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1701.07861v2_100

http://arxiv.org/abs/1701.07861v2

are predominantly localized and occasionally hop to a new location. Finally, the maximum cluster density creates a crystal-like structure, albeit not necessarily entirely symmetric due to the randomly generated b_ij terms in the adaptive dynamics.The motion of individual clusters is heavily constrained by its neighbours via mutual repulsion, while the collective motion of an ensemble of clusters is limited by the carrying capacity function. Thus, phenotypic saturation leads to a state in which

1701.07861v2_101

http://arxiv.org/abs/1701.07861v2

phenotypic saturation leads to a state in which the coevolving clusters are strongly constrained evolutionarily by the other clusters in the community, and hence to coevolutionary equilibrium dynamics. Some empirical support for an initial increase in the complexity of evolutionary dynamics with the number of species in an ecosystem comes from the laboratory evolution experiments of <cit.>, who showed that the speed of adaptation to novel environments is higher in bacterial species that are

1701.07861v2_102

http://arxiv.org/abs/1701.07861v2

is higher in bacterial species that are part of microbial communities with a small number of competitors than when evolving in monoculture. However, our results are seemingly in contrast to previous theoretical results about the effect of diversity on evolutionary dynamics <cit.>. These authors essentially argued that while a single species is free to evolve in response to changes in the environment, evolution of the same species is more constrained in a community of competitors, in which other

1701.07861v2_103

http://arxiv.org/abs/1701.07861v2

in a community of competitors, in which other species are more likely to evolutionarily occupy new niches. Hence diversity is expected to slow down evolution.However, these models only describe evolution in 1-dimensional phenotypes, and may thus miss the complexity arising in higher-dimensional spaces. Moreover, even in higher-dimensional spaces, the arguments for evolutionary slowdown presented in <cit.> essentially correspond to our observation of a slow-down when diversity reaches

1701.07861v2_104

http://arxiv.org/abs/1701.07861v2

observation of a slow-down when diversity reaches saturation, at which point evolutionary change in each species is indeed constrained due to competing species occupying all available niches. Our approach also needs to be distinguished from approaches based primarily on ecological dynamics, as in <cit.>. In these approaches, emerging ecological communities are also modelled by periodically adding new species, but there is no underlying phenotype space that would determine competitive

1701.07861v2_105

http://arxiv.org/abs/1701.07861v2

phenotype space that would determine competitive interactions. Instead, every time a new species added, its interaction coefficients with the already existing species are chosen according to a specific, randomized procedure. This leads to interesting results, such as saturating levels of diversity after initially fast and fluctuating increases from low levels of diversity. However, since there is no underlying phenotype space, this approach does not reveal the evolutionary dynamics of

1701.07861v2_106

http://arxiv.org/abs/1701.07861v2

does not reveal the evolutionary dynamics of continuous phenotypes, and in particular, it does not yield any information about the effects of the dimension of phenotype space on the evolutionary dynamics or on the amount of diversity at saturation.There has been much interest in recent years in determining the effects of phylogenetic relationships on the functioning of ecosystems (e.g. <cit.>). The intuitive notion is that phylogenetic information has predictive power for ecological

1701.07861v2_107

http://arxiv.org/abs/1701.07861v2

information has predictive power for ecological interactions if recently diverged species are more likely to interact than those that diverged long ago. More specifically, <cit.> have argued that phylogenetic information is most likely to be relevant for ecosystem dynamics if ecological interactions are based on phenotypic matching, so that species with more similar trait values are more likely to interact strongly. Our models have a component of phenotypic matching due to the Gaussian part of

1701.07861v2_108

http://arxiv.org/abs/1701.07861v2

phenotypic matching due to the Gaussian part of the competition kernel, but they also have a strong component of different types of interactions due to the “random” part of the competition kernel given by the coefficients b_ij. As we have shown, it is this non-Gaussian part of the competition kernel that causes the complicated coevolutionary dynamics, and it is this complexity in turn that makes phylogenetic signal largely irrelevant in our models. A full phylogenetic analysis of the

1701.07861v2_109

http://arxiv.org/abs/1701.07861v2

our models. A full phylogenetic analysis of the macroevolutionary dynamics generated by our models is beyond the scope of this work, but we can provide some basic insights based on the complicated evolutionary dynamics in phenotype space that the different phenotypic clusters (species) perform when there is an intermediate number of clusters in the coevolving community. An example of this is shown in the movie in Figure A1A. Here, after an initial phase of diversification, the community

1701.07861v2_110

http://arxiv.org/abs/1701.07861v2

initial phase of diversification, the community contains 12 coevolving clusters. These clusters move on a complicated evolutionary trajectory, with each cluster undergoing large evolutionary changes without further diversification. No matter what the phylogenetic relationship between these clusters (as given by their emergence from the single initial cluster), it is clear that because of the large evolutionary fluctuations in phenotype space of each cluster (species), there will be no

1701.07861v2_111

http://arxiv.org/abs/1701.07861v2

space of each cluster (species), there will be no consistent correlation between phylogenetic relationship and phenotypic distance. Even if there were such a correlation (positive or negative) at a particular point in time, it would change over time due to the large evolutionary fluctuations of each cluster over time. This is illustrated in Figure A1B, which showsthat no persistent correlation pattern between phylogenetic and phenotypic distance should be expected in communities with an

1701.07861v2_112

http://arxiv.org/abs/1701.07861v2

should be expected in communities with an intermediate amount of diversity. In particular, recently diverged species are not more likely to interact than those diverged less recently, because the evolving community has a short “phenotypic memory” due to complicated evolutionary dynamics. However, when further diversification is allowed, so that the system reaches its saturation level of diversity, the coevolving community not only becomes more diverse, but the evolutionary dynamics slows down,

1701.07861v2_113

http://arxiv.org/abs/1701.07861v2

but the evolutionary dynamics slows down, leading to ever smaller phenotypic fluctuations. In particular, new clusters emerging towards the end of the assembly of the evolutionarily stable community will stay phenotypically closer to their phylogenetically most closely related clusters, i.e., to their parent or sister species. Therefore, in the last phase of community assembly a positive correlation between phylogenetic and phenotypic distance can be expected to build up at least to some

1701.07861v2_114

http://arxiv.org/abs/1701.07861v2

can be expected to build up at least to some extent. This is illustrated in Figure A1B. Thus, weak phylogenetic signals are expected to develop towards the end of community assembly.Regarding adaptive radiations, two observations emerge from our models. The first concerns the classical notion that rates of diversification should decline over the course of a radiation <cit.>, a pattern that seems to have good empirical support <cit.>. Our models confirm this pattern of declining rates of

1701.07861v2_115

http://arxiv.org/abs/1701.07861v2

models confirm this pattern of declining rates of diversification (Figure 5).The second observation is that rates of evolution should generally slow down with an increase in diversity. This should not only be true when different ecosystems are compared (Figures 3,4), but also during an adaptive radiation in a single evolving community (Figure 5). Thus, we would expect the evolutionary dynamics to be faster and more complicated early in an adaptive radiation, and to slow down and eventually

1701.07861v2_116

http://arxiv.org/abs/1701.07861v2

radiation, and to slow down and eventually equilibrate late in the radiation. This corresponds to the so-called “early-burst” model of macroevolution <cit.> in the context of adaptive radiations. This model predicts that when lineages enter novel “adaptive zones”<cit.>, such as novel ecological niches, evolutionary rates in the lineage should be fast initially and then slow down as the adaptive zone gets filled with diverse phenotypes. <cit.> found little evidence for the early-burst model when

1701.07861v2_117

http://arxiv.org/abs/1701.07861v2

little evidence for the early-burst model when analyzing a large set of data from many different clades. Nevertheless, these authors noted that younger clades have higher rates of evolution than older clades, which points to the fact that evolutionary rates may slow down with clade age. Moreover, few clades in their data set correspond to the type of very fast adaptive radiation envisaged and observed in our models, and they did not consider high-dimensional phenotypes. Finally, <cit.> note

1701.07861v2_118

http://arxiv.org/abs/1701.07861v2

high-dimensional phenotypes. Finally, <cit.> note that groups with a larger proportion of sympatric species early in their history would be more likely to exhibit an early-burst pattern. In our models, adaptive radiations occur in complete sympatry and indeed produce the early burst pattern. According to <cit.>, the jury on early-burst models is still out, and in fact substantial evidence for this model has accumulated in recent years. For example, <cit.> reported an early burst in body size

1701.07861v2_119

http://arxiv.org/abs/1701.07861v2

<cit.> reported an early burst in body size evolution in mammals, <cit.> observed an early-burst pattern in the evolution of bill shape during adaptive radiation in seabirds,<cit.> and <cit.> reported early-burst patterns in morphological and functional evolution in cichlids, and <cit.> described patterns of early bursts in the evolution of dinosaur morphology. <cit.> have incorporated the early-burst concept into a macroevolutionary perspective in which over very long evolutionary time scales,

1701.07861v2_120

http://arxiv.org/abs/1701.07861v2

in which over very long evolutionary time scales, rare but substantial phenotypic bursts alternate with more stationary periods of bounded phenotypic fluctuations, somewhatreminiscent of the concept of punctuated equilibrium <cit.> when applied to rates of phenotypic evolution <cit.>. We think that the models presented here could provide a microevolutionary basis for such a perspective if they are extended by considering evolutionary change in the dimension of the phenotype space that

1701.07861v2_121

http://arxiv.org/abs/1701.07861v2

in the dimension of the phenotype space that determines ecological interactions. Such an extended theory would have three time scales: a short, ecological time scale, an intermediate time scale at which co-evolution and single diversifications take place in a given phenotype space, and a long time scale at which the number of phenotypic components increases (or decreases). Our hypothesis would then be that in such systems, periods of bounded evolutionary fluctuations near diversity saturation

1701.07861v2_122

http://arxiv.org/abs/1701.07861v2

fluctuations near diversity saturation levels for a given dimension of phenotype space would alternate with bursts of rapid evolutionary change, brought about by an evolutionary increase in phenotypic dimensions and the subsequent increase in diversity and acceleration in evolutionary rates until a new saturation level is reached. The resulting long-term evolutionary dynamics would thus show periods of relative phenotypic stasis alternating with periods of fast evolution. This picture would fit

1701.07861v2_123

http://arxiv.org/abs/1701.07861v2

periods of fast evolution. This picture would fit very well with the “blunderbass” pattern envisaged in <cit.>. These authors proposed that the intermittent bursts in evolutionary rates are caused by lineages encountering novel “adaptive zones” <cit.>. Novel adaptive zones would correspond to the opening up of new habitats or new resources, which would in turn correspond to new phenotypes that determine use of the novel adaptive zone. Alternatively, novel adaptive zones could also be generated

1701.07861v2_124

http://arxiv.org/abs/1701.07861v2

novel adaptive zones could also be generated by the emergence of novel sets of regulatory mechanisms allowing novel uses of already existing habitats and resources (as e.g. when a trade-off constraint is overcome through gene duplication). In either case, novel adaptive zones would correspond to an increase in the dimensionality of ecologically important phenotypes. It is interesting to note that such intermittent burst patterns have in fact been observed in phylogenetic data, and that they

1701.07861v2_125

http://arxiv.org/abs/1701.07861v2

been observed in phylogenetic data, and that they seem to be connected to novel, ecologically important phenotypes. <cit.> have shown that evolutionary rates in echinoids reveal at least two instances of rapidly accelerating and subsequently declining evolutionary rates, i.e., two intermittent bursts. Moreover, these bursts appear to be associated with the evolution of novel feeding strategies <cit.>. Also, <cit.> have shown that an evolutionary burst occurs in the dinosaur-bird transition, and

1701.07861v2_126

http://arxiv.org/abs/1701.07861v2

burst occurs in the dinosaur-bird transition, and it is tempting to conjecture that this burst was caused by the increase in phenotype dimensionality due to the proliferation of flight capabilities.There is also good empirical support for our finding that the level at which diversity saturates increases with the dimension of phenotype space. <cit.> has argued that essentially, the high number ofdifferent ecologically relevant traits is the basis for the spectacular radiations of cichlids in

1701.07861v2_127

http://arxiv.org/abs/1701.07861v2

for the spectacular radiations of cichlids in African lakes. In conjunction with ecological opportunity, genetic and phenotypic flexibility, which appears to be at least in part due to gene duplications, has allowed this group of fish to reach a much higher diversity than other groups, such as cichlids in rivers or whitefish in arctic lakes, in which fewer phenotypes appear to be ecologically relevant<cit.>. In this context, we note that incorporating the evolution of the dimension of phenotype

1701.07861v2_128

http://arxiv.org/abs/1701.07861v2

the evolution of the dimension of phenotype space may also shed light on the ongoing debate about whether diversity saturates over evolutionary time or not <cit.>. It seems that the answer could be “yes and no”: diversity saturates in the intermediate term for a given dimension of phenotype space, but does not saturate in the long term if the dimension of phenotype space increases over long evolutionary time scales, thus generating recurrent increases in saturation levels.Our study has a number

1701.07861v2_129

http://arxiv.org/abs/1701.07861v2

in saturation levels.Our study has a number of limitations that should be addressed in future research. It is currently impractical to perform the statistical analysis presented here for phenotype spaces with dimensions higher than 4 due to computational limitations. Our results indicate that the diversity saturation level, i.e., the maximal number of coexisting phenotypic clusters, increases rapidly with the dimension d of phenotype space, which makes simulations of communities at saturation

1701.07861v2_130

http://arxiv.org/abs/1701.07861v2

makes simulations of communities at saturation levels unfeasible. Nevertheless, we expect the salient result that coevolutionary dynamics slow down as communities reach the saturation level to be true in any dimension as long as the Gaussian component of competition in (<ref>) affects all phenotypic directions. Also, in our approach we have assumed that the phenotypes determining competitive interactions are the same for intra- and inter-specific competition. This may be a fair assumption for

1701.07861v2_131

http://arxiv.org/abs/1701.07861v2

competition. This may be a fair assumption for closely related species, such as those coevolving in an adaptive radiation. However, for competition in more general ecosystems it may also be relevant to assume that from a total set of d phenotypes, different subsets determine competition within a species and competition with various other species. In addition, to describe general ecosystems and food webs, it will be important to include not just competitive interactions, but also predator-prey

1701.07861v2_132

http://arxiv.org/abs/1701.07861v2

competitive interactions, but also predator-prey and mutualistic interactions, each again determined by potentially high-dimensional phenotypes. Also, throughout we have assumed a simple unimodal form of the carrying capacity to represent the external environment. More complicated forms of the carrying capacity, and hence of the external fitness landscape will likely generate even richer patterns of coevolutionary dynamics and diversification. Finally, we have assumed throughout that evolving

1701.07861v2_133

http://arxiv.org/abs/1701.07861v2

Finally, we have assumed throughout that evolving populations are well-mixed, and it will be interesting so see how the results generalize to spatially structured ecosystems. All these extensions remain to be developed. We are of course aware of the fact that we did not include genetic mixing due to sexual reproduction in our models, and our method of describing diversification by simply adding new phenotypic clusters, although fairly standard, does not take into account the actual process of

1701.07861v2_134

http://arxiv.org/abs/1701.07861v2

does not take into account the actual process of speciation. In sexual populations, adaptive diversification due to disruptive selection, as envisioned here, requires assortative mating, and the conditions for the evolution of various types of assortative mating, as well as for the likelihood of speciation once assortment is present, have been studied extensively (e.g. <cit.>). A general, if crude conclusion from this work is that when there is enough disruptive selection for diversification to

1701.07861v2_135

http://arxiv.org/abs/1701.07861v2

disruptive selection for diversification to occur in asexual models, then it is likely that adaptive speciation also occurs in the corresponding sexual models, although factors such as the strength of assortment, population size and linkage disequilibrium may become important. It would in principle be possible to incorporate sexual reproduction into the models presented here, e.g. along the lines of <cit.>. Our previous results <cit.> indicate that adaptive diversification is generally more

1701.07861v2_136

http://arxiv.org/abs/1701.07861v2

that adaptive diversification is generally more likely in high-dimensional phenotype spaces, and we think that the present models serve well as a first approximation to study adaptive diversification and coevolutionary dynamics in evolving communities.Ultimately, the applicability and relevance of our models for understanding macroevolutionary patterns in nature depends in part on being able to determine evolutionary rates of high-dimensional phenotypes from phylogenetic data, which appears to

1701.07861v2_137

http://arxiv.org/abs/1701.07861v2

from phylogenetic data, which appears to be a difficult problem <cit.>. Nevertheless, overall we think that our approach of incorporating microevolutionary processes based on ecological interactions in high-dimensional phenotype spaces into statistical models for macroevolutionary dynamics has the potential to shed new light on a number of fundamental conceptual questions in evolutionary biology. § ACKNOWLEDGMENTSM. D. was supported by NSERC (Canada). I. I. was supported by FONDECYT grant

1701.07861v2_138

http://arxiv.org/abs/1701.07861v2

(Canada). I. I. was supported by FONDECYT grant 1151524 (Chile). Both authors contributed equally to this work. § CORRELATION BETWEEN PHYLOGENETIC AND PHENOTYPIC DISTANCEFor each pair of clusters (species) in an evolving community we define the phylogenic distance between them, Pg, as the number of links in the path between them on the phylogenic tree. To measure this distance, we add the following scheme to our evolutionary algorithm: * The system is initialized with a single cluster. * Each

1701.07861v2_139

http://arxiv.org/abs/1701.07861v2

is initialized with a single cluster. * Each cluster splitting event produces two offspring separated by the distance 2. The distance between an offspring and all its existing neighbours is incremented by one. * When two recently split cluster that failed to diverge are merged, the distance between the newly produced common cluster and each of its neighbours is calculated as the minimum of the distances of the two merged clusters minus one. This reflects the observation that merging events only

1701.07861v2_140

http://arxiv.org/abs/1701.07861v2

reflects the observation that merging events only happen with newly split clusters. As a result, at any given time we know phylogenic distances between all pairs of clusters currently present in the system. To quantify the relation between the phenotypic and phylogenic similarity, we compute the correlation C between phylogenetic and phenotypic distance as follows:C=⟨ [Pg - ⟨ Pg ⟩ ][X - ⟨ X ⟩] ⟩/σ_Pgσ_X, where Ph and X are phylogenic and phenotypic distances between clusters, ⟨…⟩ define the

1701.07861v2_141

http://arxiv.org/abs/1701.07861v2

distances between clusters, ⟨…⟩ define the average over all pairs of clusters present in the system and σ_Pg and σ_X are the standard deviations of distances.The above scheme allows us to track the correlation between phylogenetic and phenotypic distance over time, as illustrated in Figure A1. Fig. A1A shows the time dependence of C for the simulation shown in Video 1, and in Fig. A1B shows the time dependence of C for the simulation shown in Video 2. During the early phase of community

1701.07861v2_142

http://arxiv.org/abs/1701.07861v2

in Video 2. During the early phase of community assembly the correlation C rapidly decays due to complicated coevolutionary dynamics of the emerging clusters. When the diversity of the coevolving community is kept intermediate (by setting the parameter m_C to intermediate values, as in Video 1), the correlation between phylogenetic and phenotypic distance itself undergoes fluctuations around 0 (Fig. A1A). This is because the clusters in the community with intermediate diversity undergo large

1701.07861v2_143

http://arxiv.org/abs/1701.07861v2

with intermediate diversity undergo large phenotypic fluctuations while their phylogenetic relationship is constant, because no further diversification (or extinction) occurs. However, when the diversity is allowed to reach saturation levels (by setting m_C to a large value, as in Video 2), a positive correlation between phylogenetic and phenotypic distance develops in the final stages of community assembly, i.e., as the coevolving community reaches the saturation diversity and hence undergoes

1701.07861v2_144

http://arxiv.org/abs/1701.07861v2

the saturation diversity and hence undergoes much smaller phenotypic fluctuations (Fig. A1B). Note that the correlation is still close to 0 during the early stages of community assembly, but some correlation remains at the end due to clusters emerging in the last phase of community assembly, which tend to stay phenotypically closer to their sister species because evolutionary dynamics become slow and stable.§ INDIVIDUAL-BASED SIMULATIONSIndividual-based realizations of the model were based on

1701.07861v2_145

http://arxiv.org/abs/1701.07861v2

realizations of the model were based on the Gillespie algorithm <cit.> and consisted of the following steps: * The system is initialized by creating a set of K_0 ∼ 10^3 - 10^4 individuals with phenotypes_k∈𝐑^d localized around the initial position _0 with a small random spread |_k - _0|∼10^-3.* Each individual k has aconstant reproduction rate _̊k=1 and a death rate _̣k=∑_ l ≠ k A(_l,_k)/[K_0K(_k)], as defined by the logistic ecological dynamics.* The total update rate is given by the sum of

1701.07861v2_146

http://arxiv.org/abs/1701.07861v2

The total update rate is given by the sum of all individual rates, U=∑_k (_̊k+_̣k). * The running time t is incremented by a random number t drawn from the exponential distribution P( t)= U exp (- tU).* A particular birth or death event is randomly chosenwith probability equal to the rate of this event divided by the total update rate U. Ifa reproduction event is chosen, the phenotype of an offspring is offset from the parental phenotype by asmall mutation randomly drawn from a uniform

1701.07861v2_147

http://arxiv.org/abs/1701.07861v2

by asmall mutation randomly drawn from a uniform distribution with amplitude = 10^-3 - 10^-2.* The individual death rates _̣k and the total update rate U are updatedto take into account the addition or removal of an individual.* Steps 4-6 are repeated until t reaches a specified end time. The movie in Video A2 shows the dynamics of the individual-based model corresponding to the adaptive dynamics simulation shown in Video A1, which is the same as the scenario used for Video 2 in the main

1701.07861v2_148

http://arxiv.org/abs/1701.07861v2

same as the scenario used for Video 2 in the main text(note that the movie in Video A1 runs for t=1200 time units, whereas the movie in Video 2 runs for t=400 time units). 1cm 1 cm§ PARTIAL DIFFERENTIAL EQUATION MODELSA deterministic large-population limit of the individual-based model is obtained as the partial differential equation (PDE) ∂ N(, t)/∂ t = N(, t)( 1 - ∫α(, ) N (, t) dy/K())+D∑_i=1^d ∂^2 N(, t)/∂ x_i^2, whereN(, t) is the population distribution at time t <cit.>. The second term

1701.07861v2_149

http://arxiv.org/abs/1701.07861v2

distribution at time t <cit.>. The second term of the right hand side is a diffusion term that describes mutations,with the diffusion coefficient typically set to D∼ 10^-4 - 10^-3. Local maxima of the solution N(x,t) can be interpreted as positions of the centers of the phenotypic clusters. Their dynamics are shown in Video A3. For any given scenario, the corresponding adaptive dynamics solution can be used to determine the single- or few-cluster trajectory, and hence to approximately determine

1701.07861v2_150

http://arxiv.org/abs/1701.07861v2

trajectory, and hence to approximately determine the region occupied by the system in phenotype space over time. Note that the deterministic PDE model is invariant with regard to the coordinate change → -, and hence its solutions must be symmetric with regard to simultaneous reflection on all coordinate axes. To numerically solve the PDE model (<ref>)wechose a lattice noticeably larger than the corresponding adaptive dynamics attractor. The number of bins B in each dimension of this lattice is

1701.07861v2_151

http://arxiv.org/abs/1701.07861v2

of bins B in each dimension of this lattice is strongly constrained by memory limitations: An efficient implementation requires computing and storing an array of B^d× B^d values of the competition kernel α(_i, _j) for the pairwise interactions between all pairs of sites i and j.With B=25 -30 to achieve a reasonable spatial resolution, the memory constraint makes the PDE implementation feasible only for d=2,3. The movie in Video A3 shows the dynamics of the partial differential equation model

1701.07861v2_152

http://arxiv.org/abs/1701.07861v2

of the partial differential equation model corresponding to the scenarios shown in Videos A1 and A2. § SCALING RELATIONSHIP FOR THE DIVERSITY AT SATURATIONThe number of clusters at the diversity saturation level, M_,d, can be estimated to be proportional to the volume of the available phenotype space with the linear dimension L, divided by the volume occupied by each cluster, which has a typical linear size : M__a,d≈ C_L^d/^d. Hence, the following scalingrelationships hold:

1701.07861v2_153

http://arxiv.org/abs/1701.07861v2

Hence, the following scalingrelationships hold: M__a,d=M__b,d(_b/_a)^d and M_,d_1=M_,d_2^d_1/d_2,where _a and _b denote different strengths of competition, and C_ is a constant of order 1 that takes into account the “imperfect packing” occurring whenand L have similar magnitude.Based on this, the equilibrium level of diversity is expected to increase exponentially with increasing dimension of phenotype space (as illustrated Figure 1), and with increasing frequency-dependence (i.e., decreasing

1701.07861v2_154

http://arxiv.org/abs/1701.07861v2

increasing frequency-dependence (i.e., decreasing ). In general, diversity is only maintained if ≲1, which is roughly the scale of the phenotypic range set by the carrying capacity given by eq. (5) in the main text. § SPECIFIC SETS OF COEFFICIENTS USED The following set of coefficients b_ij determining the competition kernel were used for Figures 5A in the main text and for the movies. [0.4070.4980.287; -0.199 -1.102 -0.305;1.387 -0.8960.341 ] The following set of coefficients b_ij determining

1701.07861v2_155

http://arxiv.org/abs/1701.07861v2

following set of coefficients b_ij determining the competition kernel were used for Figure 5B in the main text: [ -1.2890.6820.217 -0.093; -0.223 -0.0350.697 -0.117; -0.5630.434 -0.953 -0.198;0.1190.3980.1830.530 ] 2cm evolution

1701.07941v2_0

http://arxiv.org/abs/1701.07941v2

Computationally Efficient Market Simulation Tool for Future Grid Scenario Analysis Shariq Riaz, Graduate Student Member, IEEE, Gregor Verbič, Senior Member, IEEE, and Archie C. Chapman, Member, IEEE Shariq Riaz, Gregor Verbič and Archie C. Chapman are with the School of Electrical and Information Engineering, The University of Sydney, Sydney, New South Wales, Australia. e-mails: (shariq.riaz, gregor.verbic, archie.chapman@sydney.edu.au). Shariq Riaz is also with the Department of Electrical

1701.07941v2_1

http://arxiv.org/abs/1701.07941v2

Riaz is also with the Department of Electrical Engineering, University of Engineering and Technology Lahore, Lahore, Pakistan.December 30, 2023

1701.07941v2_2

http://arxiv.org/abs/1701.07941v2

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1701.07941v2_3

http://arxiv.org/abs/1701.07941v2

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1701.07941v2_4

http://arxiv.org/abs/1701.07941v2

This paper proposes a computationally efficient electricity market simulation tool (MST) suitable for future grid scenario analysis. The market model is based on a unit commitment (UC) problem and takes into account the uptake of emerging technologies, like demand response, battery storage, concentrated solar thermal generation, and HVDC transmission lines. To allow for a subsequent stability assessment, the MST requires an explicit representation of the number of online generation units, which

1701.07941v2_5

http://arxiv.org/abs/1701.07941v2

of the number of online generation units, which affects powers system inertia and reactive power support capability. These requirements render a full-fledged UC model computationally intractable, so we propose unit clustering, a rolling horizon approach, and constraint clipping to increase the computational efficiency. To showcase the capability of the proposed tool, we use a simplified model of the Australian National Electricity Market with different penetrations of renewable generation. The

1701.07941v2_6

http://arxiv.org/abs/1701.07941v2

penetrations of renewable generation. The results are verified by comparison to a more expressive and computationally-intensive binary UC, which confirm the validity of the approach for long term future grid studies. Electricity market, future grid, electricity market simulation tool, optimization, scenario analysis, unit commitment, stability assessment, inertia, loadability. [A01]𝒞Set of consumers c. [A02]𝒢Set of generators g, 𝒢^ = 𝒢^𝓈𝓎𝓃∪𝒢^ℛℰ𝒮. [A03]𝒢^𝓈𝓎𝓃Set of synchronous generators,

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http://arxiv.org/abs/1701.07941v2

[A03]𝒢^𝓈𝓎𝓃Set of synchronous generators, 𝒢^𝓈𝓎𝓃⊆𝒢. [A04]𝒢^ℛℰ𝒮Set of renewable generators, 𝒢^ℛℰ𝒮⊆𝒢. [A05]𝒢^𝒞𝒮𝒯Set of concentrated solar thermal generators, 𝒢^𝒞𝒮𝒯⊆𝒢^syn. [A06]𝒢^𝓇Set of synchronous generators in region r, ⋃_𝒢^𝓇= 𝒢^. [A07]ℋSet of sub-horizons h. [A08]ℒSet of power lines l, ℒ = ℒ^𝒜𝒞∪ℒ^ℋ𝒱𝒟𝒞. [A09]ℒ^𝒜𝒞Set of AC power lines, ℒ^𝒜𝒞⊆ℒ. [A10]ℒ^ℋ𝒱𝒟𝒞Set of HVDC power lines, ℒ^ℋ𝒱𝒟𝒞⊆ℒ. [A11]𝒩Set of nodes n. [A12]𝒩^𝓇Set of nodes in region r. [A13]𝒫Set of prosumers p. [A14]ℛSet of regions r.

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[A13]𝒫Set of prosumers p. [A14]ℛSet of regions r. [A15]𝒮Set of storage plants s. [A16]𝒯Set of time slots t.[D01]s_g,tNumber of online units of generator g, s_g,t∈{0,1} in BUC and s_g,t∈ℤ_+ in MST. [D02]u_g,tInteger startup status variable of a unit of generator g, u_g,t∈{0,1} in BUC and u_g,t∈ℤ_+ in MST. [D03]d_g,tInteger shutdown status variable of a unit of generator g, d_g,t∈{0,1} in BUC and d_g,t∈ℤ_+ in MST. [D04]δ_n,tVoltage angle at node n. [D05]p_l,t^Power flow on line l. [D08]Δ

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node n. [D05]p_l,t^Power flow on line l. [D08]Δ p_l,tPower loss on line l. [D09]p_g,tPower dispatch of generator g. [D10]p_p,t^g+/-Grid/feed-in power of prosumer p. [D11]p_s,tPower flow of storage plant s. [D12]p_p,t^bBattery power flow of prosumer p. [D13]e_g,tThermal energy stored in TES of generator g ∈𝒢^𝒞𝒮𝒯. [D14]e_s,tEnergy stored in storage plant s. [D15]e_p,t^bBattery charge state of prosumer p. [P]c_g^fix/varFix/variable cost of a unit of generator g. [P]c_g^su/sdStartup/shutdown cost

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of generator g. [P]c_g^su/sdStartup/shutdown cost of a unit of generator g. [P]p_c,t^Load demand of consumer c. [P]p_p,t^Load demand of prosumer p. [P]p_n,t^rPower reserve requirement of node n. [P]x/xMinimum/maximum limit of variable x. [P]U_gTotal number of identical units of generator g. [P]r^+/-_gRamp-up/down rate of a unit of generator g. [P]τ^u/d_gMinimum up/down time of a unit of generator g. [P]t̃Time slot offset index. [P]ΔtTime resolution. [P]B_lSusceptance of line l. [P]p_g,t^RESMax.

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[P]B_lSusceptance of line l. [P]p_g,t^RESMax. output power of renewable generator g ∈𝒢^RES. [P]p_g,t^CSTMax. thermal power capture by generator g ∈𝒢^CST. [P]H_gInertia of a unit of generator g. [P]S_gMVA rating of a unit of generator g. [P]H_n,tMinimum synchronous inertia requirement of node n. [P]η_xEfficiency of component x. [P]p_p,t^pvAggregated PV power of prosumer p. [P]λFeed-in price ratio.[I]ŝ_gNumber of online units of generator g ∈𝒢^syn at start of horizon. [I]p̂_gPower dispatch of

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at start of horizon. [I]p̂_gPower dispatch of generator g at start of horizon. [I]û_g,tMinimum number of units of generator g ∈𝒢^syn required to remain online for time t<τ_g^u. [I]d̂_g,tMinimum number of units of generator g ∈𝒢^syn required to remain offline for time t<τ_g^d. [I]ê_gEnergy stored in TES of g ∈𝒢^𝒞𝒮𝒯 at start of horizon. [I]ê_sEnergy stored in storage plant s at start of horizon. [I]ê_p^bBattery state of charge for prosumer p at start of horizon.§ INTRODUCTIONPower systems

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p at start of horizon.§ INTRODUCTIONPower systems worldwide are moving away from domination by large-scale synchronous generation and passive consumers. Instead, in future grids[We interpret a future grid to mean the study of national grid type structures with the transformational changes over the long-term out to 2050.] new actors, such as variable renewable energy sources (RES)[For the sake of brevity, by RES we mean “unconventional” renewables like wind and solar, but excluding conventional

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like wind and solar, but excluding conventional RES, like hydro, and dispatchable unconventional renewables, like concentrated solar thermal.], price-responsive users equipped with small-scale PV-battery systems (called prosumers), demand response (DR), and energy storage will play an increasingly important role. Given this, in order for policy makers and power system planners to evaluate the integration of high-penetrations of these new elements into future grids, new simulation tools need to

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into future grids, new simulation tools need to be developed.Specifically, there is a pressing need to understand the effects of technological change on future grids, in terms of energy balance, stability, security and reliability, over a wide range of highly-uncertain future scenarios. This is complicated by the inherent and unavoidable uncertainty surrounding the availability, quality and cost of new technologies (e.g. battery or photo-voltaic system costs, or concentrated solar thermal (CST)

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system costs, or concentrated solar thermal (CST) generation operating characteristics) and the policy choices driving their uptake. The recent blackout in South Australia <cit.> serves as a reminder that things can go wrong when the uptake of new technologies is not planned carefully. Future grid planning thus requires a major departure from conventional power system planning, where only a handful of the most critical scenarios are analyzed. To account for a wide range of possible future

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To account for a wide range of possible future evolutions, scenario analysis has been proposed in many industries, e.g. in finance and economics <cit.>, and in energy <cit.>. In contradistinction to power system planning, where the aim is to find an optimal transmission and/or generation expansion plan, the aim of scenario analysis is to analyze possible evolution pathways to inform power system planning and policy making. Given the uncertainty associated with long-term projections, the focus

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associated with long-term projections, the focus of future grid scenario analysis is limited only to the analysis of what is technically possible, although it might also consider an explicit costing <cit.>. In more detail, existing future grid feasibility studies have shown that the balance between demand and supply can be maintained even with high penetration of RESs by using large-scale storage, flexible generation, and diverse RES technologies <cit.>. However, they only focus on balancing

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<cit.>. However, they only focus on balancing and use simplified transmission network models (either copper plate or network flow; a notable exception is the Greenpeace pan-European study <cit.> that uses a DC load flow model). This ignores network related issues, which limits these models' applicability for stability assessment. To the best of our knowledge, the Future Grid Research Program, funded by the Australian Commonwealth Scientific and Industrial Research Organisation (CSIRO) is the

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Industrial Research Organisation (CSIRO) is the first to propose a comprehensive modeling framework for future grid scenario analysis that also includes stability assessment. The aim of the project is to explore possible future pathways for the evolution of the Australian grid out to 2050 by looking beyond simple balancing. To this end, a simulation platform has been proposed in <cit.> that consists of a market model, power flow analysis, and stability assessment, Fig. <ref>. The platform has

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assessment, Fig. <ref>. The platform has been used, with additional improvements, to study fast stability scanning <cit.>, inertia <cit.>, modeling of prosumers for market simulation <cit.>,impact of prosumers on voltage stability <cit.>, and power system flexibility using CST <cit.> and battery storage <cit.>. In order to capture the inter-seasonal variations in the renewable generation, computationally intensive time-series analysis needs to be used. A major computational bottleneck of the

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be used. A major computational bottleneck of the framework is the market simulation.Within this context, the contribution of this paper is to propose a unified generic market simulation tool (MST) based on a unit commitment (UC) problem suitable for future grid scenario analysis, including stability assessment. The tool incorporates the following key features: * market structure agnostic modeling framework, * integration of various types and penetrations of RES and emerging demand-side

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and penetrations of RES and emerging demand-side technologies, * generic demand model considering the impact of prosumers, * explicit network representation, including HVDC lines, using a DC power flow model, * explicit representation of the number of online synchronous generators, * explicit representation of system inertia and reactive power support capability of synchronous generators, * computational efficiency with sufficient accuracy. The presented model builds on our existing

1701.07941v2_24

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The presented model builds on our existing research <cit.> and combines all these in a single coherent formulation.In more detail, to reduce the computational burden, the following techniques are used building on the methods proposed in <cit.>: * unit clustering, * rolling horizon approach, * constraint clipping.The computational advantages of our proposed model are shown on a simplified 14-generator model of the Australian National Energy Market (NEM) as a test grid <cit.>. Four cases for

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(NEM) as a test grid <cit.>. Four cases for different RES penetration are run for one to seven days horizon length, and computational metrics are reported. To reflect the accuracy of the proposed MST, system inertia and voltage stability margins are used as a benchmark. In simulations, RES and load traces are taken from the National Transmission Network Developed Plan (NTNDP) data, provided by the Australian Energy Market Operator (AEMO) <cit.>.The remainder of the paper is organized as

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remainder of the paper is organized as follows: Literature review and related work are discussed in Section II, while Section III details the MST. A detailed description of the simulation setup is given in Section IV. In Section V results are analyzed and discussed in detail. Finally, Section VI concludes the paper. § RELATED WORKIn order to better explain the functional requirements of the proposed MST, we first describe the canonical UC formulation. An interested reader can find a

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UC formulation. An interested reader can find a comprehensive literature survey in <cit.>.§.§ Canonical Unit Commitment FormulationThe UC problem is an umbrella term for a large class of problems in power system operation and planning whose objective is to schedule and dispatch power generation at minimum cost to meet the anticipated demand, while meeting a set of system-wide constraints. In smart grids, problems with a similar structure arise in the area of energy management, and they are

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in the area of energy management, and they are sometimes also called UC <cit.>. Before deregulation, UC was used in vertically integrated utilities for generation scheduling to minimize production costs. After deregulation, UC has been used by system operators to maximize social welfare, but the underlying optimization model is essentially the same.Mathematically, UC is a large-scale, nonlinear, mixed-integer optimization problem under uncertainty. With some abuse of notation, the UC

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uncertainty. With some abuse of notation, the UC optimization problem can be represented in the following compact formulation <cit.>:minimize_𝐱_c, 𝐱_bf_c(𝐱_c) + f_b(𝐱_b)subjecttog_c(𝐱_c) ≤𝐛 g_b(𝐱_b) ≤𝐜 h_c(𝐱_c) + h_b(𝐱_b)≤𝐝𝐱_c∈ℝ^+, 𝐱_b∈{ 0,1 } Due to the time-couplings, the UC problem needs to be solved over a sufficiently long horizon. The decision vector 𝐱 = {𝐱_c, 𝐱_b} for each time interval consist of continuous and binary variables. The continuous variables, 𝐱_c, includegeneration dispatch

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variables, 𝐱_c, includegeneration dispatch levels, load levels, transmission power flows, storage levels, and transmission voltage magnitudes and phase angles. The binary variables, 𝐱_b, includes scheduling decisions for generation and storage, and logical decisions that ensure consistency of the solution. The objective (<ref>) captures the total production cost, including fuel costs, start-up costs and shut-down costs. The constraints include, respectively: dispatch related constraints such as

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dispatch related constraints such as energy balance, reserve requirements, transmission limits, and ramping constraints (<ref>); commitment variables, including minimum up and down, and start-up/shut-down constraints (<ref>); and constraints coupling commitment and dispatch decisions, including minimum and maximum generation capacity constraints (<ref>).The complexity of the problem stems from the following: (i) certain generation technologies (e.g. coal-fired steam units) require long start-up

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coal-fired steam units) require long start-up and shut-down times, which requires a sufficiently long solution horizon; (ii) generators are interconnected, which introduces couplings through the power flow constraints; (iii) on/off decisions introduce a combinatorial structure; (iv) some constraints (e.g. AC load flow constraints) and parameters (e.g. production costs) are non-convex; and (v) the increasing penetration of variable renewable generation and the emergence of demand-side

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generation and the emergence of demand-side technologies introduce uncertainty. As a result, a complete UC formulation is computationally intractable, so many approximations and heuristics have been proposed to strike a balance between computational complexity and functional requirements. For example, power flow constraints can be neglected altogether (a copper plate model), can be replaced with simple network flow constraints to represent critical inter-connectors, or, instead of (non-convex)

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inter-connectors, or, instead of (non-convex) AC, a simplified (linear) DC load flow is used. §.§ UC Formulations in Existing Future Grid StudiesIn operational studies: the nonlinear constraints, e.g. ramping, minimum up/down time (MUDT) and thermal limits are typically linearized; startup and shutdown exponential costs are discretized, and; non-convex and non-differentiable variable cost functions are expressed as piecewise linear function <cit.>. In planning studies, due to long horizon

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<cit.>. In planning studies, due to long horizon lengths, the UC model is simplified even further. For example: combinatorial structure is avoided by aggregating all the units installed at one location <cit.>; piecewise linear cost functions and constraints are represented by one segment only; some costs (e.g. startup, shutdown and fix costs) are ignored; a deterministic UC with perfect foresight is used, and; non-critical binding constraints are omitted <cit.>[An interested reader can refer to

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omitted <cit.>[An interested reader can refer to <cit.> for a discussion on binding constraints elimination for generation planning.]. To avoid the computational complexity associated with the mixed integer formulation, a recent work <cit.> has proposed a linear relaxation of the UC formulation for flexibility studies, with an accuracy comparable to the full binary mixed integer linear formulation. In contrast to operation and planning studies, the computational burden of future grid scenario

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the computational burden of future grid scenario analysis is even bigger, due to a sheer number of scenarios that need to be analyzed, which requires further simplifications. For example, the Greenpeace study <cit.> uses an optimal power flow for generation dispatch and thus ignores UC decisions. Unlike the Greenpeace study, the Irish All Island Grid Study <cit.> and the European project e-Highway2050 <cit.> ignore load flow constraints altogether, however they do use a rolling horizon UC, with

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however they do use a rolling horizon UC, with simplifications. The Irish study, for example doesn't put any restriction on the minimum number of online synchronous generators to avoid RES spillage, and the e-Highway2050 study uses a heuristics to include DR. The authors of the e-Highway2050 study, however, acknowledge the size and the complexity of the optimization framework in long term planning, and plan to develop new tools with a simplified network representation <cit.>.In summary, a UC

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network representation <cit.>.In summary, a UC formulation depends on the scope of the study. Future grid studies that explicitly include stability assessment bring about some specific requirements that are routinely neglected in the existing UC formulations, as discussed next.§ MARKET SIMULATION TOOL §.§ Functional RequirementsThe focus of our work is stability assessment of future grid scenarios. Thus, MST must produce dispatch decisions that accurately capture the kinetic energy stored in

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accurately capture the kinetic energy stored in rotating masses (inertia), active power reserves and reactive power support capability of synchronous generators, which all depend upon the number of online units and the respective dispatch levels. For the sake of illustration, consider a generation plant consisting of three identical (synchronous) thermal units, with the following characteristics: (i) constant terminal voltage of 1pu; (ii) minimum technical limit P_min = 0.4pu; (iii) power

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technical limit P_min = 0.4pu; (iii) power factor of 0.8; (iv) maximum excitation limit E_fd^max = 1.5pu; and (v) normalized inertia constant H = 5. We further assume that in the over-excited region, the excitation limit is the binding constraint, as shown in Fig. <ref>. Observe that the maximum reactive power capability depends on the active power generated, and varies between Q_n at P_max = 1pu and Q_max at P_min. We consider three cases defined by the total active power generation of the

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by the total active power generation of the plant: (i) 0.8pu, (ii) 1.2pu, and (iii) 1.6pu.The three scenarios correspond to the rowsin Fig. <ref>, which shows the active power dispatch level P, reactive power support capability Q, online active power reserves R, and generator inertia H.The three columns show feasible solutions for three different UC formulations: all three units are aggregated into one equivalent unit (AGG), standard binary UC (BUC) when each unit is modeled individually, and

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(BUC) when each unit is modeled individually, and the proposed market simulation tool (MST). A detailed comparison of the three formulations is given in Section V. Although the results are self-explanatory, a few things are worth emphasizing. In case (i), aggregating the units into one equivalent unit (AGG) results in the unit being shut down due to the minimum technical limit. The individual unit representation (BUC), on the other hand, does allow the dispatch of one or two units, but with