneous equation satisfied by $f_{h}(r)$ is
given by

$f_{h}^{\prime\prime}(r)+\frac{12M}{4Q^{2}-3Mr}f_{h}^{\prime}(r)+\frac{3M\gamma^{2}r^{2}}{4Q^{2}-3Mr}f_{h}(r)=0,$
(3.8)

at the large-$r$ limit. Notice that when $Q=0$, we recover Eq. (2.14).
Unfortunately, this equation can not be solved analytically. But due to the
existence of $\left(4Q^{2}-3Mr\right)$ in the denominator, it is possible to
expand the above equation at large $r$ one more time and find a solution and
discuss the boundary conditions. By doing so, we arrive at

$f_{h}^{\prime\prime}(r)-\frac{\left(4Q^{2}+3Mr\right)\gamma^{2}}{3M}f_{h}(r)=0.$
(3.9)

Here, $\gamma^{2}$ is given by Eq. (2.15). The solution reads,

$f_{h}(r)=A^{\prime}\text{Ai}\left(\frac{4Q^{2}+3Mr}{3M}\left|\gamma^{2}\right|^{1/3}\right)+B^{\prime}\text{Bi}\left(\frac{4Q^{2}+3Mr}{3M}\left|\gamma^{2}\right|^{1/3}\right),$
(3.10)

where Ai and Bi are the Airy functions of the first and second kinds, and
$A^{\prime}$ and $B^{\prime}$ are some constants. In the case of
$\gamma^{2}>0$, and at large $r$,

$\displaystyle f_{h}^{(+)}(r)$ $\displaystyle\sim$
$\displaystyle\left(\frac{4Q^{2}+3Mr}{3M}\gamma^{2/3}\right)^{1/4}\times$
$\displaystyle\left\\{A^{\prime}\exp\left[-\frac{2}{3}\left(\frac{4Q^{2}+3Mr}{3M}\gamma^{2/3}\right)^{3/2}\right]+B^{\prime}\exp\left[\frac{2}{3}\left(\frac{4Q^{2}+3Mr}{3M}\gamma^{2/3}\right)^{3/2}\right]\right\\},$

and so we must set $B^{\prime}=0$ to ensure the AdS boundary conditions are
satisfied. As a result, no ghost excitations can propagate to infinity. If
$\gamma^{2}<0$, then the homogeneous solution asymptotically takes the
following form at large $r$:

$\displaystyle f_{h}^{(-)}(r)$ $\displaystyle\sim$
$\displaystyle\left(\frac{4Q^{2}+3Mr}{3M}\gamma^{2/3}\right)^{-1/4}\times$
$\displaystyle\left\\{A^{\prime}\sin\left[\frac{2}{3}\left(\frac{4Q^{2}+3Mr}{3M}\gamma^{2/3}\right)^{3/2}\right]+B^{\prime}\cos\left[\frac{2}{3}\left(\frac{4Q^{2}+3Mr}{3M}\gamma^{2/3}\right)^{3/2}\right]\right\\}.$

Here also the solution oscillates rapidly and therefore does not approach AdS
asymptotically, and so we set $A^{\prime}=B^{\prime}=0$. As the uncharged
case, to avoid any oscillating behavior at infinity, we restrict the solutions
to the constraint $\gamma^{2}>0$. Observe that all the leading asymptotic
corrections to the metric come from the solution (3.7), while the
contributions from the homogeneous equation are extremely subleading. Hence,
the term proportional to $A^{\prime}$ above can be discarded from the
asymptotic expansion.

###  3.2 Near-horizon solution

In a similar way with the uncharged case, by making a Taylor expansion around
the horizon, one finds the following equations at two lowest-order of
expansion:

$-GM-32G^{2}\lambda\kappa_{g}^{3}\Xi^{3}+\frac{r_{h}^{3}}{l^{2}}+\frac{GQ^{2}}{r_{h}}=0,$
(3.13) $\frac{3r_{h}^{2}}{l^{2}}-2\kappa_{g}\Xi
r_{h}-\frac{GQ^{2}}{r_{h}^{2}}=0,$ (3.14)

which solving them yields the following values for the surface gravity, as
well as mass as a function of horizon radius and charge of black strings,

$\kappa_{g}=\frac{1}{2\Xi
r_{h}^{3}}\left(\frac{3r_{h}^{4}}{l^{2}}-GQ^{2}\right),$ (3.15)
$M=\frac{r_{h}^{3}}{Gl^{2}}\left[1+\frac{Gl^{2}Q^{2}}{r_{h}^{4}}-\frac{4G^{2}l^{2}\lambda}{r_{h}^{12}}\left(\frac{3r_{h}^{4}}{l^{2}}-GQ^{2}\right)^{3}\right].$
(3.16)

Using these expressions, we are able to check the first law of black string
thermodynamics in the presence of the Maxwell field in the next section.

###  3.3 Thermodynamics

Using the entropy (2.28) and $f^{\prime}(r_{h})=2\Xi\kappa_{g}$, where
$\kappa_{g}$ is given by (3.15), we find the following quantity for the Wald
entropy per unit length for charged solutions:

${\cal S}=\frac{\pi
r_{h}^{2}\Xi}{2Gl}\left[1-\frac{12G^{2}\lambda}{r_{h}^{8}}\left(\frac{3r_{h}^{4}}{l^{2}}-GQ^{2}\right)^{2}\right].$
(3.17)

The Hawking temperature can easily be obtained in terms of charge and the
horizon radius by multiplying the surface gravity (3.15) with a factor
$1/2\pi$,

$T=\frac{1}{4\pi r_{h}^{3}\Xi}\left(\frac{3r_{h}^{4}}{l^{2}}-GQ^{2}\right).$
(3.18)

The total mass and angular momentum per unit length for charged solutions can
also be found by inserting Eq. (3.16) into the Eqs. (2.37) and (2.38),

${\cal
M}=\frac{r_{h}^{3}}{8Gl^{3}}\left(3\Xi^{2}-1\right)\left[1+\frac{Gl^{2}Q^{2}}{r_{h}^{4}}-\frac{4G^{2}l^{2}\lambda}{r_{h}^{12}}\left(\frac{3r_{h}^{4}}{l^{2}}-GQ^{2}\right)^{3}\right],$
(3.19) ${\cal
J}=\frac{3r_{h}^{3}}{8Gl^{2}}\Xi\sqrt{\Xi^{2}-1}\left[1+\frac{Gl^{2}Q^{2}}{r_{h}^{4}}-\frac{4G^{2}l^{2}\lambda}{r_{h}^{12}}\left(\frac{3r_{h}^{4}}{l^{2}}-GQ^{2}\right)^{3}\right].$
(3.20)

In the following we are going to find the electric potential and total charge
of black strings. The electric potential $\Phi$, measured at infinity with
respect to the horizon, is defined by [29]

$\Phi=A_{a}\chi^{a}\lvert_{r\rightarrow\infty}-A_{a}\chi^{a}\lvert_{r=r_{h}},$
(3.21)

where, as already mentioned, $\chi$ is the null generator of the horizon.
Calculating the above expression yields the following value for electric
potential

$\displaystyle\Phi=\frac{q}{4\pi\Xi r_{h}}=\frac{Q}{2\sqrt{\pi}\Xi r_{h}}.$
(3.22)

Finally, we calculate the electric charge of the black string. To this end, we
first determine the electric field by considering the projections of the
electromagnetic field tensor on a special hypersurface. The normal vectors to
such a hypersurface are

$u^{0}=\frac{1}{N},u^{r}=0,u^{i}=-\frac{V^{i}}{N},$ (3.23)

where $N$ and $V^{i}$ are the lapse function and shift vector, respectively.
The electric field is also given by $E^{\mu}=g^{\mu\rho}F_{\rho\nu}u^{\nu}$.
The electric charge per unit length of black strings can be found by
calculating the flux of the electric field at infinity, yielding

$\displaystyle{\cal Q}=\frac{q\Xi}{2l}=\frac{\sqrt{\pi}Q\Xi}{l}.$ (3.24)

Using the above expressions, one can easily show that the first law of
thermodynamics for charged rotating black strings,

$d{\cal M}=Td{\cal S}+\Omega d{\cal J}+\Phi d{\cal Q},$ (3.25)

is fully satisfied in the context of ECG theory.

##  4 Conclusions

In this paper we have constructed generalizations of the charged rotating
black strings in four-dimensional ECG with AdS asymptotic and studied some of
their properties.

We have shown that the theory admits solutions with a single function $f(r)$,
which is given by a nonlinear second-order differential equation and studied
some of their thermodynamic properties which can be obtained analytically. The
new solutions represent the first nontrivial four-dimensional generalizations
of the asymptotically AdS charged rotating black strings in higher-order
gravity, whose thermodynamic properties can be computed exactly. Using those
results, we have been able to check analytically that the solutions exactly
satisfy the first law of thermodynamics for black strings in both charged and
uncharged cases.

Let us close with some possible future explorations. It would be interesting
to perform this kind of study, in higher dimensions. Finding the solutions in
the presence of nonlinear Born-Infeld electrodynamics is also an interesting
feature. Finally, a study of the solutions including Einsteinian quartic
gravity would be perhaps worth pursuing.

## Acknowledgements

We would like to thank the authors of Ref. [47] for developing the excellent
_Mathematica_ package “xTras,” which we have used extensively for symbolic
calculations.

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