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Δ δ̣ ρ̊ π α γ → σ β̱ ϵ Γ ω łλ ϕ ψ μ τ χ̧1cm Diversity and coevolutionary dynamics in high-dimensional phenotype spaces Michael Doebeli^∗ & Iaroslav Ispolatov^∗∗^∗ Departments of Zoology and Mathematics, University of British Columbia,6270 University Boulevard, Vancouver B.C. Canada, V6T 1Z4; doebeli@zoology.ubc.ca ^∗∗ Departamento de Fisica, Universidad de Santiago de ChileCasilla 302, Correo 2, Santiago, Chile; jaros007@gmail.com December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================10mm 10 mm Supporting Material: Appendix A 10 mm RH: Diversity and coevolutionary dynamics 10 mm Corresponding author: Michael Doebeli, Department of Zoology and Department of Mathematics,University of British Columbia, 6270 University Boulevard, Vancouver B.C. Canada, V6T 1Z4, Email: doebeli@zoology.ubc.ca.Abstract 0.3cm We study macroevolutionary dynamics by extending microevolutionary competition models to long time scales. It has been shown that for a general class of competition models, gradual evolutionary change in continuous phenotypes (evolutionary dynamics) can be non-stationary and even chaotic when the dimension of the phenotype space in which the evolutionary dynamics unfold is high. It has also been shown that evolutionary diversification can occur along non-equilibrium trajectories in phenotype space.We combine these lines of thinking by studying long-term coevolutionary dynamics of emerging lineages in multi-dimensional phenotype spaces. We use a statistical approach to investigate the evolutionary dynamics of many different systems. We find: 1) for a given dimension of phenotype space, the coevolutionary dynamics tends to be fast and non-stationary for an intermediate number of coexisting lineages, but tends to stabilize as the evolving communities reach a saturation level of diversity; and 2) the amount of diversity at the saturation level increases rapidly (exponentially) with the dimension of phenotype space.These results have implications for theoretical perspectives on major macroevolutionary patterns such as adaptive radiation, long-term temporal patterns of phenotypic changes, and the evolution of diversity. 1 cm Keywords: Long-term evolution | Diversity and stability | Adaptive radiation § INTRODUCTIONOne of the fundamental problems in evolutionary biology is to understand how microevolutionary processes generate macroevolutionary patterns. In particular, the emergence of macroevolutionary changes in the speed of evolution <cit.>, and of macroevolutionary changes in patterns of species diversity <cit.> have long been of great interest. For example, <cit.> have recently proposed that over macroevolutionary time scales, relatively short intermittent bursts of high rates of evolutionary change should alternate with long periods of bounded phenotypic fluctuations.Also, there is much discussion about whether species diversity saturates over evolutionary time in a given environment <cit.>. Phylogenetic analysis has been used to shed light on these questions <cit.>, but mechanistic models in which short-term ecological interactions are extrapolated to yield long-term patterns of diversity and evolutionary change have only recently been developed. Most of these models have been used to study the long-term evolution of diversity by analyzing processes of community assembly emerging from short-term ecological dynamics <cit.>. In particular, these papers have mainly focussed on how diversity changes over time, but not on how the nature of the coevolutionary dynamics of a given set of coexisting species changes as the diversity changes. In fact, in all these models, the evolutionary dynamics for a fixed amount of diversity, i.e., for a given set of species, converge to an equilibrium. However, if one wants to understand macroevolutionary changes in the “tempo and mode” <cit.> of evolution, one not only needs to consider how diversity changes over evolutionary time, but also how such changes in diversity affect the nature of evolutionary dynamics <cit.>. Indeed, there is evidence from evolution experiments with microbes that evolutionary dynamics in more diverse communities are qualitatively different from the evolutionary dynamics in less diverse communities <cit.>. Here we present a theoretical investigation of the questions of how diversity affects the complexity of coevolutionary dynamics.In general, the number of different phenotypes that affect ecological and evolutionary processes is an important quantity. For example,determining the dimensionality of niche space in ecological food webs is a classical problem <cit.>, and it has recently been shown that including more phenotypic dimensions in models for community assembly has a strong effect on the structure of the emerging food webs <cit.>. Implicitly, the importance of the dimension of phenotype space is also acknowledged in phylogenetic research through the notion of “adaptive zones” <cit.>. In particular, it is thought that much of the extant diversity has evolved as a consequence of lineages entering new adaptive zones, which can be interpreted from the phenotypic perspective as an increase in the dimension of phenotype space. In general, given the large number of phenotypic properties that determine an individual's life history and ecology in almost any species, one would expect that ecological interactions are generally determined by many phenotypic properties, and that selection pressures emerging from ecological interactions in turn affect many phenotypes simultaneously. For example, comprehensive modelling of the metabolic network in E. coli cells comprises more than 2000 reactions <cit.>. These reactions are in turn controlled by thousands of genes in a complicated interaction network whose exact workings are largely unknown. Nevertheless, many of the genes contributing to this network of metabolic reactions will be under selection in any given environmental setting, and as a consequence, a large number of phenotypic properties have the potential to undergo evolutionary change. It is generally not known how exactly these phenotypic properties impinge on birth and death rates of individual organisms, and hence what exactly the ecological selection pressures are on these properties. Nevertheless, it seems clear that in general, many phenotypes will evolve at the same time, i.e., that evolution generally takes place in high-dimensional phenotype spaces.We have recently argued that if evolution takes place in high-dimensional phenotype spaces, then the evolutionary dynamics, that is, the phenotypic change over evolutionary time, can be very complicated, i.e., non-stationary and often chaotic <cit.>. In low-dimensional phenotype spaces, non-equilibrium evolutionary dynamics are less likely. However, if a species evolving on a simple attractor gives rise to diversification, the effectivedimensionality of the evolving system increases, as the species that emerge from diversification coevolve, driven by both intra- and interspecific ecological interactions.Thus the total dimensionality of the resulting dynamical system describing multispecies coevolution is the number of species times the dimensionality of the phenotype space in which each species evolves. Based on our earlier results <cit.>, one could then expect that due to the increase in dimensionality, diversificationleads to more complicated evolutionary dynamics in each of the coevolving species. On the other hand, as a multispecies community becomes more diverse and evolves towards saturation, the available niches tend to get filled, and hence evolutionary change has to become highly coordinated between interacting species and thus constrained, potentially leading to simplified evolutionary dynamics. It is thus unclear how the nature of the evolutionary dynamics changes as the pattern of diversity changes during community assembly.We investigate these issues by applying the framework of adaptive dynamics <cit.>to a general class of competition models. The main question we address is, how does the complexity of long-term coevolutionary dynamics depend on the diversity of the coevolving community? We show that in low-dimensional phenotype spaces, there is a humped-shaped relationship between diversity and the complexity of evolutionary dynamics: in communities with low diversity, coevolutionary dynamics are often simple, i.e., stationary in the long-time limit; for intermediate degrees of diversity, non-stationary (complex) coevolutionary dynamics are common, and each of the species in the community evolves on a complicated trajectory in phenotype space; and for high amounts of diversity, coevolutionary dynamics become simple again, i.e., stationary. In particular, as communities reach diversity saturation, e.g. through adaptive diversification <cit.>, coevolutionary dynamics change from complex to simple. Our results are relevant for a number of issues concerning patterns of macroevolution. For example, the results suggest that during processes of adaptive radiation <cit.>, evolutionary dynamics are more complicated early in the radiation than late in the radiation, a pattern that corresponds to the “early-burst” perspective of macroevolution that has attracted much attention in recent years <cit.>. Our results also show that the level at which diversity saturates depends on the dimensionality of phenotype space, with higher dimensions allowing for more diversity. This observation is in accordance with data from radiations in fishes <cit.> and points to the possibility of a microevolutionary mechanism for the “blunderbass theory" of temporal patterns of macroevolutionary changes and diversification <cit.>: if evolution operates on the dimension of phenotype space on a very slow time scale, then on shorter time scales diversity may saturate and thereby generate relatively stationary evolutionary dynamics, whereas on longer time scales the dimension of phenotype space may increase, e.g. due to gene duplications, thus generating a new burst of non-equilibrium (co-)evolutionary dynamics until the diversity reaches a new saturation level. Such patterns of intermittent bursts have recently been found in the phylogenies of birds and echinoids <cit.>, and the bursts have been attributed to the evolution of flight capabilities and of novel feeding techniques, respectively, both of which can be interpreted as an increase in the dimensionality of the relevant phenotype space. This perspective may also shed light on the question of whether diversity saturates or not <cit.>: diversity may saturate for a given dimension of phenotype space, but evolutionary innovation in the form of new phenotypic dimensions may intermittently generate room for additional bouts of evolutionary diversification. § METHODS §.§ Single-cluster adaptive dynamicsAs in <cit.>, we study a general class of models for frequency-dependent competition in which ecological interactions are determined by d-dimensional phenotypes, where d≥1.For simplicity, we consider homogeneous systems, so no spatial coordinates are included.The ecological interactions are described by a competition kernel α(𝐱, 𝐲) and by a carrying capacity K(𝐱), where 𝐱,𝐲∈ℝ^d are the d-dimensional continuous phenotypes of competing individuals. The competition kernel αmeasures thecompetitive impact that an individual of phenotype 𝐱 has on an individual of phenotype 𝐲, and we assume thatα(𝐱, 𝐱)=1 for all 𝐱. Assuming logistic ecological dynamics, K(𝐱) is then the equilibrium density of a population that is monomorphic for phenotype 𝐱. The adaptive dynamics of the phenotype 𝐱 is a system of differential equations for d𝐱/dt. To derive the adaptive dynamics, one defines the invasion fitness f(𝐱, 𝐲) as the per capita growth rate of a rare mutant phenotype 𝐲 in the monomorphic resident 𝐱 population that is at its ecological equilibrium K(𝐱):f(𝐱, 𝐲) = 1 - α(𝐱, 𝐲) K(𝐱)/K(𝐲). The expression for the invasion fitness reflects the fact that the growth rate of the mutant 𝐲 is negatively affected by the effective density experienced by the mutant, α(𝐱, 𝐲) K(𝐱), discounted by the carrying capacity K(𝐲) of the mutant (see <cit.> for more details).Note that f(𝐱, 𝐱)=0 for all 𝐱. The invasion fitness f(𝐱, 𝐲) gives rise to the selection gradients in the i=1,...,d phenotypic components: s_i(𝐱) ≡∂ f(𝐱, 𝐲)/∂ y_i |_𝐲=𝐱 =- ∂α(𝐱, 𝐲)/∂ y_i |_𝐲=𝐱 + ∂ K(𝐱)/∂ x_i1/K(𝐱), The selection gradients in turn define the adaptive dynamics as a system of differential equations on phenotype space ℝ^d, which is given byd𝐱/dt = 𝐌(𝐱)·𝐬(𝐱).Here 𝐬(𝐱) is the column vector (s_1(𝐱),...,s_d(𝐱)), and 𝐌(𝐱) is the mutational variance-covariance matrix. In this matrix, the diagonal elements contain information about the size and rate of mutations in each of the phenotypic dimensions, whereas the off-diagonal elements contain information about the covariance between mutations in two different phenotypic dimensions. This matrix essentially captures “evolvability” of a population andgenerally depends on the current resident phenotype 𝐱, and influences the speed and direction of evolution. For simplicity, we assume here that this matrix is the identity matrix. For more details on the derivation of the adaptive dynamics (<ref>) we refer to a large body of primary literature (e.g. <cit.>). We note that the adaptive dynamics (<ref>) can be derived analytically as a large-population limitof an underlying stochastic, individual-based model that is again defined based on the competition kernel α(𝐱, 𝐲) and the carrying capacity K(𝐱) <cit.>. Specifically, here we consider a class of systems that are defined by competition kernels of the form(𝐱,𝐲)=exp [∑_i,j=1^d b_ij(x_i-y_i)x_j -∑_i=1^d(x_i-y_i)^2/2_i^2]. Here the coefficients b_ij in the first sum on the right hand side are arbitrary and correspond to the simplest form of a generic, non-symmetric competition kernel that can generate non-stationary evolutionary dynamics. It can be interpreted as the lowest-order (non-trivial) term from a Taylor expansion of an unknown non-symmetric competition function. Adaptive dynamics of asymmetric competition has been studied quite extensively (e.g. <cit.>), and is necessary to generate single-species non-equilibrium dynamics in high-dimensional phenotype spaces <cit.>. The second sum on the right hand side represents “Gaussian competition”, according to which the competitive impact between individuals increases with phenotypic similarity between the competing individuals. The parameters _i measure how fast the effect of competition declines as phenotypic distance in the i-component increases. For the carrying capacity we assume K(𝐱)=exp(-∑_i^d x_i^4/4). This implies that the carrying capacity imposes an element of stabilizing selection for the phenotype 𝐱=0, at which the carrying capacity is maximal. Thus, the frequency-dependent component of selection is generated by the competition kernel, whereas the frequency-independent component of selection is due to the carrying capacity. With these assumptions, the adaptive dynamics (<ref>) become d x_i/dt=∑_i=1^d b_ijx_j - x_i^3,i=1,...,d. We note thatthe terms -x_i^3 in (<ref>) are due to the carrying capacity and serve to contain the trajectories of (<ref>) in a bounded domain of phenotype space. Also, the Gaussian part of the competition kernel does not affect the adaptive dynamics of monomorphic populations, i.e., the _i do not appear in (<ref>), because the Gaussian part always has a maximum at the current resident, and hence the corresponding first derivative in the selection gradient (<ref>) is 0. The system of ODEs(<ref>) describes the trajectory of an evolving monomorphic population in phenotype space ℝ^d. In <cit.> we have shown that for general competition kernelssuch trajectories can be very complicated, particularly when the dimension d is large. With complex evolutionary dynamics, trajectories can be quasi-periodic or chaotic, and typically visit many different regions of phenotype space over evolutionary time. When d is low the dynamics tend to be simpler, and often converge to an equilibrium attractor.We can assess the likelihood of equilibrium dynamics for a given dimension d by choosing the d^2 coefficientsb_ij in (<ref>) randomly and independently, e.g. from a normal distribution with mean 0 and variance 1, solving the resulting adaptive dynamics (<ref>) and checking whether it converges to an equilibrium. If this is done repeatedly, we can approximate the probability of equilibrium dynamics as the fraction of runs that converged to an equilibrium. For d=1 the probability of equilibrium dynamics is of course 1, and for d=2,3,4, the resulting probabilities of equilibrium dynamics are approximately 85%, 81% and 74%, respectively. These are the dimensions that we will primarily use in the analysis presented below, but we note that the probability of equilibrium dynamics goes to 0 for large d <cit.>. §.§ Multi-cluster adaptive dynamicsHere we are interested in the question of how diversification and subsequent coexistence of species (also called phenotypic clusters or simply clusters through the text) affects the evolutionary dynamics. While the Gaussian term in the competition kernel (<ref>) does not affect the adaptive dynamics of single monomorphic populations, this term is crucial for determining whether evolutionary diversification occurs. For one-dimensional phenotype spaces (d=1) this is very well known and is encapsulated in the concept of evolutionary branching <cit.>. An evolutionary branching point is an equilibrium point of (<ref>) that is both an attractor for the adaptive dynamics and a fitness minimum. The reason that such points exist in the competition models considered here is precisely that the Gaussian term does not affect the adaptive dynamics, but does affect the curvature of the fitness landscape, i.e., the second derivative of the invasion fitness (<ref>). In particular, small enough _i's in the Gaussian term will make any equilibrium point a fitness minimum, and hence will give rise to evolutionary diversification. Evolutionary branching in scalar traits has been described in a plethora of different models (for an overview we refer to Eva Kisdi's website at the Department of Mathematics and Statistics at the University of Helsinki, http://www.mv.helsinki.fi/home/kisdi/addyn.htm). In high-dimensional phenotype spaces, equilibrium points of (<ref>) can also be fitness minima along some directions in phenotype space. For this to happen the Hessian matrix of second derivatives of the invasion fitness (<ref>), evaluated at the equilibrium, must have positive eigenvalues. Indeed, in higher dimensional phenotype spaces the conditions for the existence of positive eigenvalues of this Hessian matrix, and hence for diversification, generally become less stringent <cit.>. Importantly, evolutionary diversification can also occur from non-equilibrium adaptive dynamics trajectories <cit.>. If the adaptive dynamics (<ref>) exhibit non-equilibrium dynamics, the crucial quantity determining whether diversification occurs is again the Hessian matrix of second derivatives of the invasion fitness (<ref>), but now restricted to the subspace of phenotype space that is orthogonal to the selection gradient <cit.>. Essentially, diversification can occur in orthogonal directions in which this Hessian has positive curvature, and hence in which the invasion fitness has a minimum. Because the population is still evolving along the selection gradient, elucidating the exact conditions for diversification requires a careful analysis <cit.>. In the present context, the implication of these results is that, just as with equilibrium adaptive dynamics, diversification can occur along non-equilibrium trajectories of (<ref>) if the _i in the Gaussian term of the competition kernel are small enough, i.e., if the frequency dependence generated by Gaussian competition is strong enough <cit.>. To investigate the process of diversification and the subsequent coevolutionary dynamics, we extend the adaptive dynamics (<ref>) to several coexisting phenotypic clusters as follows. We assume that an evolving community consists of m monomorphic populations, each given by a phenotype 𝐱_r, r=1,...,m, with phenotypic components x_ri, i=1,...d (where d is the dimension of phenotype space). Let N_r be the population density of cluster 𝐱_r. Then the ecological dynamics of the m clusters are given by the system of logistic differential equations d N_r(t)/ d t = N_r( t)( 1 - ∑_s=1^m (𝐱_s, 𝐱_r) N_s ( t)/K(𝐱_r)),r=1,...,m. Let N_r^*, r=1,...,m denote the equilibrium of system (<ref>) (more generally, for the purposes of deriving the adaptive dynamics, the quantities N_r^* are suitable time averages of population densities over the ecological attractor of (<ref>); however, our extensive numerical simulations indicated that (<ref>) always converges to an equilibrium). Making the traditional adaptive dynamics assumption that ecological dynamics occur on a faster time scale than evolutionary dynamics, we calculate the invasion fitness function in cluster r based on the densities N_r^* of the various clusters:f(𝐱_1,...,𝐱_m,𝐱_r')=1 - ∑_s=1^m (𝐱_s,𝐱_r')N_s^*/K(𝐱_r'). Here 𝐱_1,...,𝐱_m describe the phenotypic state of the resident population, and 𝐱_r' denotes the mutant trait in cluster r, r=1,...,m. Taking the derivative of (<ref>) with respect to 𝐱_r' and evaluating it at the resident, 𝐱_r'=𝐱_r, yields the components of the selection gradient 𝐬_r for the cluster r as: s_ri= ∑_sN_s^*(-1/K(𝐱_r).∂(𝐱_s,𝐱_r')/∂x_ri'|_𝐱_r'=𝐱_r +(𝐱_s,𝐱_r)/K^2(𝐱_r)∂K(𝐱_r)/∂ x_ri),i=1,...,d. For coevolutionary adaptive dynamics, one has to take into account that the rate of mutations in each evolving phenotypic cluster is proportional to the current population size of that cluster <cit.>, and hence that the speed of evolution is influenced by the population size. In the single-cluster system such consideration only rescales time without affecting the geometry of the trajectory and thus is usually ignored. However, in the multi-cluster system, instead of assuming that the mutational process is described by the identity matrix as in (<ref>), we now assume that in each cluster r, the mutational variance-covariance matrix M_r is a diagonal matrix with entries N_r^*. This generates the following d· m differential equations describing the adaptive dynamics in the coevolving community: d x_ri/dt= N_r^* s_ri, i=1,…,d, r=1,…, m. For the multicluster adaptive dynamics, the equation (<ref>,<ref>,<ref>) replacetheir single-cluster analogs (<ref>,<ref>,<ref>). It is important to note that the Gaussian part of the competition kernelnot only affects whether diversification occurs, but in contrast to the adaptive dynamics of single monomorphic populations, the Gaussian term will indeed affect the coevolutionary adaptive dynamics (<ref>) of the phenotypic clusters that coexist after diversification has occurred, because it affects both the ecological dynamics (<ref>) and the selection gradient (<ref>). §.§ Numerical procedureTo study diversification and subsequent multi-cluster adaptive dynamics, we implemented the following iterative numerical scenario:0.5 cm Step 1: Each simulation run is initiated with a randomly generatedd × d matrix of the coefficientsb_ij forthe competition kernel (<ref>). The coefficients are drawn from a Gaussian distribution with zero mean and d^-1/2 variance. As explained in <cit.>, this is done to keep the sum of the d terms ∑_j=1^d b_ij x_j in (<ref>) of orderx_i, i.e. independent of d. Then a certain number of clusters, given by a parameter m_0, each with population size of order 1, are randomly placednear the phenotype 0, i.e., near the maximum of the carrying capacity. 0.5 cm Step 2: For a given set of phenotypic clusters, the population dynamics of all clusters is solved using the ecological dynamics (<ref>). The system of differential equations is integrated using a 4th-order Runge-Kutta algorithm for ∼ 10^3 time steps of duration dt ∼ 10^-2 to ensure convergence to the equilibrium (or, in case there is no such convergence, to ensure a correct calculation of the time average of the various population densities). If the population density of a givencluster falls below the threshold N_min∼10^-8, the cluster is eliminated from the system. During the ecological dynamics the evolutionary dynamics is frozen and evolutionary time does not advance.0.5 cm Step 3: After calculating the N_r^*, r=1,...,m (where m is the current number of clusters), the adaptive dynamics of the phenotypes of the clusters is advanced via (<ref>,<ref>) using a 4th-order Runge-Kutta algorithm with a typical time-step d∼ 10^-2, by which the evolutionary time is advanced as well. After this evolutionary time step, the ecological dynamics are recalculated, potentially preceded by the following step 4, which is only performed if the corresponding time condition is satisfied. 0.5 cm Step 4: The level of diversity, i.e., the number of clusters in the system, is controlled as follows. Each _c time units the distances between clusters are assessed. If the distance between two or more clusters is below a thresholdx ∼ 10^-3, these clusters are merged, preserving the total population size of the merged clusters and the position of their centre of mass.Immediately after this comparison step, the total number of clusters is compared to the target number of clusters, which is given by a system parameter m_max. If the current number of clusters is below m_max, a new cluster is created by randomly picking an existing cluster,splitting it in half and separating the two new clusters in a random direction in phenotype space bythe distanceof the merging threshold, x. 0.5 cm Step 5: In our simulations, we take measurements at regular time intervals (ranging from _m∼ 1 - 10 time units). One of the main quantities of interest is the average per capita evolutionary speed v in the evolving community,which is the average of the norms of the vectors of trait variation (evolution) rates in each cluster, weighted by the cluster population size,computed as v=∑_r=1^mN_r√(∑_i=1^d (d x_ri/dt)^2)/∑_r=1^m N_rThis quantity is a strong indicator of the nature of the evolutionary dynamics of the coevolving system. In particular, our very extensive numerical simulations indicate that when the average speed falls below 10^-2, then the system eventually exhibits equilibrium evolutionary dynamics. In contrast, when the average evolutionary speed remains high, the coevolving system tends to exhibit complicated, non-equilibrium dynamics, with the majority of the clusters exhibiting large fluctuations in phenotype space over evolutionary time. An example of such non-equilibrium coevolution is given in the next section. Other measurements include the position and population size of all clusters in the system,and the number of “distinct” clusters separated by a “visible” distance X=0.1. These measurements can also be averaged over time. 1cm For any given simulation run initiated by step 1 above, steps 2-5 were repeated iteratively until a specified final simulation time is reached, or until evolution comes to a halt, which by our definition occurs when the average evolutionary speed falls below a threshold, v<10^-4.Our general approach consisted of simulating many different systems according to the above scheme, and then computing statistical characteristics such as the fraction of runs that result in non-equilibrium dynamics, or the average evolutionary speed as a function of the level of diversity (see Results section). One crucial feature of our algorithm is the periodic generation of new clusters in step 4, which mimics diversification events, i.e., evolutionary branching. Diversification is thus modeled by simply adding new phenotypic clusters at certain points in time and close to existing clusters. This mimics the sympatric split of an ancestral lineage. Sympatric diversification is a theoretically robust phenomenon <cit.> and our procedure represents a shortcut for this phenomenon necessitated by computational feasibility. If such splitting is not feasible given the current ecological circumstance, the new cluster will not diverge phenotypically from the ancestor, and hence will be merged again with the ancestor (see below). Alternatively, newly generated clusters may go extinct ecologically. In either case, speciation was not successful. Thus, in our models it is the ecological circumstances that determine whether speciation can occur or not, but the process of speciation itself (i.e., the splitting) is performed in a simplified manner. If speciation is successful and the newly generated clusters diverge and persist ecologically, then diversity has increased (unless other clusters go extinct). We note that by construction, the maximal level of diversity in a given simulation run, i.e., the number of different clusters, cannot exceed the parameter m_max. Therefore, this parameter allows us to control the level of diversity in a given simulation.There are in principle other, less artificial ways to model diversification. In particular, stochastic, individual-based based models and partial differential equation models <cit.> have been used to describe the evolutionary dynamics of phenotype distributions. In such models, diversification is an emergent property that is reflected in the formation of new modes in the evolving phenotype distributions. While these techniques are very useful in general, they are currently not computationally feasible for the statistical approach that we employed here, which requires systematic simulation of many different systems. Also, they would not allow for control of the level of diversity, as the number of phenotypic modes would simply be an emergent property of the evolving system. Nevertheless, we have used these alternative techniques to illustrate the robustness of salient results using particular examples. A more detailed description of these techniques is given in theAppendix. Another alternative would be to assume that new clusters (species) are assigned phenotypes that are chosen randomly in phenotype space, rather than close to an existing cluster. This could correspond to immigration of new species into an existing community. However, we would not expect this to affect our main results, because with complicated evolutionary dynamics, the initial phenotypic position of a given cluster becomes irrelevant after some time. Finally, we note that the merging of clusters (species) is done solely for computational reasons and has no biological meaning (apart from designating organisms that are closely related and phenotypically very close as belonging to the same species). Merging of clusters only occurs shortly after a new cluster is seeded close to an existing one, and only if the new cluster does not diverge from the existing one (i.e., only if the ecological conditions for diversification are not satisfied). If divergence is successful, the clusters will never again get close enough to other clusters to be merged because of the repelling force of frequency-dependent competition. Thus, the only function of merging is to prevent the number of clusters from artificially becoming very large.§ RESULTSThe parameter that controls the level of diversity in our simulations is m_max, which is the maximal number of different phenotypic clusters allowed to be present at any point in time in an evolving community (see step 4 in the Methods section). Our first result is obtained by allowing this parameter to be very large, so that we can estimate the number of clusters that eventually coexist by simply running the simulations for a long time and recording the number of clusters at which the diversity equilibrates.We denote by M_,d the equilibrium number of clusters for a given phenotypic dimension d and strength of the Gaussian componentin the competition kernel (<ref>).We found that such equilibrium level of diversity increases exponentially withthe dimension d of phenotype space, and decreases with the strength (Figure 1).Here and below we assume for simplicity that the _i are the same in all phenotypic directions, _i= for i=1,...,d. In theAppendix we indicate scaling relationships that hold for M_,d as functions of the parametersand d. In general, diversity is only maintained if ≲1, which is roughly the scale of the phenotypic range set by the carrying capacity (<ref>). Only if ≲1, the equilibrium level of diversity increases exponentially with increasing dimension of phenotype space, Figure 1. Our main results are now obtained based on the observation that by fixing the parameter m_max at a value ≤ M_,d for a given d and , the community will typically evolve to a diversity level m_cluster of approximately m_max. That is, if the diversity is constrained to be below the maximal level of diversity possible for a given set of parameters, then the diversity will typically evolve to the value set by the constraint. Note that this is an “average” statement about many simulations runs, i.e., many different choices of the coefficients b_ij and stochastic realizations of cluster splitting. While some simulation runs will result in a diversity that is lower than m_max (which may reflect an intrinsic state of the system for the given set of coefficients, or a long-living metastable state which has not yet reached its full diversity), most runs will evolve to the level of diversity that is prescribed by this parameter.This allows us to then assess, for a given level of diversity, the nature of the coevolutionary dynamics that unfolds in communities with that level of diversity. Two paradigmatic examples are shown in Figure 2. We first set the level of diversitym_max=12, which is far below the saturation level M_σ,d for the given system. Starting from very few clusters the diversity quickly evolves to the level set by m_max, and the coexisting clusters then exhibit complicated, non-stationary evolutionary dynamics, with all clusters undergoing sustained and irregular fluctuations in phenotype space (Fig. 2a). This type of complicated dynamics is characterized by average evolutionary speeds v>10^-2. In the same system, but now with a value of m_max that lies above the saturation level M_σ,d, the diversity evolves to the saturation level, at which the community consists of ca. 30 coexisting phenotypic clusters (Fig. 2b). In this saturated state, the average evolutionary speed is much lower than 10^-2, and the community exhibits much more stationary coevolutionary dynamics (that would eventually converge to a coevolutionary equilibrium). Moreover, the saturated community exhibits a characteristic pattern of over-dispersion in phenotype space due to competitive repulsion caused by the Gaussian component of the competition kernel (see also Fig. A1 in the Appendix).To obtain a more systematic characterization of the coevolutionary dynamics as a function of the diversity of the evolving community, we ran, for a given dimension of phenotype space d and strength of competition , 100 simulations with randomly chosen coefficients b_ij for each m_max=1,...,M, where M is some number that is larger than the saturation level of diversity M_,d. For each run, we recorded the average per capita evolutionary speed v and the number of phenotypic clusters, i.e., the level of diversity, present at the end of 1000 evolutionary time units (averaged over the last 4 time units). We classified the dynamics into equilibrium dynamics if the average speed v was <10^-2, and non-equilibrium dynamics otherwise. As mentioned earlier, this was based on individual inspection of many simulation that ran longer than 1000 time units, which showed that the threshold 10^-2 is a very good indicator of whether the coevolutionary system eventually equilibrates.Our main results are shown in Figures 3 and 4. The general pattern is that the probability of non-equilibrium dynamics increases as diversity increases from single-cluster communities to communities with a few clusters (Figure 3).For intermediate diversity, the fraction of non-equilibrium dynamics remains high. For communities with high diversity, the fraction of non-equilibrium dynamics starts to decrease, and almost almost all communities with a diversity close to the saturation level M_,dexhibit equilibrium coevolutionary dynamics.To illustrates these trends, wegive a more detailed account of the average velocities v defined in (<ref>) in the coevolving communities (Fig. 4). It shows that there is an exponential decrease in the average speed as the diversity increases, and that there is a substantial fraction of low-diversity communities that exhibit equilibrium dynamics.The exact shape of these patterns depends on d and(Figures 3 and 4), but whenever diversification is possible, the overall trend is that non-equilibrium dynamics are most likely at intermediate levels of diversity, and that high levels of diversity tend to generate equilibrium coevolutionary dynamics. The patterns shown in Figs. 3 and 4 are based on many different simulated communities with different levels of diversity. However, similar patterns can be observed in simulations of single communities as they evolve from low to high diversity, i.e., as they undergo an adaptive radiation. Such a radiation, starting from a single phenotypic cluster, is shown in Fig. 5A.Over time the evolving community becomes more diverse due to adaptive diversification, and as a consequence the nature of the coevolutionary dynamics of the community changes. In the example shown in Figure 5A, the coevolutionary dynamics are fast for low to intermediate levels diversity, and then slow down as the community acquires more and more species, until eventually the community reaches a coevolutionary equilibrium at the diversity saturation level. Again, the slowdown of the evolutionary speed during an adaptive radiation appears to occur exponentially with an increase in diversity. This can also be seen by running a given community defined by a given set of coefficients b_ij for different values of the parameter m_max, determining the level of diversity possible in the evolving community. The evolutionary speed exponentially decreases with the diversity given by m_max (Fig. 5B). We currently do not have a mechanistic explanation for the exponential decay in evolutionary rates with increasing diversity. It is informative to watch the process of diversification and subsequent evolutionary slowdown unfold dynamically. To verify that the observed dynamical pattern is not an artifact of the adaptive dynamics approximation, we performed the individual-based and partial differential equation simulations of the same system. The movies in Videos in the Appendix, corresponding to the scenario used for Figures 2B and 5A, confirm that all three methods produce qualitatively similar evolutionary pictures. The detailed descriptions of the individual-based and partial differential equation methods are given in theAppendix. Another interesting, although perhaps not so surprising observation for single adaptive radiations concerns the rate of accumulation of new species in the evolving ecosystem. Figure 5C shows the number of species as a function of time during the adaptive radiation scenario used for Figure 5A, illustrating that the rate of diversification is highest at the beginning of the radiation, and then slows down as the community evolves towards the diversity saturation level. The details of these dynamics depend on system parameters, and in particular on the rate at which new species are introduced into the system, but the qualitative behaviour of diversification rates, which are initially high and then slow down, is common to all adaptive radiations generated by our models.§ DISCUSSION We investigated the expected long-term evolutionary dynamics resulting from competition for resources in models for gradual evolution in high-dimensional phenotype spaces. In reality, most organisms have many different phenotypic properties that impinge on their ecological interactions in generally complicated ways, and here we assumed that multi-dimensional phenotypes determine logistic ecological dynamics through the competition kernel and the carrying capacity. We then used a coevolutionary adaptive dynamics algorithm to extend the ecological dynamics to macroevolutionary time scales, and we used a statistical approach to capture general properties of the ensuing evolutionary dynamics.If the negative frequency-dependence generated by the competition kernel is strong enough, competition results in repeated adaptive diversification, and hence in communities of coevolving phenotypic species. By randomly choosing many different competition kernels, we showed that the complexity of the coevolutionary dynamics in such communities is expected to be highest for intermediate levels of phenotypic diversity. In particular, as the evolving communities increase in diversity towards the saturation level, i.e., the maximal number of different species that can coexist, the evolutionary dynamics becomes simpler, and communities at the saturation level are expected to exhibit a coevolutionary equilibrium. We also showed that the diversity saturation level increases exponentially with the dimension of phenotype space.We have used a statistical approach to determine the expected long-term evolutionary dynamics resulting from competition for resources. We have assumed that multi-dimensional phenotypes determine logistic ecological dynamics through the competition kernel and the carrying capacity, and we then used a coevolutionary adaptive dynamics algorithm to extend the ecological dynamics to macroevolutionary time scales.If the negative frequency-dependence generated by the competition kernel is strong enough, competition results in repeated adaptive diversification, and hence in communities of coevolving phenotypic species. By randomly choosing many different competition kernels, we showed that the complexity of the coevolutionary dynamics in such communities is expected to be highest for intermediate levels of phenotypic diversity. In particular, as the evolving communities increase in diversity towards the saturation level, i.e., the maximal number of different species that can coexist, the evolutionary dynamics becomes simpler, and communities at the saturation level are expected to exhibit a coevolutionary equilibrium. We also showed that the diversity saturation level increases exponentially with the dimension of phenotype space. Our interpretation of these findings is that in low-dimensional phenotype spaces such as the ones considered here, evolutionary dynamics of single species are expected to converge to an equilibrium <cit.>. However, as diversity increases, the different phenotypic clusters will “push” each other around evolutionarily due to frequency-dependent competition. This occurs mostly due to the repulsive nature of pairwise interaction induced by the Gaussian term in the competition kernel (<ref>): clusters that move further apart decrease competition felt from each other. For example,a splitting of a cluster stuck in an attractive fixed point of the adaptive dynamics creates two offspring which may become moving again if the repulsion between clusters is stronger than the attraction of the fixed point.As long as diversity is not very high, i.e., as long as there is enough available niche or unoccupied phenotype space, this typically results in non-equilibrium coevolutionary dynamics, thus leading to an increase in evolutionary complexity with phenotypic diversity. As the diversity keeps increasing towards saturation levels, which for each phenotypic dimension isdetermined roughly by the ratio of the widths of the carrying capacity and the competition kernel (see Video 2), the available carrying capacity niche gets filled, so that the evolving clusters “have nowhere to go” evolutionarily. An analogy with gas-liquid-solid phase transitions may illustrate this in the following way: As in the dynamics of molecules, the adaptive dynamics of phenotypic clusters contains a pairwise-repulsive term, which originates from the Gaussian term in the competition kernel. A few-cluster regime qualitatively corresponds to the gas phase, when the range of the repulsive interaction is significantly less than the typical distance between clusters. As the number and thus density of clusters increases, the repulsive interaction becomes more relevant, constraining the individual motion of clusters and resulting in a liquid-like behaviour, where clusters 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 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 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 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 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 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 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 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 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 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 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 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 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, 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 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 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 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 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 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 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, 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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: 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 ). 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 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

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

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 Engineering, University of Engineering and Technology Lahore, Lahore, Pakistan.December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== 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 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 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, 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. [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 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 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. 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 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 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 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 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) 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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) 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 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 <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 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 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 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 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 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 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 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 significantly different operational characteristics. In cases (ii) and (iii), the total inertia in the AGG formulation is much higher, which has important implications for frequency stability. A similar observation can be made for the reactive power support capability, which affects voltage stability. Also, dispatching power from all three units results in a significantly higher active power reserve. And last, a higher reactive power generation due to a lower P reduces the internal machine angle, which improves transient stability.In conclusion, a faithful representation of the number of online synchronous machines is of vital importance for stability assessment. An individual unit representation, however, is computationally expensive, so the computational burden should be reduced, as discussed in the following section. Next, an explicit network representation is required. An AC load flow formulation, however, is nonlinear (and non-convex), which results in an intractable mixed-integer nonlinear problem. Therefore, we use a DC load flow representation with a sufficiently small voltage angle difference on transmission lines. Our experience shows that an angle difference of 30 results in a manageable small number of infeasible operating conditions that can be dealt with separately.§.§ Computational Speedup The MST is based on the UC formulation using constant fixed, startup, shutdown and production costs. To improve its computational efficiency, the dimensionality of the optimization problem is reduced employing: (i) unit clustering <cit.> to reduce the number of variables needed to represent a multi-unit generation plant; (ii) a rolling horizon approach <cit.> to reduce the time dimension; and (iii) constraint clipping to remove most non-binding constraints.§.§.§ Unit ClusteringLinearized UC models are computationally efficient for horizons of up to a few days, which makes them extremely useful for operational studies. For planning studies, however, where horizon lengths can be up to a year, or more, these models are still computationally too expensive. Our work builds on the clustering approach proposed in <cit.>, where identical units at each generation plant are aggregated by replacing binary variables with fewer integer variables. The status of online units, startup/shutdown decisions and dispatched power are tracked by three integer variables and one continuous variable per plant per period, as opposed to three binary and one continuous variable per unit per period. Further clustering proposed in <cit.> is not possible in our formulation because of the explicit network representation required in the MST. §.§.§ Rolling Horizon Solving the UC as one block, especially for long horizons, is computationally too expensive. This can be overcome by breaking the problem into several smaller intervals called sub-horizons <cit.>. To ensure accuracy and consistency of the solution, a proper overlap between sub-horizons is maintained and the terminating state of the previous sub-horizon is used as the initial condition of the next sub-horizon. The minimum sub-horizon length depends on the time constants associated with the decision variables. While these might be in the order of hours for thermal power plants, they can be significantly longer for energy storage. Large-scale hydro dams, for example, require horizon lengths of several weeks, or even months. In our research, however, the sub-horizon length is up to a few days to cater for thermal energy storage (TES) of CST plants and battery storage. The optimization of hydro dams is not explicitly considered, however it can be taken into account heuristically, if needed.§.§.§ Constraint ClippingThe size of the problem can be reduced by removing non-binding constraints, which doesn't affect the feasible region. For instance, an MUDT constraint on a unit with an MUDT less than the time interval is redundant[This is especially the case when the time resolution is coarse. In our studies, the time step is one hour. In operational studies, where the resolution can be as short as five minutes, constraint clipping is less useful.]. Similarly, a ramp constraint for flexible units is redundant if the time step is sufficiently long. With a higher RES penetration, in particular, where backup generation is provided by fast-ramping gas turbines, this technique can significantly reduce the size of the optimization problem, and hence improves the computational performance due to a larger number of units with higher ramp rates and smaller MUDTs. It should be noted that optimization pre-solvers might not able to automatically remove these constraints. §.§ MST UC Formulation§.§.§ Objective functionThe objective of the proposed MST is to minimize total generation cost for all sub-horizons h:Ωminimize ∑_t∈𝒯^∑_g∈𝒢^( c_g^fix s_g,t +c_g^su u_g,t + c_g^sdd_g,t +c_g^varp_g,t), where Ω = {s_g,t,u_g,t,d_g,t,p_g,t, p_s,t, p_l,t} are the decision variables of the problem, and c_g^fix, c_g^su, c_g^sd, and c_g^var are fixed, startup, shutdown and variable cost, respectively. As typically done in planning studies <cit.>, <cit.>, the costs are assumed constant to reduce the computational complexity. The framework, however, also admits a piece-wise linear approximation proposed in <cit.>.§.§.§ System ConstraintsSystem Constraints[All the constraints must be satisfied in all time slots t, however, for sake of notational brevity, this is not explicitly mentioned.] include power balance constraints, power reserve and minimum synchronous inertia requirements.Power balance: Power generated at node n must be equal to the node power demand plus the net power flow on transmission lines connected to the node:∑_g∈𝒢_n^p_g,t = ∑_c ∈𝒞_np_c,t^ + ∑_p ∈𝒫_n p_p,t^g+ - ∑_p ∈𝒫_n p_p,t^g- + ∑_s ∈𝒮_np_s,t + ∑_l ∈ℒ_n(p_l,t +Δ p_l,t),where 𝒢_n, 𝒞_n, 𝒫_n, 𝒮_n, ℒ_n represent respectively the set of generators, consumers, prosumers[Price-responsive users equipped with small-scale PV-battery systems.], utility storage plants and lines connected to node n. Power reserves: To cater for uncertainties, active power reserves provided by synchronous generation g ∈𝒢^syn are maintained in each region r:∑_g ∈{ (𝒢^syn-𝒢^CST) ∩𝒢^r} (p_g s_g,t - p_g,t) +∑_g ∈{𝒢^CST∩𝒢^r}min(p_g s_g,t - p_g,t,e_g,t-p_g,t)≥∑_n ∈𝒩_r^p_n,t^r.For synchronous generators other than concentrated solar thermal (CST), reserves are defined as the difference between the online capacity and the current operating point. For CST, reserves can either be limited by their online capacity or energy level of their thermal energy system (TES). Variable s_g,t in (<ref>) represents the total number of online units at each generation plant, and 𝒢^r and 𝒩_r represent the sets of generators and nodes in region r, respectively.Minimum synchronous inertia requirement:To ensure frequency stability, a minimum level of inertia provided by synchronous generation must be maintained at all times (more details are available in <cit.>) in each region r:∑_g ∈{𝒢^syn∩𝒢^r}^ s_g,tH_g S_g ≥∑_n ∈𝒩_r^H_n,t. §.§.§ Network constraintsNetwork constraints include DC power flow constraints and thermal line limits for AC lines, and active power limits for HVDC lines.Line power constraints:A DC load flow model is used for computational simplicity for AC transmission lines[A sufficiently small (∼30) voltage angle difference over a transmission line is used to reduce the number of nonconvergent AC power flow cases.]:p_l,t^x,y = B_l(δ_x,t - δ_y,t),l ∈ℒ^𝒜𝒞,where the variables δ_x,t and δ_y,t represent voltage angles at nodes x ∈𝒩 and y ∈𝒩, respectively.Thermal line limits: Power flows on all transmission lines are limited by the respective thermal limits of line l:|p_l,t|≤p_l,where p_l represents the thermal limit of line l.§.§.§ Generation constraintsGeneration constraints include physical limits of individual generation units. For the binary unit commitment (BUC), we adopted a UC formulation requiring three binary variables per time slot (on/off status, startup, shutdown) to model an individual unit. In the MST, identical units of a plant are clustered into one individual unit <cit.>.This requires three integer variables (on/of status, startup, and shutdown) per generation plant per time slot as opposed to three binary variables per generation unit per time slot in the BUC, as discussed in Section III.B of A Computationally Efficient Market Model for Future Grid Scenario Studies.Generation limits:Dispatch levels of a synchronous generator g are limited by the respective stable operating limits:s_g,tp_g≤ p_g,t≤ s_g,tp_g, g ∈𝒢^syn.The power of RES[For the sake of brevity, by RES we mean “unconventional” renewables like wind and solar, but excluding conventional RES, like hydro, and dispatchable unconventional renewables, like concentrated solar thermal.] generation is limited by the availability of the corresponding renewable resource (wind or sun):s_g,tp_g≤ p_g,t≤ s_g,tp_g,t^RES, g ∈𝒢^RES. Unit on/off constraints:A unit can only be turned on if and only if it is in off state and vice versa:u_g,t-d_g,t=s_g,t-s_g,t-1, t ≠ 1,g ∈𝒢^syn. In a rolling horizon approach, consistency between adjacent time slots is ensured by:u_g,t-d_g,t=s_g,t - ŝ_g, t =1,g ∈𝒢^syn,where ŝ_g is the initial number of online units of generator g. Equations (<ref>) and (<ref>) also implicitly determine the upper bound of u_g,t and d_g,t in terms of changes ins_g,t.Number of online units: Unlike the BUC, the MST requires an explicit upper bound on status variables:s_g,t≤U_g. Ramp-up and ramp-down limits: Ramp rates of synchronous generation should be kept within the respective ramp-up (<ref>), (<ref>) and ramp-down limits (<ref>), (<ref>):p_g,t - p_g,t-1≤ s_g,tr^+_g, t ≠ 1, g ∈{𝒢^syn | r^+_g < p_g},p_g,t - p̂_g≤ s_g,tr^+_g, t =1, g ∈{𝒢^syn | r^+_g < p_g},p̂_g - p_g,t≤ s_g,t-1r^-_g,t ≠ 1, g ∈{𝒢^syn | r^-_g < p_g},p̂_g - p_g,t≤ŝ_gr^-_g, t =1, g ∈{𝒢^syn | r^-_g < p_g}.In the MST, a ramp limit of a power plant is defined as a product of the ramp limit of an individual unit and the number of online units in a power plant s_g,t. If s_g,t is binary, these ramp constraints are mathematically identical to ramp constraints of the BUC.If a ramp rate multiplied by the length of the time resolution Δt is less than the rated power, the rate limit has no effect on the dispatch, so the corresponding constraint can be eliminated. Constraints explicitly defined for t=1 are used to join two adjacent sub-horizons in the rolling-horizon approach.Minimum up and down times:Steam generators must remain on for a period of timeτ_g^u once turned on (minimum up time):s_g,t≥∑_t̃=τ_g^u-1^0 u_g,t-t̃, t ≥τ_g^u, g ∈{𝒢^syn | τ_g^u > Δt},s_g,t≥∑_t̃=t-1^0u_g,t-t̃+ û_g,t, t < τ_g^u,g ∈{𝒢^syn | τ_g^u > Δt}. Similarly, they must not be turned on for a period of timeτ_g^d once turned off (minimum down time):s_g,t≤U_g - ∑_t̃=τ_g^d-1^0 d_g,t-t̃, t ≥τ_g^d,g ∈{𝒢^syn | τ_g^d > Δt},s_g,t≤U_g - ∑_t̃=t-1^0d_g,t-t̃ - d̂_g,t, t < τ_g^d,g ∈{𝒢^syn | τ_g^d > Δt}. Similar to the rate limits, if the minimum up and down times are smaller than the time resolution Δt, the corresponding constraints can be eliminated.Due to integer nature of discrete variables in the MST, the definition of the MUDT constraints in the RH approach requires the number of online units for the last τ^u/d time interval to establish the relationship between the adjacent sub-horizons. If the τ_g^u/d is smaller than time resolution Δt, then these constraints can be eliminated. §.§.§ CST constraints:CST constraints include TES energy balance and storage limits.TES state of charge (SOC)determines the TES energy balance subject to the accumulated energy in the previous time slot, thermal losses, thermal power provided by the solar farm and electrical power dispatched from the CST plant:e_g,t=η_ge_g,t-1+p_g,t^CST-p_g,t, t ≠ 1,g ∈𝒢^CST,e_g,t=η_gê_g+p_g,t^CST-p_g,t,t=1,g ∈𝒢^CST, where, p_g,t^CST is the thermal power collected by the solar field of generator g ∈𝒢^CST. TES limits: Energy stored is limited by the capacity of a storage tank:e_g≤e_g,t≤e_g, g ∈𝒢^CST. §.§.§ Utility storage constraintsUtility-scale storage constraints include energy balance, storage capacity limits and power flow constraints. The formulation is generic and can capture a wide range of storage technologies.Utility storage SOC limits determine the energy balance of storage plants:e_s,t=η_se_s,t-1+p_s,t, t ≠ 1,e_s,t=η_sê_s+p_s,t, t=1. Utility storage capacity limits: Energy stored is limited by the capacity of storage plant s:e_s≤e_s,t≤e_s. Charge/discharge rates limit the charge and discharge powers of storage plant s:p_s^- ≤p_s,t≤p_s^+,where p_s^- and p_s^+ represent the maximum power discharge and charge rates of a storage plant, respectively.§.§.§ Prosumer sub-problemThe prosumer sub-problem captures the aggregated effect of prosumers. It is modeled using a bi-level framework in which the upper-level unit commitment problem described above minimizes the total generation cost, and the lower-level problem maximizes prosumers' self-consumption. The coupling is through the prosumers' demand, not through the electricity price, which renders the proposed model market structure agnostic. As such, it implicitly assumes a mechanism for demand response aggregation. The Karush-Kuhn-Tucker optimality conditions of the lower-level problem are added as the constraints to the upper-level problem, which reduces the problem to a single mixed integer linear program.The model makes the following assumptions: (i) the loads are modeled as price anticipators; (ii) the demand model representing an aggregator consists of a large population of prosumers connected to an unconstrained distribution network who collectively maximize self-consumption; (iii) aggregators do not alter the underlying power consumption of the prosumers; and (iv) prosumers have smart meters equipped with home energy management systems for scheduling of the PV-battery systems, and, a communication infrastructure is assumed that allows a two-way communication between the grid, the aggregator and the prosumers. More details can be found in <cit.>.Prosumer Objective function:Prosumers aim to minimize electricity expenditure:p_p^g+/–, p_p^bminimize∑_t∈𝒯^ p_p,t^g+ - λ p_p,t^g-,where λ is the applicable feed-in price ratio. In our research, we assumed λ = 0, which corresponds to maximization of self-consumption.The prosumer sub-problem is subject to the following constraints:Prosumer power balance:Electrical consumption of prosumer p, consisting of grid feed-in power, p_p,t^g-, underlying consumption, p_p,t^, and battery charging power, p_p,t^b, is equal to the power taken from the grid, p_p,t^g+, plus the power generated by the PV system, p_p,t^pv:p_p,t^g+ + p_p,t^pv =p_p,t^g- +p_p,t^ + p_p,t^b. Battery charge/discharge limits:Battery power should not exceed the charge/discharge limits:p_p^b-≤p_p,t^b≤p_p^b+, where p_b^- and p_b^+ represent the maximum power discharge and charge rates of the prosumer's battery, respectively. Battery storage capacity limits:Energy stored in a battery of prosumer p should always be less than its capacity:e_p^b≤e_p,t^b≤e_p^b. Battery SOC limits: Battery SOC is the sum of thepower inflow and the SOC in the previous period:e_p,t^b = η_p^be_p,t^b +p_p,t^b,t ≠ 1, e_p,t^b = η_p^bê_p^b+p_p,t^b, t=1, where ê_p^b represents the initial SOC and is used to establish the connection between adjacent sub-horizons.§ SIMULATION SETUPThe case studies provided in this section compare the computational efficiency of the proposed MST with alternative formulations. For detailed studies on the impact of different technologies on future grids, an interested reader can refer to our previous work <cit.>.§.§ Test SystemWe use a modified 14-generator IEEE test system that was initially proposed in <cit.> as a test bed for small-signal analysis. The system is loosely based on the Australian National Electricity Market (NEM), the interconnection on the Australian eastern seaboard. The network is stringy, with large transmission distances and loads concentrated in a few load centres. Generation, demand and the transmission network were modified to meet future load requirements. The modified model consists of 79 buses grouped into four regions, 101 units installed at 14 generation plants and 810 transmission lines.§.§ Test CasesTo expose the limitations of the different UC formulations, we have selected a typical week with sufficiently varying operating conditions. Four diverse test cases with different RES penetrations are considered.First, RES0 considers only conventional generation, including hydro, black coal, brown coal, combined cycle gas and open cycle gas. The generation mix consists of 2.31 hydro, 39.35 of coal and 5.16 of gas, with the peak load of 36.5. To cater for demand and generation variations, 10 reserves are maintained at all times. The generators are assumed to bid at their respective short run marginal costs, based on regional fuel prices <cit.>.Cases RES30, RES50, RES75 consider, respectively, 30, 50 and 75 annual energy RES penetration, supplied by wind, PV and CST. Normalized power traces for PV, CST and wind farms (WFs) for the 16-zones of the NEM are taken from the AEMO's planning document <cit.>. The locations of RESs are loosely based on the AEMO's 100% RES study <cit.>. §.§ Modeling AssumptionsPower traces of all PV modules and wind turbines at one plant are aggregated and represented by a single generator. This is a reasonable assumption given that PV and WF don't provide active power reserves, and are not limited by ramp rates, MUDT, and startup and shutdown costs, which renders the information on the number of online units unnecessary.Also worth mentioning is that RES can be modeled as negative demand, which can lead to an infeasible solution. Modeling RES (wind and solar PV) as negative demand is namely identical to preventing RES from spilling energy. Given the high RES penetration in future grids, we model RES explicitly as individual generators.Unlike solar PV and wind, CST requires a different modeling approach. Given that CST is synchronous generation it also contributes to spinning reserves and system inertia. Therefore, the number of online units in a CST plant needs to be modeled explicitly.An optimality gap of 1% was used for all test cases. Simulation were run on Dell OPTIPLEX 9020 desktop computer with Intel(R) Core(TM) i7-4770 CPU with 3.40 clock speed and 16B RAM.§ RESULTS AND DISCUSSIONTo showcase the computational efficiency of the proposed MST, we first benchmark its performance for different horizon lengths against the BUC formulation employing three binary variables per unit per time slot and the AGG formulation where identical units at each plant are aggregated into a single unit, which requires three binary variables per plant per time slot.We pay particular attention to the techniques used for computational speedup, namely unit clustering, rolling horizon, and constraint clipping. Last, we compare the results of the proposed MST with BUC and AGG formulations for voltage and frequency stability studies.§.§ Binary Unit Commitment (BUC) We first run the BUC for horizon lengths varying from one to seven days, Fig. <ref> (top).As expected, with the increase in the horizon length, the solution time increases exponentially. For a seven-day horizon, the solution time is as high as 25000 (7). Observe how the computational burden is highly dependent on the RES penetration. The variability of the RES results in an increased cycling of the conventional thermal fleet, which increases the number of on/off decisions and, consequently the computational burden. In addition to that, a higher RES penetration involves an increased operation of CST. This poses an additional computational burden due to the decision variables associated with TES that span several time slots. In summary, the computational burden of the BUC renders it inappropriate for scenario analysis involving extended horizons.§.§ Aggregated Formulation (AGG)Aggregating identical units at a power plant into a single unit results in a smaller number of binary variables, which should in principle reduce the computational complexity.Fig. <ref> confirms that this is mostly true, however, for RES50-HL7 the computation time is higher than in the BUC formulation. The reason for that is that, in this particular case, the BUC formulation has a tighter relaxation than the AGG formulation and, consequently, a smaller root node gap. Compared to the MST formulation, with a similar number of variables than the AGG formulation, the MST has considerably shorter computation time due to a smaller root node gap.In terms of accuracy, the AGG formulation works well for balancing studies <cit.>. On the other hand, the number of online synchronous generators in the dispatch differs significantly from the BUC, which negatively affects the accuracy of voltage and frequency stability analysis, as shown later. Due to a large number of online units in a particular scenario, a direct comparison of dispatch levels and reserves from each generator is difficult. Therefore, we compare the total number of online synchronous generators, which serves as a proxy to the available system inertia. Fig. <ref> shows the number of online generators of four different RES penetration levels for a horizon length of seven days. For most of the hours there is a significant difference between the number of online units obtained from the BUC and the AGG formulation.In conclusion, despite its computational advantages, the AGG formulation is not appropriate for stability studies due to large variations in the number of online synchronous units in the dispatch results. In addition to that, the computational time is comparable to the BUC in some cases. We now evaluate the effectiveness of the techniques for the computational speedup.§.§.§ Unit ClusteringIn unit clustering, binary variables associated with the generation unit constraints are replaced with a smaller number of integer variables, which allows aggregating several identical units into one equivalent unit, but with the number of online units retained. This results in a significant reduction in the number of variables and, consequently, in the computational speedup. Compared to the BUC, the number of variables in the MST with this technique alone reduces from 24649 to 5990 for RES75 with a horizon length of seven days. Therefore, the solution time for RES75-HL7 reduces from 25000 in the BUC to 450 in MST with unit clustering alone.§.§.§ Rolling Horizon ApproachA rolling horizon approach splits the UC problem into shorter horizons. Given the exponential relationship between the computational burden and the horizon length, as discussed in Section <ref>, solving the problem in a number of smaller chunks instead of in one block results in a significant computational speedup. The accuracy and the consistency of the solution are maintained by having an appropriate overlap between the adjacent horizons. However, the overlap depends on the time constants of the problem. Long term storage, for example, might require longer solution horizons. The solution times for different RES penetrations are shown in Table <ref>. Observe that in the RES75 case, the effect of rolling horizon is much more pronounced, which confirms the validity of the approach for studies with high RES penetration.§.§.§ Constraint ClippingEliminating non binding constraints can speedup the computation even further. Table <ref> shows the number of constraints for different scenarios with and without constraint clipping. Observe that the number of redundant constraints is higher in scenarios with a higher RES penetration. The reason is that a higher RES penetration requires more flexible gas generation with ramp rates shorter than the time resolution (one hour in our case). Note that the benefit of constraint clipping with a shorter time resolution will be smaller. §.§ MST Computation Time and AccuracyThe proposed MST outperforms the BUC and AGG in terms of the computational time by several orders of magnitude, as shown in Fig. <ref> (bottom). The difference is more pronounced at higher RES penetration levels. For RES75, the MST is more than 500 times faster than the BUC. In terms of the accuracy, the MST results are almost indistinguishable from the BUC results, as evident from Fig. <ref> that shows the number of online synchronous units for different RES penetration levels. Minor differences in the results stem from the nature of the optimization problem. Due to its mixed-integer structure, the problem is non-convex and has therefore several local optima. Given that the BUC and the MST are mathematically not equivalent, the respective solutions might not be exactly the same. The results are nevertheless very close, which confirms the validity of the approach for the purpose of scenario analysis. The loadability and inertia results presented later further support this conclusion. §.§ Stability AssessmentTo showcase the applicability of the MST for stability assessment, we analyze system inertia and loadability that serve as a proxy to frequency and voltage stability, respectively. More detailed stability studies are covered in our previous work, including small-signal stability <cit.>, frequency stability <cit.>, and voltage stability <cit.>.§.§.§ System inertiaFig. <ref> (bottom) shows the system inertia for the BUC, AGG and the proposed MST, respectively, for RES0. Given that the inertia is the dominant factor in the frequency response of a system after a major disturbance, the minuscule difference between the BUC and the MST observed in Fig. <ref> validates the suitability of the MST for frequency stability assessment. The inertia captured by the AGG, on the other hand, is either over or under estimated and so does not provide a reliable basis for frequency stability assessment.§.§.§ Loadability AnalysisThe dispatch results from the MST are used to calculate power flows, which are then used in loadability analysis[The loadability analysis is performed by uniformly increasing the load in the system until the load flow fails to converge. The loadability margin is calculated as the difference between the base system load and the load in the last convergent load flow iteration.]. Fig. <ref> (top) shows loadability margins for the RES0 scenario for different UC formulations. Observe that the BUC and the MST produce very similar results. The AGG formulation, on the other hand, gives significantly different results. From hours 95 to 150, in particular, the AGG results show that the system is unstable most of the time, which is in direct contradiction to the accurate BUC formulation.Compared to the inertia analysis, the differences between the formulations are much more pronounced. Unlike voltage, frequency is a system variable, which means that it is uniform across the system. In addition to that, inertia only depends on the number of online units but not on their dispatch levels. Voltage stability, on the other hand, is highly sensitive both to the number of online units and their dispatch levels, which affects the available reactive power support capability, as illustrated in Fig. <ref>.Close to the voltage stability limit, the system becomes highly nonlinear, so even small variations in dispatch results can significantly change the power flows and, consequently, voltage stability of the system. One can argue that in comparison to BUCthe proposed MST result in the more conservative loadability margin, although this is not always the case (around hour 85, the MST is less conservative). § CONCLUSIONThis paper has proposed a computationally efficient electricity market simulation tool based on a UC problem suitable for future grid scenario analysis. The proposed UC formulation includes an explicit network representation and accounts for the uptake of emerging demand side technologies in a unified generic framework while allowing for a subsequent stability assessment. We have shown that unit aggregation, used in conventional planning-type UC formulations to achieve computational speedup, fails to properly capture the system inertia and reactive power support capability, which is crucial for stability assessment. To address this shortcoming, we have proposed a UC formulation that models the number of online generation units explicitly and is amenable to a computationally expensive time-series analysis required in future grid scenario analysis. To achieve further speedup, we use a rolling horizon approach and constraint clipping.The effectiveness of the computational speedup techniques depends on the problem structure and the technologies involved so the results cannot be readily generalized. The computational speedup varies between 20 to more than 500 times, for a zero and 75% RES penetration, respectively, which can be explained by a more frequent cycling of the conventional thermal units in the high-RES case. The simulation results have shown that the computational speedup doesn't jeopardize the accuracy. Both the number of online units that serves as a proxy for the system inertia and the loadability results are in close agreement with more detailed UC formulations, which confirms the validity of the approach for long term future grid studies, where one is more interested in finding weak points in the system rather than in a detailed analysis of an individual operating condition. IEEEtran

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

bgadway@illinois.edu Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA Ultracold atoms in optical lattices offer a unique platform for investigating disorder-driven phenomena. While static disordered site potentials have been explored in a number of optical lattice experiments, a more general control over site-energy and off-diagonal tunneling disorder has been lacking. The use of atomic quantum states as “synthetic dimensions” has introduced the spectroscopic, site-resolved control necessary to engineer new, more tailored realizations of disorder. Here, by controlling laser-driven dynamics of atomic population in a momentum-space lattice, we extend the range of synthetic-dimension-based quantum simulation and present the first explorations of dynamical disorder and tunneling disorder in an atomic system. By applying static tunneling phase disorder to a one-dimensional lattice, we observe ballistic quantum spreading as in the case of uniform tunneling. When the applied disorder fluctuates on timescales comparable to intersite tunneling, we instead observe diffusive atomic transport, signaling a crossover from quantum to classical expansion dynamics. We compare these observations to the case of static site-energy disorder, where we directly observe quantum localization in the momentum-space lattice.Ballistic, diffusive, and arrested transport in disordered momentum-space lattices Bryce Gadway December 30, 2023 ==================================================================================Over the past two decades, dilute atomic gases have become a fertile testing ground for the study of localization phenomena in disordered quantum systems <cit.>. They have allowed for some of the earliest and most comprehensive studies of Anderson localization of quantum particles <cit.>, strongly interacting disordered matter <cit.>, and many-body localization <cit.>. Still, the emulation of many types of disorder relevant to real systems - e.g., crystal strain and dislocation, site vacancies, interstitial and substitutional defects, magnetic disorder, and thermal phonons - will require new types of control that go beyond traditional methods based on static disorder potentials <cit.>.The recent advent of using atomic quantum states as synthetic dimensions has broadened the cold atom toolkit with the spectroscopic, site-resolved control of field-driven transitions <cit.>. This technique has aided the study of synthetic gauge fields <cit.>, and its spatial and dynamical control offers a prime way to implement specifically tailored, dynamical realizations of disorder that would otherwise be difficult to study. However, current studies based on internal states <cit.> have been limited to a small number of sites along the synthetic dimension, inhibiting the study of quantum localization in the presence of disorder.Here, we employ our recently-developed technique of momentum-space lattices <cit.> to perform the first studies of tailored and dynamical disorder in synthetic dimensions. Our approach introduces several key advances to cold atom studies of disorder: the achievement of pure off-diagonal tunneling disorder, the dynamical variation of disorder, and site-resolved detection of populations in a disordered system. For the case of tunneling disorder, we examine the scenario in which only the phase of tunneling is disordered. As expected for a one-dimensional system with only nearest-neighbor tunneling, these random tunneling phases are of zero consequence when applied in a static manner. When this phase disorder fluctuates on timescales comparable to intersite tunneling, however, we observe a crossover from ballistic to diffusive transport <cit.>. We compare to the case of static site-energy disorder, observing Anderson localization at the site-resolved level.Our bottom-up approach <cit.> to Hamiltonian engineering is based on the coherent coupling of atomic momentum states to form an effective synthetic lattice of sites in momentum space (see Fig. <ref>). This approach may be viewed as studying transport in an artificial dimension <cit.> of discrete spatial eigenstates <cit.> (as opposed to a bounded set of atomic internal states <cit.>) through resonant or near-resonant field-driven transitions.Starting with ^87Rb Bose-Einstein condensates of ∼5 × 10^4 atoms, we initiate dynamics between 21 discrete momentum states by applying sets of counter-propagating far-detuned laser fields (wavelength λ = 1064 nm, wavevector k = 2π/λ), specifically detuned to address multiple two-photon Bragg transitions, as depicted in Fig. FIG:fig1(a-b). Our spectrally-resolved control of the individual Bragg transitions permits a local control of the system parameters, similar to that found in photonic simulators <cit.>. Unique to our implementationis the direct and arbitrary control of tunneling phases <cit.>, and the realized tight-binding model is depicted in Fig. FIG:fig1(c). Here, we use this capability to explore the dynamics of cold atoms subject to disordered and dynamical arrangements of tunneling elements.Specifically, we explore disorder arising purely in the phase of nearest-neighbor tunneling elements. In higher dimensions, such disordered tunneling phases would give rise to random flux patterns that mimic the physics of charged particles in a random magnetic field <cit.>. In 1D, however, the absence of closed tunneling paths renders any static arrangement of tunneling phases inconsequential to the dynamical and equilibrium properties of the particle density. Time-varying phases, however, can have a nontrivial influence on the system's dynamical evolution.We engineer annealed, or dynamically varying, disorder of the tunneling phases and study its influence through the atoms' nonequilibrium dynamics following a tunneling quench. Our experiments begin with all population restricted to a single momentum state (site). We suddenly turn on the Bragg laser fields, quenching on the (in general) time-dependent effective HamiltonianĤ(τ) ≈ -t ∑_n(e^i φ_n (τ)ĉ^†_n+1ĉ_n + h.c.) + ∑_n ε_n ĉ^†_n ĉ_n,where τ is the time variable, t is the (homogeneous) tunneling energy, and ĉ_n (ĉ^†_n) is the annihilation (creation) operator for the momentum state with index n (momentum p_n = 2nħ k). The tunneling phases φ_n and site energies ε_n are controlled through the phases and detunings of the two-photon momentum Bragg transitions, respectively. After a variable duration of laser-driven dynamics, we perform direct absorption imaging of the final distribution of momentum states, which naturally separate during 18 ms time of flight. Analysis of these distributions, including determination of site populations through a multi-Gaussian fit, is as described in Ref. <cit.>.As a control, we first examine the case of no disorder, with all site-energies set to zero and uniform, static tunneling phases φ_n(τ) = φ. Figure  FIG:fig2(a,i) shows the evolution of the 1D momentum distribution, obtained from time-of-flight images integrated along the axis normal to the imparted momentum, displaying ballistic expansion characteristic of a continuous-time quantum walk. For times before the atoms reflect from the open boundaries of the 21-site lattice, we find good qualitative agreement between the observed momentum distributions and the expected form P_n = |J_n (ϑ)|^2, where J_n is the Bessel function of order n and ϑ = 2 τ t/ħ. Figure FIG:fig2(b,i) shows the (symmetrized) momentum profile at time τ∼ 3 ħ/t along with the Bessel function distribution for ϑ = 5.4.In comparison, Fig. FIG:fig2(a,ii) shows the case of zero site energies and static, random tunneling phases φ_n ∈ [0,2π). The dynamics are nearly identical to the case of uniform tunneling phases. This is consistent with the expectation that any pattern of static tunneling phases in 1D is irrelevant for the dynamics of the effective tight-binding model realized by our controlled laser coupling. For this case, Fig. FIG:fig2(b,ii) shows the (symmetrized) momentum profile at τ∼ 2.5 ħ/t along with the Bessel function distribution for ϑ = 5.35.While static phase disorder has little impact on the quantum random walk dynamics, we may generally expect that controlled random phase jumps or even pseudorandom variations of the phases should inhibit coherent transport, mimicking random phase shifts induced through interaction with a thermal environment. To probe such behavior, we implement dynamical phase disorder by composing each tunneling phase φ_n from a broad spectrum of oscillatory terms with randomly-defined phases θ_n,i but well-defined frequencies ω_i, the weights of which are derived from an ohmic bath distribution. Specifically, the dynamical tunneling phases take the formφ_n(τ) = 4π∑_i = 1^N S(ω_i) cos (ω_i τ + θ_n,i) / ∑_i = 1^N S(ω_i),where S(ω) = (ħω / k_B T) exp[-(ħω / k_B T)], the θ_n,i are randomly chosen from [0, 2π ), and T is an artificial temperature scale that sets the range of the frequency distribution. In this discrete formulation of φ_n(τ), we include N=50 frequencies ranging between zero and 8 k_B T / ħ. The frequency spectrum and dynamics for one tunneling phase φ_n(τ) are shown in Fig. FIG:fig2(c) for the case of k_BT/t = 1.Figure FIG:fig2(a,iii) displays the population dynamics in the presence of this dynamical disorder, characterized by an effective temperature k_B T/t = 0.66(1) and averaged over three independent realizations of the disorder. The dynamics no longer feature ballistically separating wavepackets, instead displaying a broad, slowly spreading distribution peaked near zero momentum. A clear deviation of the (symmetrized) momentum distribution from the form P_n = |J_n (ϑ)|^2 describing the previous quantum walk dynamics can be seen in Fig. FIG:fig2(b,iii) (shown at the time τ∼ 3.8 ħ/t). The displayed Gaussian population distribution gives much better agreement, consistent with spreading governed by an effectively classical or thermal random walk.Lastly, while no influence of static tunneling phase disorder is expected in 1D, the effect of static site-energy disorder is dramatically different. Here, with homogeneous static tunneling terms, we explore the influence of pseudorandom variations of the site energies governed by the Aubry-André model <cit.>. With an irrational periodicity b=(√(5)-1)/2, the site energies ε_n = Δcos(2 π b n + ϕ) do not repeat, and are governed by a pseudorandom distribution. For an infinite system, this Aubry-André model with diagonal disorder features a metal-insulator transition at the critical disorder strength Δ_c = 2 t. The expansion dynamics for the strong disorder case Δ / t = 5.9(1) are shown in Fig. FIG:fig2(a,iv), with population largely restricted to the initial, central momentum order. The exponentially localized distribution of site populations (symmetrized and averaged over all profiles in the range τ∼ 5-6.3 ħ/t) is shown in Fig. FIG:fig2(b,iv), along with an exponential distribution with localization length ξ = 0.6 lattice sites. Analogous population distributions (again symmetrized and averaged over the same time range) are shown for the cases of weaker disorder [Δ/t = 0.98(1), 1.96(3), 3.05(4), 4.02(9)] in Fig. FIG:fig2(d), exhibiting an apparent transition to exponential localization for Δ/t ≳ 2.For all of the explored cases, we study these expansion dynamics in greater detail in Fig. <ref>. Figure FIG:fig3(a) examines the momentum-width (σ_p) dynamics of the atomic distributions for the cases of static and dynamic random phase disorder. For static phase disorder, we observe a roughly linear increase of σ_p until population reflects from the open system boundaries, while dynamical phase disorder leads to sub-ballistic expansion. In particular, for time τ measured in units of ħ/t and momentum-width σ_p in units of the site separation 2 ħ k, these two cases agree well with the displayed theory curves for ballistic and diffusive expansion, having the forms σ_p = √(2)τ and σ_p = √(2 τ), respectively (with the latter curve shifted by 0.35 ħ /t). To explore these two different expansions more quantitatively, we fit the momentum variance V_p ≡σ_p^2 to a power-law V_p(τ) = ατ^γ, performing a linear fit to variance dynamics on a double logarithmic scale as shown in Fig. FIG:fig3(c). The fit-determined expansion exponents γ for the cases of static and dynamically disordered tunneling phases are 2.05(2) and 1.27(2), respectively. These values are roughly consistent with a coherent, quantum random walk for the case of static tunneling phases and an incoherent, nearly diffusive random walk for the case of dynamical phase disorder.The observed transport dynamics cross over from ballistic to diffusive as the effective thermal energy scale k_B T approaches the coherent tunneling energy t, matching our expectation that randomly-varying tunneling phases can mimic the random dephasing induced by a thermal environment. We note that similar classical random walk behavior has been seen previously for both atoms and photons, due to irreversible decoherence <cit.> and thermal excitations <cit.>. However, this is the first observation based on reversible engineered “noise” of a Hamiltonian parameter. These observations of a thermal random walk suggest that annealed disorder may provide a means of mimicking thermal fluctuations and studying thermodynamical properties <cit.> of simulated models using atomic momentum-space lattices, and by extension other nonequilibrium experimental platforms such as photonic simulators.We also analyze the full expansion dynamics for the case of static site energy disorder in Figs. FIG:fig3(b,d). For homogeneous static tunnelings and thus zero disorder (Δ/t =0), we observe momentum-width dynamics similar to the case of static random tunneling phases, but with one distinct difference: while σ_p features a linear increase for random static phases, it increases in a step-wise fashion for uniform tunneling phases. This slight disagreement is a byproduct of the underlying laser-driven dynamics that give rise to the effective tight-binding model described by Eq. <ref>. The Bragg laser field 2 (see Fig. <ref>) features a comb of 20 discrete, equally-spaced frequencies, each of which primarily addresses a single Bragg transition. Weak off-resonant coupling terms conspire to produce this step-like behavior in the case of equal-phase driving, while this behavior is mostly absent for random tunneling phases.Evolution of the momentum-width (σ_p) for the site-energy disorder cases of Δ/t = 0.98(1), 2.47(3), 5.9(1) are also shown in Fig. FIG:fig3(b). We observe the reduction of expansion dynamics with increasing disorder, with nearly arrested dynamics in the strong disorder limit. More quantitatively, fits of the variance dynamics as shown in Fig. FIG:fig3(d) reveal sub-ballistic, nearly diffusive expansion for intermediate disorder [γ = 1.00(2) for Δ/t = 0.98(1)], giving way to a nearly vanishing expansion exponent for strong disorder [γ = 0.12(6) for Δ/t = 5.9(1)].The extracted expansion exponents for all of the explored cases are summarized in Fig. FIG:fig3(e). For static site-energy disorder (red circles), while longer expansion times than those explored (τ≲ 6.3 ħ/t) would better distinguish insulating behavior from sub-ballistic and sub-diffusive expansion, a clear trend towards arrested transport (γ∼ 0) is found for Δ/t ≫ 1. Combined with the observation of exponential localization of the site populations in Fig. FIG:fig2(b,iv) and Fig. FIG:fig2(d), these observations are consistent with a crossover in our 21-site system from metallic behavior to quantum localization for Δ/t ≳ 2.Our observations of a crossover from ballistic expansion (γ∼ 2) to nearly diffusive transport (γ∼ 1) for randomly fluctuating tunneling phase disorder are also summarized in Fig. FIG:fig3(e). In the experimentally-accessible regime of low to moderate effective thermal energies (k_B T / t ≲ 1), our experimental data points (blue squares) match up well with numerical simulation (open black circles). For the magnitude of tunneling energy used in these experiments, we are restricted from exploring higher effective temperatures (k_B T /t ≳ 1), as rapid variations of the tunneling phases introduce spurious spectral components of the Bragg laser fields that could drive undesired transitions. Simulations in this high-temperature regime suggest that the expansion exponent should rise back up for increasing temperatures, saturating to a value γ∼ 2. This results from the fact that the time-averaged phase effectively vanishes when the timescale of pseudorandom phase variations is much shorter than the tunneling time.The demonstrated levels of local and time-dependent control over tunneling elements and site energies in our synthetic momentum-space lattice have allowed us to perform first-of-their-kind explorations of annealed disorder in an atomic system. Such an approach based on synthetic dimensions should enable myriad future explorations of engineered Floquet dynamics <cit.> and novel disordered lattices <cit.>. 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http://arxiv.org/abs/1701.07493v1

http://arxiv.org/abs/1701.07836v2

" daniel.reinholz@sdsu.edu Department of Mathematics & Statistics, San Diego State University, San D(...TRUNCATED)

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" []inancsahin@ankara.edu.trDepartment of Physics, Faculty of Sciences, Ankara University, 06100 Tan(...TRUNCATED)

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"Department of Electrical Engineering, Technion - Israel Institue of Technology, IsraelDepartment of(...TRUNCATED)

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"Department of Astronomy, Space Sciences Building, Cornell University, Ithaca, NY 14853, USA; We rep(...TRUNCATED)

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