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1701.07941v2_44

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

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

1701.07941v2_45

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

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

1701.07941v2_46

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

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

1701.07941v2_47

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

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

1701.07941v2_48

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

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

1701.07941v2_49

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

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

1701.07941v2_50

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

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

1701.07941v2_51

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

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

1701.07941v2_52

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

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

1701.07941v2_53

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

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

1701.07941v2_54

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

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

1701.07941v2_55

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

= {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

1701.07941v2_56

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

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,

1701.07941v2_57

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

∈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 -

1701.07941v2_58

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

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

1701.07941v2_59

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

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

1701.07941v2_60

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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

1701.07941v2_61

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

∈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

1701.07941v2_62

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

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

1701.07941v2_63

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

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

1701.07941v2_64

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

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

1701.07941v2_65

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

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 <

1701.07941v2_66

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

- 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

1701.07941v2_67

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

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

1701.07941v2_68

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

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

1701.07941v2_69

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

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

1701.07941v2_70

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

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

1701.07941v2_71

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

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

1701.07941v2_72

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

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

1701.07941v2_73

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

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

1701.07941v2_74

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

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

1701.07941v2_75

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

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

1701.07941v2_76

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

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

1701.07941v2_77

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

+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 =

1701.07941v2_78

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

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

1701.07941v2_79

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

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

1701.07941v2_80

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

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,

1701.07941v2_81

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

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

1701.07941v2_82

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

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

1701.07941v2_83

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

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

1701.07941v2_84

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

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)

1701.07941v2_85

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

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

1701.07941v2_86

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

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

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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

1701.07941v2_88

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

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

1701.07941v2_89

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

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

1701.07941v2_90

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

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

1701.07941v2_91

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

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

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

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

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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.

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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

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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

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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

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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,

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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,

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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

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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

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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

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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

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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

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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

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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

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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,

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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

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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 ∈

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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

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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

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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

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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

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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

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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,

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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

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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 ≳

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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

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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,

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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

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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

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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

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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

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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

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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

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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