Arnold N'goran
email: arnold.ngoran@mines-paristech.fr

Bruno Daugrois
email: bruno.daugrois@bertin.fr

Marc Lotteau
email: marc.lotteau@bertin.fr

Sophie Demassey
email: sophie.demassey@mines-paristech.fr

Optimal engagement and operation of a grid-connected PV/battery system

Keywords: microgrids, optimal scheduling, mathematical programming, genetic algorithms, numerical simulation

Microgrid operation scheduling is a widely studied topic, but the performances of the solution algorithms proposed in the literature are rarely discussed and, in practice, scheduling is often only loosely optimized. In this paper, we present a practical context of coordination between a microgrid and an utility grid which concerns hundreds of microgrids worldwide. We adapt three distinct optimization strategies -mixed integer quadratic programming, simulation-based genetic algorithm and expertbased heuristic -and empirically compare their strengths. Our results emphasize the benefit of seeking for optimality, while the inaccuracies of the system dynamic model are compensated by periodic recomputation.

I. INTRODUCTION

With the deregulation of power generation and the decarbonization targets set by countries worldwide, distributed generation with grid-connected renewable power plants are developing rapidly. Ranging from small solar rooftop systems to large power stations combining several sources, independent power systems feed power to the utility grid, ideally in a coordinated way: the independent producer seeks to maximize profit while respecting the constraints imposed by the grid operator to ensure network reliability and avoid voltage and frequency stability issues. Different coordination models exist to ensure equilibrium between the intermittent and unpredictable nature of renewable sources and the required stability of the power supply. A two-stage model is considered in the call for tenders CRE ZNI 2015 [START_REF]Appel d'offres portant sur la réalisation et l'exploitation d'installations de production d'électricité à partir de techniques de conversion du rayonnement solaire d'une puissance supérieure à 100 kwc et situées dans les zones non interconnectées[END_REF] launched by the French Energy Regulatory Commission to promote PV power generation facilities in small non-interconnected grids, mainly overseas islands. In this model, the producer first plans its injection/withdrawal, called its engagement profile, for the day after, by considering the solar irradiation forecasts. The producer must then ensure in real-time that the actual production meets the engagement to avoid high financial penalties. Storage devices, like lithium-ion battery banks, are needed to compensate forecast errors as well as PV unavailability periods to smooth the production and supply during the evening peak. Given a dynamic electricity tariff driven by the load forecast and the possibility to charge the batteries from the utility grid, the producer must then schedule the battery charge/discharge and injection/withdrawal operations over the day to maximize profit and minimize penalties. This involves solving two related optimization problems -to compute the engagement and to manage the production -which remain hard even with reasonable approximations in the system dynamic model.

Microgrid operation scheduling covers a large range of optimization problems, borrowing from economic dispatch, power flow and unit commitment, and their algorithmic solutions are widely studied (see e.g. recent surveys [START_REF] Karthik | Optimal operation of microgrids-a survey[END_REF], [START_REF] Minchala-Avila | A review of optimal control techniques applied to the energy management and control of microgrids[END_REF]). The algorithms differ according to the specific features of the considered microgrids (e.g. on/off-grid, with/without storage or controllable sources) or the optimization paradigm, primarily chosen among mathematical programming (e.g. [START_REF] Palma-Behnke | A microgrid energy management system based on the rolling horizon strategy[END_REF], [START_REF] Malysz | Milp-based rolling horizon control for microgrids with battery storage[END_REF]), dynamic programming (e.g. [START_REF] Das | Near optimal control for microgrid energy systems considering battery lifetime characteristics[END_REF], [START_REF] Jadhav | Optimal power flow in wind farm microgrid using dynamic programming[END_REF]) or population-based heuristics such as particle swarm and genetic algorithms (e.g. [START_REF] Chen | Smart energy management system for optimal microgrid economic operation[END_REF]- [START_REF] Nemati | A memory-based genetic algorithm for optimization of power generation in a microgrid[END_REF]). Most of these studies focus on the operability of the microgrids, and do not discuss the choice of the algorithm, nor its performance in terms of runtime or optimality. Comparisons of different algorithms are even rarer, and mostly on variants of a same algorithm (see e.g. [START_REF] Maulik | Optimal operation of microgrid using four different optimization techniques[END_REF]), because simulating these algorithms in a real or realistic environment can be difficult to implement. A notable exception is the comparative study in [START_REF] Hu | Energy coordinative optimization of wind-storage-load microgrids based on short-term prediction[END_REF] between a mathematical programming (MP) approach and a genetic algorithm (GA), but the experimental protocol is not detailed enough (how do the two models differ ? is MP solved at optimality ?) to explain the author's counterintuitive observation that the optimal MP solutions have lower profit than the approximated GA solutions. Finally, in practice, advanced optimization techniques are scarcely implemented and real microgrids are often managed with partial awareness of forecasts and optimality.

However, the financial gains of a microgrid rely on the scheduling algorithm and its ability to compute optimal or near-optimal solutions. This ability depends on the optimization paradigm, but is also correlated to the model granularity which impacts, in turn, the runtime performance of the algorithm. When scheduling is made in real-time, it is important to carefully choose the granularity as a trade-off between high profit solutions and low computation times.

In this paper, we study the real application context for microgrid scheduling specified in the CRE ZNI 2015 call for tenders [START_REF]Appel d'offres portant sur la réalisation et l'exploitation d'installations de production d'électricité à partir de techniques de conversion du rayonnement solaire d'une puissance supérieure à 100 kwc et situées dans les zones non interconnectées[END_REF], and which concerns about 150 winner projects of this and subsequent calls. This application is characterized by the two-phase engagement/control approach resulting in two related optimization problems, and by the penalty minimization criterion measured as a non-convex piecewise quadratic distance between two profiles. We then present a coarse-grain analytic formulation of the scheduling problems, as Mixed-Integer Quadratic Programs, and conduct experiments, on a real study-case which is representative of the CRE ZNI contractors. We compare mathematical programming (MP) with two distinctive approaches: a business-as-usual expertbased strategy (BAU) and a genetic algorithm (GA) embedding a fine-grain numerical simulator of the dynamical system. A key finding of these experiments is that, compared to the BAU strategy, the future-aware algorithms achieve substantial savings, ranging from 50 to 100 euros a day, corresponding approximately to 10% of the daily profits. This result, while predictable, should encourage microgrid operators to invest in optimization-oriented controllers. We also found that the MP solutions are always more profitable, while computed 70 times faster, than the simulation-based GA solutions. This suggests that it is preferable to optimize exactly over a rough model of the system dynamics than to optimize heuristically over a precise model. The main reason is that model inaccuracies are compensated in our experiments by the periodic recomputation of the production schedule in real-time control.

The paper is organized as follows: Section II defines the architecture of the microgrid and of the management workflow. Section III describes the three distinct optimization approaches. Section IV presents the experimental protocol, results and analysis.

II. MICROGRID MANAGEMENT

We illustrate the CRE ZNI 2015 coordination problem on a 1000 kWp PV/battery power plant installed on a Carribean island in response to the call for tenders, the same we used in our experimental protocol. The microgrid illustrated in Figure 1 is made of a PV array (E) and a battery bank (B) equipped with power converters and transformers and is connected to the public distribution power grid (G) of the island. The power flows through the microgrid and from/to the grid are depicted by red arrows. The microgrid is operated by a Model Predictive Control (MPC) implementing an Energy Management System (EMS) coupled with a Power Management System (PMS). They are made of precisely three components run at different frequencies on different discretized time horizons, each denoted as a sequence of time steps t ∈ {1, . . . , T }. The Predictor determines the engagement profile d ∈ R T for the day after with a sampling time of 1 minute (i.e. T = 1440), given: unit electricity tariff profile ρ ∈ R T + , PV generation forecast profile E ∈ R T + , and the current state x 0 of the system (including the battery state of charge). Engagement d t is defined as the stationary electric power during time step t, that is injected to the grid if positive, or drawn from the grid if negative. The predictor is run once at midnight, and possibly every 6 hours with updated forecasts. The engagement is sent to the grid operator and as input to the Tracker. The Tracker computes every 15 minutes, at time t, the production (i.e. injection or withdrawal) plan u ∈ R T -t until the end of the day with a sampling time of 1 minute, using the observed plant state x t and updated forecasts E ∈ R T -t + . The PMS attempts to apply the Tracker command u t for every minute t. It is in charge of the dispatch and communicates every second with the plant by observing the current state and PV generation and returning the real time control for the next second. The PMS relies on a rule-based system which allows to change command u t when it is not applicable.

A. The microgrid and controller architectures

B. The coordination model

The CRE ZNI 2015 coordination model [START_REF]Appel d'offres portant sur la réalisation et l'exploitation d'installations de production d'électricité à partir de techniques de conversion du rayonnement solaire d'une puissance supérieure à 100 kwc et situées dans les zones non interconnectées[END_REF] is characterized by (1) an incentive for injection in the evening in the form of higher electricity prices ρ t during a fixed 2-hour peak period, and (2) financial penalties C P t incurred at any time t when production u t does not respect engagement d t within a tolerance range of 5% of the installed power P inst . Penalties are computed as follows. Let d -= d -0.05P inst and d + = d + 0.05P inst denote the tolerance limits, then

C P t = 0 when the engagement is satisfied (d - t ≤ u t ≤ d + t ), C P t = ρ t u t in case of overproduction (u t > d + t ), and
C P t = ρ t ∆(u t , d t ) in case of underproduction (u t ≤ d - t ), with ∆(u, d) = (u-d + ) 2 -(d --d + ) 2

Pinst

. The producer's benefit at time t includes electricity sale and purchase and is thus equal to J t (u t ) = ρ t u t -C P t . Note that, for a given engagement value d t , J t is a non-concave piecewise quadratic function.

The shape of the engagement profile d is defined by the grid operator as specified bounds on the ramp rates

d t+1 -d t ∈ [α - t , α + t ] or on the power d t ∈ [β - t , β + t ]
(see values in [START_REF]Appel d'offres portant sur la réalisation et l'exploitation d'installations de production d'électricité à partir de techniques de conversion du rayonnement solaire d'une puissance supérieure à 100 kwc et situées dans les zones non interconnectées[END_REF]) . These bounds primarily differ during the evening peak period.

III. OPTIMIZATION STRATEGIES A. Mixed Integer Quadratic Programming (MP)

Linear Programming is widely used for microgrid operation scheduling for its two main benefits: (1) by using efficient off-the-shelf solvers, it requires almost no algorithmic development, and (2) it converges to optimal solutions and provides optimality certificates. In return, modelling the system dynamics (the battery operation, for example) as linear constraints involves deep simplifications. Introducing discrete variables does not help to improve the physical model accuracy but it allows to handle logical constraints and binary states. Despite the higher algorithmic complexity, discrete models of limited size can efficiently be optimized by the state-of-the-art solvers.

We adapt the standard mathematical programming formulation of the production scheduling problem to our context.

The decision variables are the power flows u between PV (P ), battery (B) and grid (D) depicted in Figure 1 and the battery state of charge soc (with initial soc 0 fixed to the currently observed state). They are related by the usual balance constraints (2), [START_REF] Minchala-Avila | A review of optimal control techniques applied to the energy management and control of microgrids[END_REF]. PV curtailment is allowed by [START_REF] Palma-Behnke | A microgrid energy management system based on the rolling horizon strategy[END_REF]. The efficiency of inverters and transformers are explicit in this modelthrough coefficients η -as well as additional equipment losses u L (5). Two vectors of binary variables are required here: by ( 7)-( 8), y B t distinguishes battery charging from discharging, as the associated efficiencies η c and η d are different, given β + c and β + d the maximum charge and discharge values; by ( 9)-( 10), y G t distinguishes injection (then u t = u M G t ≥ 0) from withdrawal (then u t = -u GB t ≤ 0). However, no additional binary variables are required to model the piecewise objective functions J t in this context, observing that overproduction is non-optimal as curtailment is allowed (thus (9) enforces u t ≤ d + t ), and that J t (u t ) = ρ t max(u t , u t -∆(u t , d t )) for any u t ≤ d + t (which is enforced by [START_REF] Maulik | Optimal operation of microgrid using four different optimization techniques[END_REF] and [START_REF] Hu | Energy coordinative optimization of wind-storage-load microgrids based on short-term prediction[END_REF] given that u = u M G -u GB and z = J/ρ). Finally, Constraints [START_REF] Hu | Energy coordinative optimization of wind-storage-load microgrids based on short-term prediction[END_REF] being concave quadratic, modern Mixed Integer Quadratic Programming solvers can directly solve this model.

max t δ t ρ t z t s.t. : (1)
soc t -soc t-1 δ t = η c (u EB t + u GB t ) - u BG t η d ∀t (2)
u EG t + η bt u BG t -u L t = u M G t ∀t (3)
u EB t + u EG t ≤ η pt E t ∀t (4)
u L t ≥ 1 -η bt η L (u EB t + u GB t + u BG t ) ∀t ( 5
)
y G t ≤ y B t , y G t ∈ {0, 1}, y B t ∈ {0, 1} ∀t (6) 0 ≤ u BG t ≤ β + d y B t ∀t (7) 0 ≤ u EB t + u GB t ≤ β + c (1 -y B t ) ∀t (8) 0 ≤ u M G t ≤ d + t y G t ∀t (9) 0 ≤ u GB t ≤ β - t (y G t -1) ∀t (10)
z t ≤ u M G t -u GB t ∀t (11)
z t ≤ u M G t -u GB t -∆(u M G t -u GB t , d t ) ∀t (12) 0 ≤ soc t ≤ 1 ∀t (13)
We optimize the engagement using the same model after setting vector d as decision variables and adding the ramp and power bounds α and β (see Section II-B) as linear constraints.

B. Simulation-Based Genetic Algorithm (GA)

A numerical software is best suited to accurately model dynamical systems like microgrids, but it would only simulate one production plan at a time. To be used in microgrid scheduling, it must be combined with an optimization strategy, guided by an estimate of the profit to generate the sequence of production plans to evaluate by the simulator. Genetic Algorithm (GA) is a favorite strategy: (1) the global performance of this metaheuristic is widely recognised, (2) it is population-based and thus can easily be interfaced to a blackbox simulator to evaluate each individual, (3) generic configurable implementations exist. Contrarily to the MP approach, however, the convergence to an optimum is not guaranteed and it does not compute upper bounds on the objective to estimate the quality of the returned solutions.

In our implementation on the production control problem, individuals are potential production plans u ∈ R T + . During the optimization, a population of P OP SIZE individuals is generated. At the first iteration, individuals are sampled from a problem knowledge-based production plan, following a Gaussian distribution. Iteratively, new individuals are generated by selection, combination and mutation of the individuals of the previous generation. At this step, only the bounds on the production variables are enforced to assert the feasibility of the individuals. Each new generated individual is then evaluated by the simulator to assert its feasibility regarding the physical constraints, then the actual profit of the individual is computed. Individuals with higher profits are used in priority to generate new individuals, while low profit individuals may be discarded from the population. After N GEN iterations, an individual with the maximum profit is returned. We use the same algorithm to optimize the engagement, by just ignoring penalties when computing the individual profits.

C. Expert system (BAU)

The third strategy is representative of expert microgrid management policies which are common in practice. From the CRE ZNI 2015 specifications and the specified ramp bounds, we can draw the general shape of a valid production profile as depicted in Figure 2. The main policy is to fully charge the battery when sun is available (typically between 6AM and 6PM) in order to satisfy the peak constraint and get the maximum profit during the peak period (between 7PM and 9PM). After having charged the battery, the remaining solar production is used to directly supply the grid. Using this policy and the knowledge of the maximum power slope and of the plateau phase start time (10AM) and end time (2PM), simple arithmetic calculations are used to derive values for the trapezoid slopes and for the peak height, thus to fully determine the engagement profile d for the day. With this strategy, the production control u is set to d + 0.05P inst during peak period and matches engagement d, elsewhere.

IV. BENCHMARK ANALYSIS A. Experimental settings

Our benchmark set is built on the real power plant described in Section II with P inst =1000kW and a battery nominal capacity of 1000kWh. For a fair comparison between the different optimization solutions, we use historical solar irradiation data in place of forecasts to erase stochasticity. Hence we built a benchmark of 24 instances by considering the irradiation profiles observed at the plant location for 2 random days each month of year 2015. Daily profiles E are given from midnight to midnight with a discretization step of one minute.

Because experiments on sampling data cannot be run on the real plant, we simulate the system with a numerical model implemented with MATLAB/Simulink instead. This reference model has empirically shown to be very accurate. The PMS thus commands this model in place of the plant. The same model is implemented as the black-box solver in the simulation-based GA.

Experiments were performed on an Intel Core i7-6700HQ OctaCore computer at 2.6 GHz and 16 Go RAM. The genetic algorithm was implemented using the Python library DEAP 1.0.2 and the mathematical model solved with the Gurobi 7.5.2 MIQCP solver using the default parameter set. The BAU strategy is implemented in Python 3.5.

B. Evaluating the modeling flaws

Our first experiment evaluates the accuracy of the MP model regarding the reference model. For that purpose, we implemented the MP model within the Tracker to optimize the production plans, and compared the conceptual optimal solution (u, soc, J) of MP with the "real" solution of the reference model, observed after applying control u through the PMS. In the standard setting, denoted P15, the Tracker is run every 15 minutes and the updated production control applied for the next 15 minutes. The conceptual solution is thus recomposed by concatenating the sequence of 15-minutes profiles. To amplify the impact of the MP model approximation, we also considered a fictional setting PP where the optimal production control is computed only once at midnight and applied the entire day. We also evaluated a simpler MP model where the equipment losses are neglected, by relaxing Constraints [START_REF] Malysz | Milp-based rolling horizon control for microgrids with battery storage[END_REF] and fixing the inverter and transformer efficiencies to 1. The settings corresponding to the two execution modes above are denoted N15 and PN, respectively.

Table I reports, for the 4 settings, the deviation between the MP conceptual solution and the observed result, evaluated through 3 numerical indicators computed in average on the 24 instances (avg) and for the worst case (max): the difference of the daily profits (∆J) and the Mean Absolute Error 1 on the battery state of charge (∆soc) and on the command (∆u). The last two columns (J) show the observed generated profit in average (avg) and the standard deviation (stddev).

We first observe that the deviations on the battery state of charge and on the production control are limited even for the worst-case PN. The approximations of the MP model are thus reasonably accurate in this context. The deviations are negligible when MP is solved repeatedly every 15 minutes (for P15 and N15) since, each time, the production plan is readjusted according to the real state of the plant at the beginning of the period. This shows how the periodic run of the Tracker makes the overall management procedure robust to the errors on the physical model in the Tracker.

1 for two time series u, v ∈ R T , M AE(u, v) = 1 T t |ut -vt|.
Incidentally, the smallest average deviations are achieved by N15 when equipment losses are fully neglected, although the model in P15 seems more realistic. Indeed, both models are approximations, and one is not theoretically better than the other. When observing the generated profits J, P15 and N15 achieve the highest profits. This is not surprising, since the MP solutions were always optimal (for the conceptual model), thus, given they are nearly feasible in P15 and N15, they are probably close to the real feasible optima. Figure 3 illustrates the impact of the model approximation in setting PP on the day where we observed the highest deviation. state of charge is under-estimated in the optimal MP solution (on the left in red) and the battery actually reaches it maximal capacity more than 1 hour before than expected. However the error does not exceed 12% of the battery capacity at any time. Observing the output power u (on the right), the absolute error oscillates between 0 and 1kW, and is thus clearly inside the tolerance range [-50kW, 50kW ].

Finally, we also experimented with a higher recomputation frequency, from 15 to 5 minutes, but we did not observe a significant improvement of the results.

C. Comparing the optimization techniques

Our second experiment compares the solutions of MP and GA, in order to evaluate the benefits of a finer model (using the simulation-based GA) or of a better guided optimization strategy (using MP). Two implementations of the Tracker, with MP and with GA, are provided. For a fair comparison, we first use the same engagement profile, obtained with GA. The MP Tracker is run in the standard setting P15, i.e. updating the production control every 15 minutes. We evaluate the solution in the reference model after applying the control through the PMS. Because, GA solutions are feasible for the reference model, since it uses the same simulator, it is not required to update the solution periodically, thus GA is run only once at midnight as the engagement. The algorithm parameters (maximum 10 generations and 400 individuals per generation) were chosen empirically as the best balance between low solution time and high profit.

First, the runtime of MP Tracker is about 3.6s in average with a standard deviation of 3.3s, while the runtime of GA Tracker is about 251s with a standard deviation of 15.2s. The MP Tracker is then about 70 times faster than GA Tracker, but both are eligible for a use on site since 15 minutes are allowed to the computation. Comparing the generated profits over the 24 instances, we see on Figure 4 that the MP Tracker (J GA-M P in blue) is always more profitable than the GA Tracker (J GA-GA in red) up to 9.7% higher (about 110e). The average daily profits are, respectively, 1006 and 975e, with a standard deviation of 156 and 136e. The difference is significant in practice and it confirms our first analysis that a detailed model of the system dynamics is not a requirement. This also highlights the benefit of solving the problem at optimality since GA did not generate the individuals correponding to the production plans obtained with MP.

Finally, we observe on Figure 4, that higher profits were obtained for 21 instances out of 24 when computing the engagement, in the Predictor, also with an MP solver (see configuration J M P -M P , in purple), but the difference with J GA-M P is tight.

D. Worth of optimization

Finally, our third experiment evaluates the worth of applying advanced optimization techniques compared to simplest BAU strategies. We thus experimented the BAU strategy described in Section III-C and observed the real production achieved when applying this command through PMS. The generated profits are depicted as J RB-RB (in grey) on Figure 4.

BAU generate profits slightly higher that GA on only 3 instances, where MP was significantly higher. On the 21 other instances, GA and MP profits were up to 100 euros higher.

V. CONCLUSION On our study case, the MP-based approach constantly generated the highest profits -often significantly higher than the two alternatives -in only few seconds of computation time. It suggests that seeking for optimality is profitable, even when considering a loose approximation of the system dynamics, thanks in particular to the periodic recomputation of the solutions. Because one main criticism of the MP approach is to rely on an expensive proprietary solver, we should note that open-source solvers could be used instead to compute optimal solutions in much less time than the 15 minutes allowed. Finally, because mathematical programming is flexible enough to handle model variants and uncertain data, it could be interesting to have the confirmation of these observations on various microgrid study cases and, in the opposite case, to understand the limits of the approach.

Fig. 1 .

 1 Fig. 1. Architecture of the microgrid

Fig. 2 .

 2 Fig. 2. A valid production profile for the CRE ZNI 2015 model.

  Figure on the left depicts the battery state of charge in the optimal MP solution soc opt (in red) and the observed profile soc real (in blue). Figure on the right depicts the production plan u in the MP optimal solution u opt (in red) and the observed profile u real (in blue). The engagement d is depicted in purple and the tolerance range in green.

Fig. 3 .

 3 Fig. 3. Profiles of state of charge (left) and production plan (right) in the MP optimal solution (red) and as observed (blue) for setting PP.

Fig. 4 .

 4 Fig. 4. Daily profit over the 24 days.

TABLE I DEVIATION

 I BETWEEN THE MP OPTIMAL SOLUTION AND THE RESULT OF THE OPTIMAL PRODUCTION CONTROL, AND GENERATED PROFITS.

		∆J (e)	∆soc (%)	∆u (kW)	J (e)
		avg	max	avg	max	avg	max	avg stddev
	PP	0.3	1.4	3.33	3.92	0.08 0.13	964.3	156.7
	P15	0.8	5.9	0.11 0.147	0.06 0.14	1006.3	158.0
	PN	61.6 81.1	5.18	6.54	4.56 5.96	970.9	152.5
	N15	0.0	0.2	0.09	0.11	0.06 0.09	1009.3	160.3


Amelia Carolina Sparavigna

The Geometry of the Roman Torino, that is to say the Varatio of Augusta Taurinorum

Keywords: Centuriaton, Orientaton of Roman colonies, Varare, Varato

This work is proposing the use of the geometry based on rectangular triangles, the varato, in the study of the planning of Roman Torino, that is, of Augusta Taurinoruma It is known that the planning of Roman towns was based on the centuriaton, a grid of parallel and perpendicular streets, where the two main axes were the Decumanus and the Cardoa By means of the varato, the angle the Decumanus was forming with the east-west directon was given by the rato of the cathet of a rectangular trianglea In the case of Torino, the town planning seems based on the rato of cathet 1:2, which is giving an azimuth of the decumanus (Via Garibaldi) of 116,565°f rom true northa This value is in good agreement with the measurement of the directon of this streeta

The planning of the Roman colonies was mainly realized, by the ancient land surveyors, by means of a method which is known as limitatio or ceoturiatioa This planning is characterized by a layout in the form of a square or rectangular grids, created by parallel and perpendicular roads or canals, which were subdividing the setlement into iosulaea These parcels of land were the house-blocks in the towns or the agricultural plots in the countrysidea In some cases these plots were allocated to Roman army veterans [START_REF] Haverield | a Ancient town -planning[END_REF]a The main axes of the grids were the decumaous maximus and the cardi maximusa The parallel roads to them were the decumaoi and the cardioesa Of course, the choice of the place of a town or colony was mainly dictated by the local environment and the presence of rivers and roadsa To subdivide and label the plots of the land, the surveyor (grimatcus) identied a central viewpoint, known as the umbilicus agri or umbilicus silia Taking positon there and assuming the same orientaton of the decumanus, the gromatcus deined the territory with the following names: ultra, the land he saw in front of him; citra, the land behind him; dextera, the land to his right; sioistra, the land to his lef [START_REF] Barthel | a Römische Limitaton in der Provinz Africa[END_REF][START_REF]dextram appellauerunt quae septentrioni subiaceret, sinistram quae ad meridianum terrae esset, ab oriente ad occasum, quod eo sol et luna spectaret, sicut quidam architect delubra in occidentem recte spectare scripserunta Aruspices altera linea ad septentrionem a meridiano diuiserunt terram, et a media ultra antca, citra postca nominauerunta[END_REF]a According to the Latn writer Frontnus [START_REF]dextram appellauerunt quae septentrioni subiaceret, sinistram quae ad meridianum terrae esset, ab oriente ad occasum, quod eo sol et luna spectaret, sicut quidam architect delubra in occidentem recte spectare scripserunta Aruspices altera linea ad septentrionem a meridiano diuiserunt terram, et a media ultra antca, citra postca nominauerunta[END_REF], who was referring Varro, the Romans inherited this manner of subdividing the land from Etruscansa We read in [START_REF]dextram appellauerunt quae septentrioni subiaceret, sinistram quae ad meridianum terrae esset, ab oriente ad occasum, quod eo sol et luna spectaret, sicut quidam architect delubra in occidentem recte spectare scripserunta Aruspices altera linea ad septentrionem a meridiano diuiserunt terram, et a media ultra antca, citra postca nominauerunta[END_REF] that the decumaous runs from irieos to iccasus, that is from the sunrise to the sunseta Of course, this is a natural manner to identfy the plots of the land by a simple orientaton from east to west and from north to south, in order to have a conventonal method to label thema Actually, this happens because the centuriaton was a process at the end of which the land was parcelled out and assigned ("divisus et adsignatus") [START_REF] Cuomo | a Divide and rule: Frontnus and Roman land-surveyinga Studies in History and Philosophy of Science Part A[END_REF]a Afer Frontnus' words, some scholars, such as Walter Barthel for instance [START_REF] Barthel | a Römische Limitaton in der Provinz Africa[END_REF], argued for an orientaton of the decumanus towards the point of the horizon where the sun was rising, secuodum cielum. Other scholars maintained that the orientaton was just dictated by the local environment or by the main roads of the area, as stressed by Le Gall and Castagnoli [START_REF] Castagnoli | a Le ricerche sui rest della centuriazionea Storia e Leteraturaa Roma 1958a[END_REF][START_REF] Gall | a Les romains et l'orientaton solairea[END_REF]a In any case, to have the regular grids of the limitaton matching the local environments, secuodum oaturam, the Roman surveyors, the Gromatci, had to rotate the gridsa Actually, these surveyors possessed some methods to adapt the grid to the land, methods that we can include under the term of varati. The literature of the Gromatci shows that the varato was based of the geometry of triangles [START_REF] Libertni | a Gromatci Veteres -Gli Antchi Agrimensori -Traduzione in italiano con comment, igure, schemi e illustrazioni a cura di Giacinto Libertni e con presentazione di Gianluca Soricellia Isttuto Di Studi Atellani, Fratamaggiore, Naples & Copernican Editons[END_REF][START_REF] Campbell | The Writngs of the Roman Land Surveyorsa Introducton, Text, Translaton and Commentary (Journal of Roman Studies Monograph 9) a London: Society for the Promoton of Roman Studies[END_REF]a Several artcles are discussing the link between geometry and Roman land surveyinga In [START_REF] Decramer | a Approche géométrique des centuriatons romainesa Les nouvelles bornes du Bled Seguia Histoire & mesure[END_REF], we ind a detailed discussion of the centuriaton, applied in partcular to the analysis of the Roman Tunisiaa In [START_REF] Congés | a Modalités pratques d'implantaton des cadastres romains : quelques aspects (Quintarios Clauderea Perpenderea CultellareaVarare : la constructon des cadastres sur une diagonale et ses traces dans le Corpus agrimensoruma Mélanges de l'école française de Rome Année[END_REF], we ind explained -with several illustratons -the methods used by the Romans and the Euclidean geometry involved in thema Let us stress that in [START_REF] Congés | a Modalités pratques d'implantaton des cadastres romains : quelques aspects (Quintarios Clauderea Perpenderea CultellareaVarare : la constructon des cadastres sur une diagonale et ses traces dans le Corpus agrimensoruma Mélanges de l'école française de Rome Année[END_REF] the author, Roth-Congès, uses the verb varare, from which varati is cominga This verb can be translated in the acton of measuring by means of triangulaton [START_REF] For | see the link htps://wwwadizionario-latnoacom/dizionario-latno-italianoaphp? lemma=VARO200 ; vara[END_REF]a In [START_REF] Peterson | a Design and Performance of the Varatoscopea BAR Internatonal Series[END_REF], it is told that the term varati, also writen uarati, refers in general to the process of oblique surveyinga It has two approaches: the v arati flumiois, by means of which the distance of an inaccessible point could be calculated by the constructon of right angled triangles, and the varati io agris divisis, that the surveyors could made by means of lines "along the hypotenuse of right angled triangles deined in a rectangular coordinate system" [START_REF] Peterson | a Design and Performance of the Varatoscopea BAR Internatonal Series[END_REF]a This second varati is the one which is statstcally analyzed in [START_REF] Peterson | a Design and Performance of the Varatoscopea BAR Internatonal Series[END_REF]a As stressed by John Peterson, when introducing his analysis, the variety of observed oblique relatonships, that is (1:1, 1:2, 2:3, 3:4 )) , is limiteda Besides the study of varati concerning the Hadrian's Wall and the Weterau Limes [START_REF] Peterson | a Design and Performance of the Varatoscopea BAR Internatonal Series[END_REF], we can ind the varati considered for the orientaton of the Roman towns in Spain [START_REF] Orila | a Las estructuras ortogonales de nueva planta en época romanaa De la varato y sus variacionesa Granada[END_REF][START_REF] Rodríguez-Antón | a The Uarato and Its Possible Use in Roman Urban Planning to Obtain Astronomical Orientatonsa[END_REF]a In [START_REF] Sparavigna | a The Geometry in the Urban Layout of the Roman Como and Verona: The Same Soluton to Diferent Problemsa SSRN Electronic Journala[END_REF][START_REF]Amelia Carolina Sparavignaa The Roman Towns and the geometry -Examples of Varatoa 2019a Zenodoa 18[END_REF], we have analyzed some Roman towns in Italy, Como, Verona, Pavia, and othersa In [START_REF]Amelia Carolina Sparavignaa The Roman Towns and the geometry -Examples of Varatoa 2019a Zenodoa 18[END_REF], Torino was shortly discusseda Here we consider this town speciicallya In the discussion here proposed, let us assume that the iosulae of the centuriaton are squaresa Then the town planning is essentally a chess board of insulaea The towns of Como and Verona, as shown in [START_REF] Sparavigna | a The Geometry in the Urban Layout of the Roman Como and Verona: The Same Soluton to Diferent Problemsa SSRN Electronic Journala[END_REF], are examples of a chessboard rotated to form, with the east-west axis, a rectangular triangle with sides 3,4 and 5 (the fundamental Pythagorean triple) a In the Figure 1, the rotated chessboard is showna

In the Figure 1 we can see the rectangular triangles (Pythagorean triple 3,4,5) which could have been used in the varati of Como and Verona [START_REF] Sparavigna | a The Geometry in the Urban Layout of the Roman Como and Verona: The Same Soluton to Diferent Problemsa SSRN Electronic Journala[END_REF]a The hypotenuses are coincident to the eastwest directona The cathet have to correspond to 3 and 4 iosulae respectvelya The two cathet give the directons of the decumaous and the cardi (in this case 4 and 3) a The angle between the decumanus and the hypotenuse is α=arctan( 3/4)=36.87°a Then, the azimuth from true north of the decumaous is 53,13°a

Let us now analyse the case of the Roman Torino, that is, Augusta Taurinorum. About the foundaton and the orientaton of Torino, we discussed in [START_REF] Sparavigna | a Discussione di alcuni artcoli sulla fondazione di Augusta Taurinoruma Zenodoa htp[END_REF][START_REF] Sparavigna | a The Orientaton of Julia Augusta Taurinorum (Torino) a[END_REF][START_REF] Sparavigna | a Julia Augusta Taurinorum, an archaeoastronomical reloada[END_REF]a Torino was probably founded in 28/27 aaCa as the cilioia Julia Augusta Taurioiruma The Roman town was placed close to the confluence of Dora and Po rivers, at the foot of the hills (see Figure 2) a In 69 AD, the colony sufered a great ire as reported by Tacitus [START_REF] Sparavigna | La note che la legione parta Zenodoa htp[END_REF]a It was probably afer this ire that the colony was subjected to some restructuring worksa The colony received a sewage system and an aqueduct, and some streets were paveda The network of the Roman sewage system was running under the decumani and cardi of the town [START_REF] Torinoa | Impianta Available at idraulicihtp[END_REF]a Under Via Garibaldi, the decumanus maximus of the Roman colony, it was running the greatest sewera Actually, we can easily appreciate that Via Garibaldi slightly slopes down from Piazza Statuto to Piazza Castello, that is, approximately from the west gate to the east gate of the Roman towna It means that, in the irst century AD, the decumanus of Augusta Taurinorum was rebuilt above the greatest sewer, which was gently sloping down to help the drainage of watersa The Roman town, its perimeter and the positon of the gates are given in the Figure 3a Zenodo. [START_REF]Amelia Carolina Sparavignaa The Roman Towns and the geometry -Examples of Varatoa 2019a Zenodoa 18[END_REF] October 2019 -DOI 10.5281/zenodo.3493368 The angle between the decumanus and the hypotenuse is α=arctan(1/2)=26,565 °a The azimuth from true north of the decumaous is 116,565°a In [START_REF] Massone | a Scheda: la misurazione dell'azimuta Giornale di Astronomia[END_REF], we ind a value of the measured azimuth of Via Garibaldi of 116,379° +/-0,002a The diference 0,186°, that is 11' of degree, is a very small diference indeeda So it seems that the geometry proposed by Gaetano Barbella could have been the varato used for Torino, to have the best agreement between the planning of the colony and the local environmenta

To conclude, let me quote an important sentencea As told in [START_REF] Boeing | Urban Spatal Order: Street Network Orientaton, Coniguraton, and Entropy[END_REF], the author of [START_REF] Kaiser | Roman Urban Street Networks: Streets and the Organizaton of Space in Four Cites[END_REF], Alan Kaiser, is telling that the "Roman Empire used standardized street grids to efciently lay out new towns and colonies during rapid imperial expansion"a Then, also Torino had its planning determined by a standardized urban layout, based on the rato 1:2a

Figure 2 :Figure 3 :

 23 Figure 2: Thanks to Acme Mapper, htps://mapperaacmeacom, we can appreciate the positon of Augusta Taurinorum, the red square, and the terrain surrounding ita