Wale Arewolo

Philipp Blechinger

Catherina Cader

Yannick Perez

Wale Arowolo
email: adewale.arowolo@u-psud.fr

Seeking workable solutions to the electrification challenge in Nigeria: Minigrid, reverse auctions and institutional adaptation

Keywords: Nigeria Power Sector, Reverse Auction, Geographical Information System (GIS), Market Design, Energy

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Only 58% of Nigerians have access to electricity with 78% access in urban areas and 39% in rural areas (IEA and World Bank, 2017). Furthermore, an estimated 80% of those with access also use an alternative source of electricity supply, mostly diesel generators due to reliability concerns (IEA, 2017). Nigeria currently has an available grid power hovering around 4,000MW for over 190 Million people. It is estimated that the Nigeria economy loses US$ 29.3 Billion annually due to the lack of adequate power supply and is estimated to have lost US$ 470 Billion in GDP since year 2000 due to under-investment in power infrastructure [START_REF] Psrip | Power Sector Recovery Implementation Program[END_REF].

Providing access to electricity is frequently considered a precondition for sustainable development [START_REF] Grimm | A First Step up the Energy Ladder? Low Cost Solar Kits and Household's Welfare in Rural Rwanda[END_REF]. Growing evidence, both micro-and macroeconomic, suggests that better electricity infrastructure significantly boosts economic growth and improves a range of development outcomes [START_REF] Alby | Firms Operating under Electricity Constraints in Developing Countries[END_REF].

Detailed geospatial modelling suggests that decentralized systems, led by solar photovoltaic in offgrid/minigrid systems, are the least-cost solution for three-quarters of the additional connections needed in Sub-Sahara Africa (IEA, 2017). In addition, renewable minigrid technologies are projected to have an especially large potential in providing universal access to electricity in Sub-Sahara Africa [START_REF] Dagnachew | The role of decentralized systems in providing universal electricity access in Sub-Saharan Africa -A model-based approach[END_REF]. Dramatic cost declines in solar and wind technologies, and now energy storage, open the door to a reconceptualization of the roles of research and deployment of electricity production, transmission, and consumption that enable a clean energy transition [START_REF] Kittner | Energy storage deployment and innovation for the clean energy transition[END_REF]. Energy storage solves the temporal mismatch by storing excess PV output in a battery for later consumption [START_REF] O'shaughnessy | Solar plus: Optimization of distributed solar PV through battery storage and dispatchable load in residential buildings[END_REF].

Moreover, the long-term perspectives of the solar PV technology are very favourable based on the ongoing technological improvements and cost reductions in PV technology, batteries and electric appliances [START_REF] Zubi | Techno-economic assessment of an off-grid PV system for developing regions to provide electricity for basic domestic needs: A 2020-2040 scenario[END_REF]. Over the last decades the growth of PV energy use was mainly driven by public incentives, but the shift to an economic driven use of solar PV electricity as one of the lowest cost electricity supplies is obvious [START_REF] Jäger-Waldau | PV Status Report 2017[END_REF].

Rapidly decreasing technology costs, increasing reliability, and a growing track record of deployments has strengthened the case for the accelerated adoption of minigrid solutions (IRENA, 2016).

Nigeria has good solar resource potentials (the annual average solar radiation varies from 12.6 MJ/m 2 /day (3.5kwhr/m 2 /day) in the coastal latitudes to about 25.2MJ/m2/day (7 kwhr/m 2 /day) in the North [START_REF] Fgn | The Nigeria Power sector investment opportunities and guidelines, Ministry of Power, Federal Republic of Nigeria[END_REF] 5 . Despite this potential, solar PV in offgrid/minigrid applications is still in its infancy. According to a recent study for the World Bank, a cumulative capacity of 236kW solar minigrid serving about 9,100 people exist in Nigeria and the operators are charging a high price between US$0.58 and US$0.61 per kWh when they do not receive capital grants [START_REF] Esmap | Minigrids In Nigeria: A case study of a promising market[END_REF]. As a potential solution, [START_REF] Arowolo | Designing Reverse Auctions for Solar Power in Nigeria as a potential energy access solution[END_REF] 6 argued in favour of reverse auctions for solar PV plus storage for the Nigeria market and discussed how the auction run can be successfully executed. The author argued that the capacity to be contracted should be determined based on reliable power demand projections, locational information from Geographical Information System (hereinafter 'GIS') and energy system simulation tools. This is to provide reliable data on the needed procurement capacity (what to bid for) and at specified locations (where). The author posited that the availability of this information is crucial to achieve a competitive auction and effectively mitigate failure risk(s).

Models addressing electricity access typically need large volumes of reliable energy-related data and information, which in most developing countries have been limited or not available [START_REF] Mentis | Spatially explicit electrification modelling insights. Applications, benefits, limitations and an open tool for geospatial electrification modelling[END_REF]. The use of innovative research methodologies such as the application of GIS and remote sensing technologies, spatial evaluation and interpolation of statistical and economic data will support and facilitate strategic energy decisions (Szabo et al, 2011). GIS are a powerful calculation and analytical tool of spatial variables (Amador and Dominguez, 2005). GIS layers can provide location specific energy-related information that has not been previously accessible to enable filling data gaps in developing countries using remote sensing techniques and allows for the assessment of spatially explicit energy demand information [START_REF] Mentis | Spatially explicit electrification modelling insights. Applications, benefits, limitations and an open tool for geospatial electrification modelling[END_REF].

Combining GIS, energy system optimization tools and market and regulation design expertise for research and analysis, this paper attempts to bridge the knowledge gaps and make contribution to the solar PV auction design and implementation literature for the Nigeria Power sector in five significant ways. First, we provide the consumer locations without electricity access (hereinafter 'non-electrified clusters') with their breakdown at the administrative decision-making levels (36 states plus FCT and the local governments of Nigeria). Second, we analysed the non-electrified clusters' population to determine the most populated ones that will provide the highest impact on solving electricity access problem with minigrid auctions. Third, we performed the load demand assessments and techno-economically optimized solar PV plus battery capacities for the selected clusters to achieve high quality power supply enhanced by some demand side management. Fourth, we determined the land requirements and ascertained the availability at the non-electrified clusters to come up with a shortlist of 233 nonelectrified clusters that require 3,280 MW solar PV plus 7,518 lithium-ion MWhr battery storage capacities. This should be the priority locations for the first sets of solar PV plus battery storage minigrid auctions in Nigeria. Finally, we provide the policy implications and implementation guidelines to fit in our proposed design with the existing market and regulatory framework to facilitate decision-making.

We thus strengthened the argument for auctions design for the Nigeria market with the critical groundwork required for minigrid/offgrid electrification planning and the successful real-life implementation as a contribution to the achievement of SDG7 'Ensuring access to affordable, reliable, sustainable and modern energy for all by 2030' [START_REF] Sdg | United Nations Global Sustainable Development Report[END_REF] for the Nigeria Power sector.

This paper is structured as follows: Section 2 is the literature review on geospatial electrification planning for Nigeria. Section 3 is the methodology. Section 4 is the results and discussions.

Section 5 is the discussion of the regulatory limitations to overcome to successfully implement our proposed solution. Section 6 is the policy implications and section 7 is the conclusion.

LITERATURE REVIEW ON GEOSPATIAL ELECTRIFICATION PLANNING FOR

NIGERIA

In this section, we provide a review of the literature that has attempted to use geospatial modelling for electrification planning for Nigeria.

In their study, Szabo et al 2011 applied GIS to determine whether solar PV, diesel generator or grid extension is the least cost electrification option in African countries. The authors' analysis shows where grid extension will be the least cost option based on a distance of 10km, 30km and 50 km from the low (LV), medium (MV) and high voltage (HV) lines respectively.

Their result in (Szabo et al, 2011 andSzabo et al, 2013) for Nigeria as seen on the map of Africa suggests grid extension as the most economical option for a large part of Nigeria.

In a World Bank study [START_REF] Afrea | Achieving universal access in the Kano Electricity Distribution Company Service Area, Africa Renewable Energy and Access Program[END_REF], the authors used the Network Planner model to determine the least-cost geospatial implementation plan for the Kano Electric Distribution Company (KEDC is one of the 11 Discos 7 in Nigeria). KEDC covers Kano, Jigawa and Katsina states. The analysis focused on the grid and offgrid roll-out to achieve universal electrification for an estimated population of 35 million people by 2030 using a weighted average household demand of 840 kWh/year. The authors concluded that grid roll-out is the least cost solution to provide power to approximately 97% of the households requiring a total cost of US$3.3 Billion 7 Disco -Distribution Companies for universal access in this area and to provide new or improved connections for 5.3 million projected households by 2030. In a similar World Bank study [START_REF] Afrea | Achieving universal access in the Kaduna Electric service Area, Africa Renewable Energy and Access Program[END_REF], for the Kaduna Electric covering four states (Kaduna, Zamfara, Kebbi and Sokoto), using a weighted average household demand of 1,330 kWh/year, the authors concluded that grid roll-out is the least cost electrification for more than 99% of the households. The authors also concluded that approximately US$3.8 Billion is required to reach about 5.8 million households for an estimated 39 million people by 2030. [START_REF] Mentis | Lighting the World: the first application of an open source, spatial electrification tool (OnSSET) on Sub-Saharan Africa[END_REF] used a spatially explicit continent-wide model (at 1km by 1km resolution) in their Open-Source Spatial Electrification Tool (OnSSET) to consider varying electricity consumption targets to derive the least cost solutions for infrastructure development and resulting generation mix for Nigeria. The authors designed ten scenarios and estimated an investment need between a low of US $11.7 Billion (20% grid, 0% minigrid and 80% standalone) and a high of US $153 Billion (78% grid,16% minigrid and 6% stand alone) to provide universal access in Nigeria. The authors' focused on a technology split between grid, minigrid and standalone solutions based on different levels of electricity access. However, the model does not provide the required macro-information to mobilize actions for initiatives such as Sustainable Energy for All or Power Africa [START_REF] Mentis | Lighting the World: the first application of an open source, spatial electrification tool (OnSSET) on Sub-Saharan Africa[END_REF].

In a related work, [START_REF] Mentis | A GISbased approach for electrification planning -A case study on Nigeria[END_REF] used GIS to analyze Nigeria on a settlement level with a grid cell size of 2.5km 2 for electrification planning using 170kWh/capita/year for rural and 350 kWh/capita/year for urban areas. In their conclusion, the authors advocated grid extension for 86% of the newly electrified population in Nigeria with 2030 as the time horizon. This will be at a cost of US$ 11.4Billion based on the assumption that there is available investment funds and human capacities.

In their paper, [START_REF] Ikejemba | Locating solar and wind parks in south eastern Nigeria for maximum population coverage. A multi-step approach[END_REF] focused on the location analysis to design a network of solar parks in Anambra State (South East Nigeria) using geographical and demographical characteristics. The authors used a square grid of 3.3km by 3.3km and assumed population is uniformly distributed around the grid points to obtain 262 suitable locations for the solar parks. The authors then made a shortlist of 71 potential locations for the installations. Furthermore, the authors assumed that households are at a distance of 30km from the nearest park and used mathematical (integer linear) programming to determine the best five solar park locations. Thereafter, they allocated the demand of each grid point to conclude that 1,157MW of generation capacity is required by 2020 in the state. [START_REF] Ohiare | Expanding electricity access to all in Nigeria: a spatial planning and cost analysis[END_REF] used the network expansion planning (Network Planner) model to estimate the cost projections for three electrification technology options: (a) offgrid (solar PV with diesel generator), (b) diesel generator minigrid and (c) grid electrification. The author recommended grid extension for 98% of the currently non-electrified communities in Nigeria at an estimated total cost of US$34.5 Billion for 125 million people by 2030. However, the author mentioned the limitation of his model to include the lack of consideration for the constantly changing nature of costs brought about by economic and technological changes. Others are the assumption that all cost conditions remain unchanged throughout the planning period, the costs excluding the technical and geographical constraints of connecting the communities and the exclusion of generation cost for the grid technology [START_REF] Ohiare | Financing Rural Energy Projects in Developing Countries: A Case Study of Nigeria[END_REF].

Knowledge gaps in the literature

Trotter et al (2017), reviewed 306 scientific journal articles on electrification planning and related implementation analysis on Sub-Sahara Africa and identified further research need as follows. First, several reviewed articles point to the general lack and unreliability of data necessary for electrification planning. Second, no single article could be identified for 12 countries and fewer than 5 articles for 36 of the 49 countries. Third, there is no single study that has planned electrification along more than 3 out of the 5 possible value chain elements.

Therefore, the authors advised that it is crucial to conduct electrification planning addressing the specific country or region where it is supposed to be implemented. Moreover, the authors concluded that it would be enlightening to understand the complete implication of a proposed alternative, from understanding the exact demand to selecting a generation technology, operation and transmission planning and the successful implementation.

Considering the research works that focus on Nigeria discussed above, AFREA (2015) and AFREA (2016) combined are limited to only seven (7) states out of the 36 states plus FCT8 in Nigeria. Furthermore, their results could only support high level planning to provide universal grid access with the investment total cost estimate for the advocated grid extension and intensification. The authors neither discussed generation source (i.e. where the power supply for the recommended grid roll-out will come from), the currently insufficient status of power supply nor focused on the detailed design of any specific generation technology for the 97% of households in their medium-term interim off-grid solution discussed. Furthermore, their research lacks discussion on the market design and regulatory framework to make their proposal compatible with the existing system to facilitate implementation. Furthermore, although [START_REF] Ohiare | Expanding electricity access to all in Nigeria: a spatial planning and cost analysis[END_REF] provided broad national coverage compared to AFREA (2015) and AFREA (2016), the author neither focused on any specific generation technology nor performed the technical scrutiny of the resource potentials at different locations to facilitate operation. [START_REF] Mentis | Lighting the World: the first application of an open source, spatial electrification tool (OnSSET) on Sub-Saharan Africa[END_REF]Szabo et al (2011) are at a low resolution of 1km by 1km while [START_REF] Mentis | A GISbased approach for electrification planning -A case study on Nigeria[END_REF] is at a grid cell size of about 2.5km 2 . The papers applied frameworks developed by the authors to provide a broad and top-level overview of Nigeria. The papers lacked indepth country discussions at the administrative levels that could facilitate informed decisions for operation and implementation. [START_REF] Ikejemba | Locating solar and wind parks in south eastern Nigeria for maximum population coverage. A multi-step approach[END_REF] work is limited to only a state (Anambra), and is at a low resolution of 3.3km by 3.3km. In addition, the operation and implementation issues to integrate their proposal with the existing system were not discussed.

Our Present Work

We attempt to bridge the knowledge gaps identified in the literature (with a focus on Nigeria) by providing the complete implication of minigrid/offgrid electrification plan with solar PV plus battery storage technology. Our research takes an interdisciplinary approach by combining GIS (for geospatial planning at a high 100m by 100m resolution), energy system optimization tools with electricity market design and regulation expertise to achieve the following 1.

Generate a robust database of the non-electrified clusters with their breakdown at the administrative levels. 2. Provide the detailed techno-economic optimizations for solar PV plus battery storage at the prioritized non-electrified clusters 3. Determine the land requirement per cluster and ascertain availability and 4. Address the implementation issues to adapt our proposed design to fit in with the existing market and regulatory framework based on in-depth understanding of the status quo in the Nigeria Power sector.

METHODOLOGY

Electrification modelling combining geospatial planning with energy system modelling requires a series of input data to generate a realistic scenario analysis [START_REF] Cader | Electrification planning with focus on hybrid minigrids -A comprehensive modelling approach for the Global South[END_REF]. Our research approach is based on a step-by-step procedure: First, consumer clusters (such as villages, community, town or city etc.) are identified. Second, the electrification status of each consumer cluster is determined with satellite night light imagery and qualitative data from the school location dataset. Third, we provide the detailed breakdown of the non-electrified clusters at the administrative levels. Fourth, the population of the clusters are determined, and the most populated non-electrified clusters were selected for further analysis. Fifth, we applied the World Bank multi-tier framework (ESMAP, 2015) to make the demand projection at the selected non-electrified clusters. Sixth, we performed techno-economic optimization to determine the solar PV and battery sizes at the selected non-electrified clusters. Seventh, we analysed the clusters with GIS to ascertain the land availability. Finally, based on the above processes, we prioritized the most populated and feasible non-electrified clusters to run solar PV plus storage minigrid auctions in Nigeria.

Identification of consumer clusters

In this section, we will discuss the process to identify the locations of the existing and potential electricity consumers (the clusters) using population raster data, school location dataset and polling units' datasets.

Population Raster data

To identify the populated areas in Nigeria, spatial demographic dataset from (Worldpop, 2015) is used as the base source [START_REF] Bertheau | Electrification modelling for Nigeria[END_REF] with population density at 100 meters spatial resolution. This dataset is the most sophisticated spatially resolved (raster9 cell pixels) dataset on population figures for Nigeria at the highest (100m by 100m) publicly available resolution.

For each pixel, an estimated population number is provided. The raster pixels were analysed, and the population density threshold is set at 7 people per pixel ('ppp') i.e 700 people per km². This threshold indicates the location of a community, town or city.

The dataset is then reclassified into a binary dataset containing values of "1" for all areas with a population equal and above 7 'ppp' and "0" for all areas with a population below 7 'ppp'. The resulting binary raster dataset containing values of "1" are converted to a vector10 dataset and a buffer zone with a radius of 500 meters is added around the cluster. The 500m is used based on the typical values for connecting locations within the same low voltage11 grid.

Finally, the populated areas are derived from the raster dataset and the consumer clusters are created. This process is illustrated below (Fig. 1).

Fig 1: Identification of populated consumer clusters from raster dataset in Nigeria

Thereafter, the (Worldpop, 2015) dataset is extended with a school dataset (NMIS, 2014). The school dataset is provided by the Nigeria Millennium Development Goal (MDG) Information System. It contains information regarding the educational facilities, number of students, facilities and the administrative information of Nigeria schools. The dataset was collected by the Government and Sustainable Engineering Lab at Columbia University (NY) to facilitate local planning at the national scale of the MDG goals. Trained enumerators using androidbased smartphones were used to collect locational information using GPS combined with data available through surveys as described in (Pokharel et al, 2014). This provides a rigorous, georeferenced baseline data of 98,667 educational facilities in Nigeria. The school point dataset is used to extend the number of the consumer cluster to achieve a higher coverage of the populated areas. Similarly, a buffer zone with a radius of 500 meters is added around each of the schools.

Polling Units Dataset

To further improve accuracy, the Worldpop and school datasets were extended with the polling units' dataset of INEC -Independent National Electoral Commission (INEC, 2012) of Nigeria.

INEC's data provides information about local polling units, the location of polling booths and further administrative information. This dataset covers approximately 130,000 polling locations including their latitude/longitude coordinates (geo-located) that match satellite imagery. It is comprehensive, since most settlements have at least one polling unit [START_REF] Afrea | Achieving universal access in the Kaduna Electric service Area, Africa Renewable Energy and Access Program[END_REF]. Furthermore, the INEC data is the highest resolution and most validated national data source currently available and there is often a close match between polling unit locations and human settlements, particularly larger villages which are easily identified in satellite imagery [START_REF] Afrea | Achieving universal access in the Kano Electricity Distribution Company Service Area, Africa Renewable Energy and Access Program[END_REF]. Similarly, a buffer zone of 500 meters is added around the polling units. The general approach is depicted below.

Fig. 2: Identification of consumer clusters

To create a reliable and comprehensive database useful for decision-making, we combined the three datasets to produce a robust database. With this newly created database, we calculated the population in every cluster by summing up the raster population pixel values within each cluster. Finally, administrative areas dataset for Nigeria obtained from [START_REF] Gadm | Global Administrative Areas -Dataset for Nigeria[END_REF], were assigned to each consumer cluster to further enrich the database with information at the administrative state and local government levels.

Determination of the electrification status

We determined the electrification status of the consumer clusters based on two steps: Night light imagery (NASA, 2015): These images are satellite data which shows night light emissions over an area at a spatial resolution of 750 meters by 750 meters. These regularly visible light emissions are usually based on street lamps or other permanent lights in the villages or communities. This high amount of light activity allows the assumption that any source of electricity exist in an area [START_REF] Cader | Electrification planning with focus on hybrid minigrids -A comprehensive modelling approach for the Global South[END_REF].

Qualitative data from the school dataset (NMIS, 2014): For each school, information is provided whether the facility is grid connected or not. Each cluster was defined as electrified where a grid-connected school is located within a region with night light emissions. Consequently, all the consumer clusters with observable night light emissions and information on grid connection from the school dataset are considered electrified by grid connection. See Appendix 6 for the map of Nigeria showing the consumer clusters with nightlight emissions and electrified schools.

Administrative breakdown of the non-electrified clusters

We selected the non-electrified clusters from our model for the entire Nigeria. Thereafter, we provide their detailed breakdown at the administrative decision-making levels (36 states and FCT and at the local Government areas in Nigeria).

Determination of Minigrid Auction sites at the non-electrified clusters

Using the non-electrified clusters identified, we performed the power demand assessment by scaling the population from >=100 to >=25,000 to determine the smallest possible number of the non-electrified clusters that captures the largest population of non-electrified people for the entire country. This would be the priority locations for the solar PV plus battery storage minigrid auctions. With this process, we identified the most populated 340 non-electrified clusters.

Load Demand projection

Load demand projections were estimated using an average household size of 6 people for rural areas in Nigeria (LSMS, 2016) and the 'Tier 5' assumptions of the World Bank multi-tier framework for minimum 23 hours daily power supply (i.e. minimum of 8.2kwh/day energy consumption) covering households power supply, productive engagements and community facilities (ESMAP, 2015). This should capture currently 'suppressed' demand and the anticipated load demand growth within the first year of installation due to the migration of people to the electrified locations, industrial and commercial activities. We thus derived the load demand projections for the most populated 340 clusters.

Energy System Optimization (Solar PV plus Lithium-ion Battery Storage) Sizing

Using HOMER (Hybrid Optimization of Multiple Energy Resources) software, we determined the optimal combination of the utility scale solar PV capacity (lifetime of 25 years) and the MWh lithium-ion battery capacity (lifetime of 15 years). We used the community load profile based on the peak power demand and the yearly energy requirement (derived from the tier 5 of the multi-tier framework) at the 340 selected clusters. Please see Appendix 5 for our assumptions. Furthermore, we performed the sensitivity analysis for five scenarios 0% (i.e. uninterrupted power supply), 5%, 10%, 15% and 20% capacity constraints. This is to ensure that for each scenario, we can analyze the different combinations of the excess energy generated and the unmet electrical load occasioned by the community load demand curve. Furthermore, the selected capacity constraints have direct techno-economic impact on the PV and battery size capacities and the Levelized Cost of Energy (hereinafter 'LCOE') calculation.

Hence, the optimized solution avoids overcapacity of solar PV panels and gives high battery autonomy. With some demand side management, it can reallocate the excess energy generated to periods with unmet load to achieve a minimum battery autonomy of 20 hours per day at the optimized LCOE. For instance, by using demand response -a reactive or preventive method to reduce, flatten or shift peak demand [START_REF] Ikpe | A means to an industrialisation end? Demand Side Management in Nigeria[END_REF]. Demand side management solutions are crucial in energy management and planning in Africa and should be incorporated in any modelling framework [START_REF] Ouedraogo | Modeling sustainable long-term electricity supply-demand in Africa Applied Energy 190[END_REF]. Furthermore, [START_REF] Mehra | Estimating the value of demand-side management in low-cost, solar micro-grids[END_REF] found that demand-side management capabilities can reduce the number of requisite solar panels and batteries due to the integration of real-time management and control.

This simulation was performed separately for each of the 340 clusters due to their different solar irradiations and ambient temperatures to decide the robust scenario per cluster.

Performance assessment using location specific parameters is very valuable since PV system cannot be considered a one-size-fits-all [START_REF] Okoye | Economic feasibility of solar PV system for rural electrification in Sub-Sahara Africa[END_REF].

Land Requirement and Availability

A concern regarding the large-scale deployment of solar energy is its potentially significant land use (NREL, 2013). Using data for utility-scale ground-mounted fixed-tilt solar PV from the US National Renewable Energy Laboratory (NREL, 2013), we used the value of 7.6 acres (30,755.87m 2 ) per MW for the installations between 1MW and 20MW and 7.5acres (30351.19m 2 ) per MW for the installations above 20MW to compute the total land requirement for each of the 340 clusters.

Thereafter, we analyzed each of the 340 clusters with a combination of the available GIS satellite imagery from Google (2018 imagery), Bing Aerial (2018 imagery), Google Streets and Open Street Maps to ascertain the availability of the total land area required on a cluster-bycluster basis. This analysis also helps to shortlist the number of clusters that should later be physical visited for onsite verification prior to the implementation of our proposed reverse auction. Based on the above processes, we provide the details of the best clusters for solar PV plus storage minigrid auctions in Nigeria.

RESULTS AND DISCUSSIONS

Non-Electrified cluster population

45,456 non-electrified clusters were identified in the entire Nigeria where an estimated 72.6 million people reside. However, about 10 million people without electricity access are out of the clusters. This population is assumed to live in small or dispersed rural settlements or have no permanent settlement locations and could not be assigned to any consumer cluster. In total, about 82.6 million people without electricity access were identified in Nigeria.

Our result is consistent in the range estimated by some renowned institutions. The World Bank

Administrative breakdown of the non-electrified clusters

Below is the map of Nigeria showing the non-electrified clusters.

Fig 3: Map of Nigeria showing the non-electrified clusters with black dots

We also provide the information on a state by state basis (36 states plus FCT) for the local governments located in the states to provide useful information to decision makers at the individual state and local government levels (please see Appendix 2). Below is a sample result of the non-electrified clusters with the breakdown at the Local Government Area (LGA) for one of the states (Adamawa). We chose this state for illustration because it is quite representative.

It has both high and low populated non-electrified clusters. Below is a snapshot of some results of the sensitivity/scenario analysis for a cluster (Lamurde -we chose to illustrate this populated cluster because it is representative with an average solar irradiation of 5.64kwhr/m 2 /day and yearly average air temperature of 26 0 C). We have thus provided a workable solution with solar PV offgrid/minigrid plus storage reverse auction to attract private investments and potentially solve the lack of funding and grid constraint/collapse problems.

Capacity

Levelized Cost of Energy (LCOE)

The LCOE of a given technology is the ratio of the lifetime costs to the lifetime electricity generation, both of which are discounted back to a common year using a discount rate that reflects the average cost of capital (IRENA, 2018a Based on our model and analysis, we understand the added cost of the battery storage to the solar PV system. Furthermore, we have provided some critical groundwork information to facilitate the auction design for the Nigeria case. Thus, we posit that it is not impossible to expect a well-designed solar plus storage auction to produce prices in the range of US$0.09-US$0.12/kWh in Nigeria. This cost will be at the low end of the current grid tariffs and can potentially achieve high quality power supply with some demand side management.

In comparison, the current grid tariff in Nigeria ( 2018) range from a low of US$0.07/kWh13 for LV residential customers in Eko Disco to a high of US$0.21/kWh for high voltage maximum demand industrial customers in Yola Disco (NERC,2015) and this power supply is only available at an average of 35.8 hours per week (LSMS, 2016). To offset these negative impacts, industrial firms in developing countries often opt for self-generation of electricity, even though it is widely considered a second-best solution [START_REF] Alby | Firms Operating under Electricity Constraints in Developing Countries[END_REF]. Most private enterprises in Nigeria are forced to resort to self-generation at a high cost to themselves and the economy at about US$0.20-0.30/kWh (World Bank, 2017). Therefore, we believe our design is techno-economically feasible for Nigeria with the right policy support frameworks addressed in the next section.

REGULATORY LIMITATIONS TO OVERCOME FOR BIDDERS

With the results obtained from our model and analysis, we have improved understanding of solar PV plus storage auction design for the Nigeria market. Furthermore, we have provided data on the clusters without electricity access as well as the most populated clusters that will have the largest impact on offgrid/minigrid solar PV plus battery storage reverse auction. In addition, we have ascertained the land availability and proved it is a techno-economically feasible potential solution to increase electricity access in Nigeria. In this section, we will discuss how to implement the solution from the market design and regulation viewpoint.

Nigerian Electricity Regulatory Commission Minigrids Regulation (NERC, 2016), is the current minigrid regulation guiding decision making in the Nigeria Power sector. Prospective bidders will read this regulation to make decisions on whether to participate or not in the auction run.

This will have direct impact on the competitiveness of the auction. Below, we will discuss the key clauses that need to be amended or repealed in the minigrid regulation.

• NERC (2016) Section 3 defined "Independent Electricity Distribution Network" or "IEDN" as distribution network not directly connected to a transmission system operated by the system operator excluding minigrids.

Our proposed offgrid/minigrid solution does not require connection to the transmission system. Therefore, this regulation should be amended. The successful bidders shall become Independent Electricity distribution network operators. Hence, the regulation should be expanded to make allowance for the utility scale offgrid/minigrids.

• NERC (2016) Section 4, Clause 1. This regulation shall apply to minigrids with Generation Capacity of up to 1MW. Within the regulations, the term minigrid is used for any isolated or interconnected minigrid generating between 0kW and 1MW of Generation Capacity.

This should be amended to redefine the minigrid capacity range. The generation capacity should not be restricted to an uppermost limit of 1MW. As shown in this paper, we have identified clusters where isolated utility-scale offgrid/minigrid between 6.45MW and 92.55MW can be installed.

• NERC (2016) Section 5 Clause 2 gives the requirement for a minigrid to have a 'generator' in its network to be operated by a minigrid operator or a third party.

The term 'generator' should be clarified. If it means a fuel (such as diesel) run power supply generator, this regulation should be repealed, as there is no justification to demand a generator for minigrid installation, it should not be a requirement. We have shown in this paper that the minigrid market design is feasible and economically viable without needing a 'generator' in offgrid application.

• NERC (2016) Section 6, Clause 3 and 4: A minigrid developer applying for a minigrid permit shall submit to the Commission an accurate description of the proposed distribution and generation system, including geographical depiction.

The accurate description of distribution, generation and geographical depiction of sites should not be the responsibility of the developers that have no hundred percent success guarantee in a bidding process. Otherwise, they may be deterred from participating considering the enormous transaction costs involved in gathering the information. The government should provide the information. The good news is that the effort in this research work has provided some of this critically required information, which the government can make available to prospective developers/bidders.

However, this does not absolve the bidders of the responsibility to conduct their due diligence.

• Perhaps, a major regulatory limitation is related to the Distribution licensee's (i.e Discos) expansion plans and minigrid deployment. NERC (2016) Section 7, Clauses 1b, c, d, e, f and g, Section 9 and 19 needs to be amended or repealed to allow an efficient development of the minigrid. Based on the Distribution licensee's expansion plans approved by the Commission, the minigrid activities will not interfere with the expansion plans into the designated unserved area of the Discos.

The Discos expansion plans are best on paper as they lack the financial viability to bring them to fruition. Currently, the dilapidated infrastructure of the Discos do not have the capacity to fully accept the loads currently fed to them, hence load rejection is a common occurrence. In addition, publicly available financial statements of some of the Discos show they are consistently making losses since privatization (they have cumulatively made a total loss of over US$ 2 Billion between 2013 and 2016 -see Appendix 4). This is 42% more than the $1.42 Billion invested by the Discos to acquire the assets during privatization. Moreover, assessing debt financing for investment is a significant challenge for the Discos as they are unable to provide convincing cashflow projection to the lenders that shows their ability to repay loans. Although, equity funding is an investment option for the Discos to explore, it seems the Disco investors are unwilling to stake in equity considering their recent threat to the government to sell off their business at discounts. contract third party investors in a bidding process to finance the procurement of meters.

However, it remains to be seen if this initiative will achieve the desired result.

In a nutshell, while the government seems to be making some efforts to address the sectoral liquidity issues, refurbish some dilapidated infrastructure and to resolve the metering problems.

It appears the Discos can barely survive further in business (as material uncertainty exists that cast significant doubts on their ability to continue as a going concern). This opinion was unanimously expressed by their auditors PWC, KPMG, EY etc. in their financial statement).

Therefore, the Discos should focus on solving their technical and financial challenges in their current operational areas in centralized generation and give room for other private players in decentralized generation in unserved areas. This will enable the centralized and decentralized generation to effectively co-exist. As the threat of grid expansion remains a major uncertainty for the investors, any assurance in this respect can improve the investment prospects [START_REF] Bhattacharyya | Minigrid based off-grid electrification to enhance electricity access in developing countries: What policies may be required?[END_REF].

On this basis, these potential investment show-stopper clauses 7, 9 and 19 and their accompanying annexes should be amended. The Commission has the power to amend or repeal, in whole or in part the provisions of the regulation as contained in Section 26 of the Regulation (NERC, 2016). Moreover, Section 71(6) of the primary legislation, Electric Power Sector Reform Act EPSRA Act. ( 2005) on the terms and conditions of licenses clearly shows that exclusivity was not intended for Discos licensees as follows: "unless expressly indicated in the license, the grant of a license shall not hinder or restrict the grant of a license to another person for a like purpose and, in the absence of such an express indication, the licensee shall not claim any exclusivity, provided that the commission may allow a licensed activity to be exclusive for all or part of the period of the license for a specific purpose, for a geographical area, or for some combination of the foregoing". Besides, there seems to be no official record till date that shows any Disco has been granted exclusivity to an unserved area. Therefore, the Regulator should amend these clauses.

Similarly, the transmission expansion plan [START_REF] Fitchner | Transmission Expansion Plan. Development of Power System Master Plan for the Transmission Company of Nigeria[END_REF] should focus on projects to refurbish the dilapidated transmission network infrastructure that causes the frequent national grid collapse (total and partial blackout) problems instead of the grid expansion projects.

POLICY IMPLICATIONS

Solar PV Minigrid Auction to attract investment and increase electricity access

Minigrid auction with solar PV plus storage is a workable scenario and a reasonable path forward to attract investment considering the current financial constraints bedevilling the Nigeria power sector. It is noteworthy that the three tariff increase scenarios proposed by the Government/World Bank's team in the Power Sector Recovery Program (PSRP) framework

were not implemented. The only option left in the framework is to increase the tariff in July 2019 while the government supports the sector with US$7.5 Billion.

In the best-case scenario, where the government provides this fund and is able to increase tariff in July 2019, the grid can be stabilized to provide about 4,000MW of electricity by refurbishing the dilapidated infrastructure to address the frequent grid collapse problems and 5,500 MW by 2021 (PSRP, 2017).

Therefore, solar PV plus storage minigrid auction can attract investment and improve electricity access through decentralized generation competitively as shown in this paper. Our thoroughly researched shortlist of 233 non-electrified consumer clusters will provide 3,280MW with expected high socio-political acceptability as it covers all states (except Lagos) in Nigeria.

Strengthening the institutional framework

Considering the depth of regulatory concerns, strengthening the regulatory framework will improve overall market performance and restore confidence to the sector [START_REF] Arowolo | Designing Reverse Auctions for Solar Power in Nigeria as a potential energy access solution[END_REF]. Critical to an enabling environment are both the policies on the books and the effectiveness of their implementation (RISE, 2016). Existence, scope and monitoring of officially approved electrification plans, utility transparency, monitoring and creditworthiness (RISE, 2016) are good starting points to strengthen the institutional framework in Nigeria.

Capacity building for sustainability

The sustainable management of a renewable energy project after its commissioning is an important aspect. Numerous implemented projects in Sub-Sahara Africa have failed to live up to their promises because effective management method /maintenance plans were not designed [START_REF] Ikejemba | Enhancing sustainable development in Sub-Sahara Africa -New integrated sustainability mechanism for securing substantial benefit of renewable energy projects[END_REF]. To ensure sustainability 25 of off-grid solutions in the long-term, the entire supply chain should be developed and properly supported. A local ecosystem should be developed for offgrid electrification activities, including local manufacturing and assembly capacity, ancillary after-sales service system, a pool of trained technicians, and demand creating services [START_REF] Bhattacharyya | Minigrid based off-grid electrification to enhance electricity access in developing countries: What policies may be required?[END_REF]. automation/control and power system engineering). In addition, there is also a need for the government to fund some trainings in a sustainable way (for instance, the NGSDP is presently self-financed by the applicants). This will empower more people to meet the skills requirements of the sector and enhance sustainable solutions. Finally, to facilitate the extensive development of solar PV in Nigeria, trends across the country show that efforts should be improved towards education through increased awareness [START_REF] Nwokocha | Photovoltaics in Nigeria. Awareness, attitude and expected benefit based on a qualitative survey across regions[END_REF].

Current efforts at the

CONCLUSION

This paper contributes to improving the economic, institutional and technical challenges in the Nigeria power sector through the lens of market design and regulation. We combined GIS at a high resolution with energy system optimization tools to bridge the knowledge gaps in the literature and to provide the critically needed information (at the administrative levels) to enhance decentralized offgrid reverse auctions with solar power plus storage in Nigeria.

We provide the consumer locations without electricity access with their breakdown at administrative decision-making levels (federal, 36 States plus FCT and local Governments of Nigeria). Thereafter, we analysed the most populated clusters that provide the most impact on solving electricity access problem in Nigeria with minigrid auction. Thereafter, we performed the load demand assessment and techno-economically optimized the solar PV plus battery capacities for the selected clusters to achieve high quality power supply with some demand side management.

Furthermore, we determined the land requirements and ascertained the availability of the land at the clusters to derive a final shortlist of 233 clusters covering 7.2 million people requiring 3,280 MW solar PV and 7,518MWhr battery capacities as the best locations for minigrid auctions in Nigeria. In addition, we showed it is a competitive solution with an LCOE from a low of US$0.18/kWh in the North to a high of US$0.25/kWh in Southern Nigeria and argued that it is not impossible to achieve competitive auction prices of US$0.09-US$0.12/kWh in Nigeria.

Besides, we discussed the regulatory limitations to overcome to make our proposed minigrid auction design adaptable to the Nigeria market. This centers on the need to amend or repeal specific sections and clauses in the current minigrid regulation (NERC, 2016). Finally, we provided the policy implications and guidelines to decision makers in the Nigeria power sector as follows. i. Solar PV plus storage minigrid auction can attract investment and improve electricity access through decentralized generation competitively. ii. Strengthening the institutional framework and iii. Capacity building for sustainability.

A limitation of this research is the ease of replicability. This work takes an interdisciplinary approach and has been specifically designed for Nigeria. Therefore, it cannot be replicated on the fly for another country. Furthermore, the determination of population and load profiles follows a simulation approach that may over or underestimate the results depending on the quality of input parameters. Hence, there is a need for physical site visit to the selected clusters to ascertain their current state before implementation of the auction. Thus, the attempt in this paper is by no means exhaustive.

Appendix 1

Non-Electrified Clusters Population Scaling

  estimated a total of 75 million people without electricity access in 2014 (SEAR, 2017), Renewable Energy Policy Network for the 21st century (REN21) estimated 98 million in 2014 (REN21, 2017), while the International Energy Agency (IEA) estimated 74 million without access in 2016 (IEA, 2017).

Fig 4 :

 4 Fig 4: Non-electrified clusters at the Local Governments in Adamawa State

Fig 5 :

 5 Fig 5: Number of clusters per population scale in the 10,000 to 25,000 people range

Figure 6 :

 6 Figure 6: Map of Nigeria showing the locations of the 340 clusters

  National Power Training Institute of Nigeria (NAPTIN) through the six months technical skills acquisition program (NTSAP) 26 for technicians, craftsmen, artisans etc., the one-year graduate skills development programs (NGSDP) 27 for engineers and the Solar PV (SPVI 28 , SPVIS 29 , MGD 30 training courses) in collaboration with the AFD and IFC is commendable. Nevertheless, there is a need for an upgraded training designed for utility scale solar PV plus storage installation and design (particularly on instrumentation,

Table 1 :

 1 Our result (on a finer scale for Nigeria) has enriched the literature by providing in-country microlevel data to increase understanding of the implication of using minigrid/offgrid solar PV plus battery storage for power generation and to facilitate system planning, market design and

		PV Size	Battery Size	Autonomy	Excess Energy	Unmet	load
	Shortage	(MW)	(MWh)	(h)	(kWh/Year)	(kWh/year)
	0%	30.29	24	19	40,970,91	5,984
	5%	8.16	27	21.4	4,160,567	384,491
	10%	8.59	19	15	5,292,016	715,879
	15%	7.07	18	14.2	3,195,211	1,099,923
	20%	8.09	15	11.9	5,391,980	1,492,812
	In total, 3,280MW solar PV capacity plus 7,518MWhr lithium-ion battery storage capacity
	requiring a total land area 12 of 100.55km 2 is required at these 233 non-electrified clusters. Our
	result shows that solar PV capacity ranges between 6.45MW and 92.55MW and lithium-ion
	battery sizes between 15MWh and 197MWh are required depending on the cluster.
	Furthermore, these 233 clusters cover approximately 7.2 Million people without electricity
	access that are in 187 different Local Governments in all the Nigeria states plus FCT (except
	Lagos).

Sensitivity/scenario Analysis for a cluster -Lamurde (5.64kWh/m 2 /day, average 26 0 C) Furthermore, we performed the geospatial analysis of the clusters based on the land requirement/availability criteria. With these processes, we derived a shortlist of the optimally and carefully analyzed 233 geographical locations for the proposed minigrid solar PV plus storage reverse auction (see Appendix 3).

AFREA (2016) analysed the investment needs for grid extension. However, funding constraints and the low wheeling capacity of the grid makes their findings challenging for stakeholders to implement. The grid's highest ever peak wheeling capacity of 5, 222 MW was achieved in December 2017

(vanguard, 2017)

. Moreover, grid collapse is often a recurrent problem. The national grid experienced six total system collapses in the first quarter of 2018

(NERC, 2018)

.

  ) have resulted in electricity prices from solar PV equal to almost a fifth of what they were in 2010, reflecting developments in the sector(IRENA, IEA and REN21, 2018).

	). Appendix 5 explains how the LCOE is
	calculated.
	The LCOE for the 233 selected non-electrified clusters (Appendix 3) range from a low of
	US$0.18/kWh in the North to a high of US$0.25/kWh in Southern Nigeria. It is noteworthy that,
	auction results in Dubai, Mexico, Peru, Chile, Abu Dhabi and Saudi Arabia obtained in 2016
	and 2017 confirm that the LCOE can be reduced to US$ 0.03/kWh from 2018 onward given

the right conditions

[START_REF] Irena | Renewable Power Generation Costs in 2017[END_REF]

. For instance, Dubai achieved (US$0.03/kWh), Abu Dhabi (US$0.024/kWh) and Saudi Arabia (US$ 0.023/kWh) (IRENA, 2018b). Hence, solar PV auction (without storage

  The support is to cover the shortfall arising from the lack of cost reflective tariffs. It is paid directly through NBET17 to the Gencos and gas companies to guarantee meeting their obligations. Besides, most Discos have received 14 https://punchng.com/fg-not-afraid-of-discos-threat-to-resell-firms/15 The total amount invested by the DISCOs to acquire 60% stake during the privatization was US$1.42 Billion. The Discos were valued at US$2.37 Billion. The Government 40% share was US$ 946 Million. NBET is Nigeria Bulk Electricity Trading Plc -the institution coordinating transactions between the Gencos and Discos some of their share of the N 213 Billion (US$ 1.07 Billion) Nigeria Electricity Market Stabilisation Funds (NEMSF)/Power intervention loan18 .It is noteworthy that the case of fund diversion has been alleged against some DISCO 19 .A case in point is IBEDC 20 whose board of directors were suspended for misappropriating the sum of 6 Billion Naira (US$19.6 Million from the NEMSF loan meant to improve its networks and reduce aggregate technical, commercial and collection losses and diversion of same to its shareholder company IEDMG 21 . IBEDC is yet to refund the money in line with the agreed terms with the Regulator Order of September 2017 22 . It should be enlightening to see the result of the proposed forensic audits of the DISCOs to understand the utilization of the funds received from the government post-privatization till date 23 . The Nigeria Electricity Market Stabilisation Funds (NEMSF) was a loan facility availed to the DISCO and other market players to cater for the debts owed by the privatized PHCN to the generation and gas supply companies for the energy and gas supply prior to the privatization in 2013. It also covers the market shortfall for the noncost reflective tariffs MYTO 2.0 until January 2015 before the commencement of the transitional market.

	The total deficit was 213 Billion Naira (US$1.07 Billion at the exchange rate 163 Naira/1US$) in December 2014
	when it was finalized, and it was designed to fully pay off the deficit (PSRIP, 2017)
	19 http://www.nercng.org/index.php/media-library/press-releases/568-nerc-suspends-board-of-directors-of-
	ibadan-disco
	20 IBEDC is Ibadan Electricity Distribution Company Plc

According to an investor in Jos Disco… "You asked me whether we are willing to quit the business. Now, please listen to me and put it down clearly that we bought our distribution company cash down for $82m in 2013; we are willing to take $72m in 24 hours and leave."

14

Although the government (which has 40% stake) in the Discos is yet to make any investments in any of them since privatization 15 . Its recent plan to invest 72 Billion Naira (US$ 235.8 Million) as a shareholder loan in all the 11 Discos to upgrade their distribution network infrastructure is a welcome development

16

. This is in addition to the ongoing N702 Billion (US $ 2.3 Billion) government payment guarantee support (January 2017 -December 2018). 16 https://www.thisdaylive.com/index.php/2018/07/31/fg-approves-tcn-to-manage-n72bn-plannedinvestment-in-power-discos/ 17

Furthermore, the government Ministries, Department and Agencies (MDAs) allegedly owe debts in unpaid electricity bills to the DISCOs have been verified as N 27 Billion Naira (US$ 88.4 million). The government has approved that the debt should be deducted from the current debts owed by the DISCOS to NBET. NBET shall then pay the equivalent sum to the GENCOs.

24

Moreover, the government has recently approved funds to the Discos through a loan facility of N37bn (US$ 121Million) to assist in closing the metering gap. By October 2018, the Discos are expected to assess the fund to 18 21 IEDMG is Integrated Energy Distribution and Marketing Group (IEDMG) Ltd 22 http://www.nercng.org/index.php/media-library/press-releases/525-nerc-fines-ibedc-n50-million 23 https://www.businessdayonline.com/exclusives/article/nerc-unveils-plan-tackle-liquidity-gaps-power-sector/ 24 https://www.thenigerianvoice.com/news/263279/communique-of-the-24th-monthly-meeting-of-thehonminister-o.html

  25 Thousands of youths and educators should be enrolled in training programs that provide hands-on experience in the design, installation, testing, operation and maintenance to acquire the skills and competencies required in the solar industry to create a sustainable pool of skilled workforce in the entire value chain. Likewise, training programs should include technical and policy training for professionals and decision makers in the sector.26 NTSAP means National Technical Skills Acquisition Program. It is a 6 months government sponsored program to train line maintenance technicians, distribution substation operators, cable jointing, electrical maintenance and fitters, electrical marketing and meter management technicians

27

NGSDP is a one year Electrical/Mechanical Engineers graduate skills development program 28 SPVI means Solar PV Installation training couse. It is a 3 week (120 hours course) 29 SPVIS means Solar PV installer course. It is a 3 week (120 hours course) 30 MGD means Solar PV Minigrid Design Course. It is a 4 week (160 hours course)

Appendix

is the Solar Resource Map of Nigeria (Solargis)6 This paper builds on the discussion in the paper[START_REF] Arowolo | Designing Reverse Auctions for Solar Power in Nigeria as a potential energy access solution[END_REF], Designing Reverse Auctions for Solar Power in Nigeria as a potential energy access solution.

FCT means Federal Capital Territory (Abuja)

Raster is a digital aerial photographs or satellites imagery made up of matrix of pixels containing a value that represents the conditions for the area covered by the pixel.

Vector use geometry (points, lines, and polygons) to represent the real world.

Low voltage (LV) cable systems up to 1 kV serve to connect buildings and carry power over short distances of not more than several hundred meters[START_REF] Europacable | An Introduction to Medium and Low Voltage Cables in Distribution Networks as support of Smart Grids[END_REF]

Since we anticipate the states, local governments and communities will provide the land for the projects, they should be promptly engaged to ensure buy-in of the proposed solutions and address potential socio-political obstacles.

Exchange rate conversion of N305.30 to 1USD (Central Official exchange rate) from the website of the Central Bank of Nigeria as at 6th March, 2018

Acknowledgement

This work partly results from authors (2) and ( 3) collaboration with the Nigeria Energy Support Programme (NESP) implemented by the Deutsche Gesellschaft fur Internationale Zusammenarbeit (GIZ) GmbH in collaboration with the Federal Ministry of Power, Works and Housing (FMPWH) Nigeria and funded by the German Government and European Union. Any conclusions or recommendations expressed in this paper do not reflect the views of GIZ-NESP or FMPWH.

The authors would like to acknowledge useful comments and suggestions from two anonymous reviewers.

SOLAR RESOURCE MAP OF NIGERIA

Map Source : https://solargis.com/maps-and-gis-data/download/nigeria/


I Bailleul

Rough integrators on Banach manifolds

We introduce a notion of p-rough integrator on any Banach manifolds which plays the role of weak geometric Hölder p-rough paths in the usual Banach space setting, for any p ě 1. The awaited results on rough differential equations driven by such objects are proved, and a canonical representation is given if the manifold is equipped with a connection.

Introduction

One of the most basic and illustrative example of a rough path is associated with the following 2-dimensional spinning signal

h n t "
1 n `cospn 2 tq, sinpn 2 tq ˘, 0 ď t ď 1.

Although h n converges uniformly to 0, this path sweeps an area t in any time interval r0, ts, independantly of n, suggesting that this sequence of paths may be really different from the zero signal. As a matter of fact, given any two vector fields V 1 , V 2 on R d , of class C 3 b , it is remarkable that the solution path x n ' to the ordinary controlled differential equation 9 x n t " `9 h n t ˘iV i `xn t converges uniformly to the solution path x ' to the ordinary differential equation

9 x t " " V 1 , V 2 ‰ px t q.
This is due to the fact that the canonical lift of h n ' as a weak geometric Hölder p-rough path, for any 2 ă p ă 3, converges in a rough paths sense to the pure area rough path with area process pt ´sq ˆ0 1 ´1 0 ˙we refer the reader to the lecture notes [START_REF] Caruana | Differential equations driven by rough paths[END_REF][START_REF] Bailleul | A flow-based approach to rough differential equations[END_REF] and [START_REF] Friz | A course on rough paths. With an introductions to regularity structures[END_REF][START_REF] Baudoin | Diffusion processes and Stochastic calculus[END_REF] for basics on rough paths and rough differential equations. This elementary example makes it clear that, when understood as controls, smooth paths may have a richer structure than expected at first sight, as the above constant path with non-trivial area process shows. Smooth paths have canonical lifts as rough paths; a smooth path in R whose rough path lift is non-canonical will be called a spinning path. While this may not be obvious, we shall see in this note that it actually makes perfect sense to talk of a spinning manifold-valued path, as an example of a manifold valued 'rough path'. One should keep in mind, however, that the classical notion of rough path only becomes really interesting when understood as a control in some differential equation. Does it make sense to think of a manifold-valued smooth path as a control? Yes. It is indeed common, in physics and differential geometry, to be given a bundle B over some manifold M , together with a connection, given by a T B-valued 1-form H on M . Lifting a smooth M -valued path γ ' into a smooth horizontal B-valued smooth path e ' is a basic procedure in which γ is used as a control in the ordinary differential equation 9 e t " Hpe t q 9 γ t defining the path e ' . What would happen if γ were a kind of spinning smooth Mvalued path? What about spinning geodesics in a Riemannian setting? The notion of p-rough integrator introduced below, and the results proved, provide a framework where one can answer such questions.

So far there has been only a few works dealing with rough paths in a geometrical setting, starting with the work [START_REF] Lyons | Flow equations on spaces of rough paths[END_REF] of Lyons and Qian. This seminal work investigated the well-posedness problem for the ordinary differential equation on the path space of a compact manifold generated by Itô vector fields, with an eye on probabilistic applications related to path space versions of the Cameron-Martin theorem and Driver's flow equation. It was enriched by the work [START_REF] Li | Smoothness of Itô maps and diffusion process on path spaces[END_REF] of Li and Lyons, showing that the Itô-Lyons solution map to a Young differential equation is Fréchet regular under appropriate conditions, when the driving signal has finite p-variation, with 1 ă p ă 2, and by the work [START_REF] Bailleul | Regularity of the Itô-Lyons map[END_REF] of the author providing a general regularity result for the Itô-Lyons solution map, for any p ě 2, in the setting of controlled paths. Another work [START_REF] Lyons | Stochastic Jacobi fields and vector fields induced by varying area on path spaces[END_REF] of Lyons and Qian addressed well-posedness issues for ordinary differential equations on path space associated with Itô vector fields obtained by varying the driving rough signal -see also [START_REF] Bailleul | Regularity of the Itô-Lyons map[END_REF] for similar results.

These works only use rough paths as an ingredient to construct some dynamics in a geometric configuration space. Cass, Litterer and Lyons made a step further in putting rough paths theory in a geometrical setting and proposed in [START_REF] Cass | Rough paths on manifolds. New trends in stochastic analysis and related topics[END_REF] a notion of rough path on a manifold extending the classical notion defined in a linear setting. In the same way as a vector field on a manifold M can be understood in an analytic/algebraic setting as a differentiation in the ring of smooth functions on the manifold M , a rough path is abstractly defined as a linear form on the space of sufficiently regular 1-forms on M , which is required to have some continuity property; call it an integrator. This functional analytic definition rests on a basic chain rule which eventually enables to understand their notion of rough path on a manifold as an equivalence class of classical rough paths, related by some chain rule under change of coordinates. This situation is exactly similar to representing a tangent vector on a d-dimensional manifold as an equivalence class of vectors in R d , indexed by local diffeomorphisms of a neighbourhood of 0 (that is local changes of coordinates), and related by a change of coordinate rule which exactly balances the changes in the numerical representation of a given 1-form α on M associated with local coordinates, so the quantity αpuq is independent of any choice of coordinates used to compute it. Their approach rests however on a notion of Lip-γ manifold which prevents its easy use even with non-compact finite dimensional manifolds, not to speak about infinite dimensional manifolds.

The ideas of [START_REF] Cass | Rough paths on manifolds. New trends in stochastic analysis and related topics[END_REF] have been reloaded in a different and more accessible form in the recent work [START_REF] Cass | Constrained rough paths[END_REF] by Cass, Driver and Litterer, in which they define a weak geometric Hölder p-rough path on a finite dimensional compact embedded submanifold of R d as an integrator, obtained by "projection" of a weak geometric Hölder p-rough path in the ambiant Euclidean space, for 2 ď p ă 3 only. Nothing is lost in working with submanifolds, and this notion is eventually shown to be independent of the embedding of the manifold in its environment, while these objects can be used to define and solve uniquely rough differential equations driven by weak geometric Hölder p-rough paths on compact manifolds. They can construct parallel transport along manifold-valued rough paths in their sense, and use it to show a one-to-one correspondence between rough paths on a finite d-dimensional manifold and rough paths on d-dimensional Euclidean space. The recent work [START_REF] Boutaib | On Lipschitz maps and their flows[END_REF][START_REF] Boutaib | A new definition of rough paths on manifolds[END_REF] of Boutaib and Lyons follow the original intuition of [START_REF] Cass | Rough paths on manifolds. New trends in stochastic analysis and related topics[END_REF] and clarify the geometric notion of Lipscthiz manifold needed to make it work. On the other hand, Driver and Semko adapt in [START_REF] Semko | Controlled Rough Paths on Manifolds I[END_REF] Gubinelli's notion of controlled path to a manifold setting, to deal with p-rough paths on manifolds with 2 ď p ă 3, by using a formalism very close to the second order calculus developed by Schwartz in his analysis of Itô calculus on manifoldssee for instance Meyer expository article [START_REF] Meyer | Géométrie stochastique sans larmes[END_REF], or Emery's book [START_REF] Emery | Stochastic Calculus in Manifolds[END_REF].

We propose in this note an elementary and flexible notion of weak geometric Hölder p-rough path on Banach manifolds that goes beyond the previous works. Roughly speaking, a weak geometric Hölder p-rough path on a manifold M is a triple made up of a vector field valued 1-form F defined on some Banach space U, a weak geometric Hölder p-rough path X over U, and the maximal solution to the rough differential on M constructed from F and X. The data of these three objects is sufficient to define and solve uniquely rough differential equations driven by "manifold-valued" rough paths, or better, by p-rough integrators. The abovementionned spinning smooth paths on M are precisely those p-rough integrators whose associated M -valued paths are smooth. Nothing else than the (smooth) manifold structure is needed to make sense of a p-rough integrator. One shows in section 3.2 that these objects have a canonical representation if the tangent bundle of M is equipped with a connection, in which case one can always choose for Banach space U in the preceeding description of a p-rough integrator the Banach space on which the manifold is modelled.

The interest of working with Banach manifolds comes from the fact that they naturally pop up in a number of geometric situations, as path or loop spaces over some finite dimensional manifold, as in the works [START_REF] Brzezniak | Stochastic Differential Equations on Banach Manifolds; applications to diffusions on loop spaces[END_REF][START_REF] Brzezniak | Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces[END_REF] of Brzezniak, Carroll and Elworthy, or the works [START_REF] Inahama | Large deviations for heat kernel measures on loop spaces via rough paths[END_REF][START_REF] Inahama | On asymptotics of Banach space-valued Itô functionals of Brownian rough paths[END_REF] of Inahama and Kawabi, or as manifolds of maps of a given finite dimensional manifold, as in the works [START_REF] Elworthy | Stochastic Differential Equations On Manifolds[END_REF][START_REF] Brzezniak | Stochastic flows of diffeomorphisms for stochastic differential equations[END_REF] of Elworthy and Brzezniak, to mention but a few works from the probability community. Note however that many interesting infinite dimensional manifolds are Fréchet manifolds, for which no theory of rough paths is presently available. As a matter of fact, working with Banach manifolds will not bring any additional difficulty along the way, as compared to working with finite dimensional manifolds. We refer the reader to the books [START_REF] Lang | Differential and Riemannian manifolds[END_REF] of Lang, and [START_REF] Abraham | Manifolds, tensors, analysis, and applications[END_REF] of Abraham, Marsden and Ratiu, for the basics of differential geometry in an infinite dimensional setting. Let just recall that it makes perfect sense in a manifold setting to say that a (Banach space or real-valued) function f of class C k , defined on the domain of some local chart ϕ of M , has bounded derivatives if the function f ˝ϕ´1 , defined on a neighbourood of 0 in some Banach space, has bounded derivatives; the bounded character will indeed not depend on the precise chart used to define it, while the bounds themselves will depend on ϕ.

Notations. We gather here a few notations that will be used throughout the note.

' We shall denote by U a generic Banach space; the notation T rps pUq will be used for the truncated tensor product of order rps, completed for some choice of tensor norms. The letter X will stand for a weak geometric Hölder prough path over U, for some p ě 2, and for the above choice of tensor norm on T rps pUq.

' We shall denote by M a Banach manifold modelled on some Banach space E. The set of continuous linear homomorphisms of E will be denoted by LpEq, and the set of continuous linear isomorphisms of E will be denoted by GLpEq.

Rough integrators on Banach manifolds

We define in this section what will play the role in a manifold setting of weak geometric Hölder p-rough paths. Rough integrators are basically defined as a triple, consisting of a traditional weak geometric Hölder p-rough path with values in some Banach space U, a T M -valued vector field valued 1-form on U, and a solution to an associated rough differential equation on the manifold. This definition, given in section 2.2, requires that we recall some facts on rough differential equations with values in manifolds; this is done in section 2.1. We explain in section 2.3 how rough integrators can be used to define and solve rough differential equations.

Rough differential equations

We adopt in this work the definition of a solution path to an M -valued rough differential equation given in [START_REF] Bailleul | Rough differential equations driven by Banach-space valued rough paths[END_REF] in a Banach space setting, as it is perfectly suited for our needs. (This definition was also adopted in Cass and Weidner's work [START_REF] Cass | s theorem for rough differential equations on manifolds[END_REF] on Hörmander's theorem for rough differential equations on manifolds.) It essentially amounts to requiring from a solution path that it satisfies some uniform Taylor-Euler expansion formulas, in the line of Davie' seminal work [START_REF] Davie | Differential equations driven by rough paths: an approach via discrete approximation[END_REF]. Let U be a Banach space, and F be a 1-form on U with values in the space of vector fields on a Banach manifold M , of class C rps`1 . Given u P U, we identify the vector field Fp¨; uq on M with its associated first order differential operator F b puq, and extend the definition of F to T rps pUq setting Fp1q " Id, and Fpu 1 b ¨¨¨b u k q " Fpu 1 q ¨¨¨Fpu k q, for 1 ď k ď rps, and u i P U, and by linearity; so Fpu 1 b ¨¨¨b u k q is a differential operator of order k. Given a weak geometric Hölder p-rough path X over U, recall that an M -valued continuous path px t q 0ďtăζ is said to solve the rough differential equation

dx t " F `xt ; Xpdtq ˘(1)
if there exists a constant a ą 1 such that, for any 0 ď s ă ζ, there exists an open neighbourhood V s of x s such that we have the Taylor-Euler expansion

f px t q " `F`X ts ˘f ˘px s q `O`| t ´s| a for
all t close enough to s for x t to belong to V s , and any function f of class C rps`1 defined on V s , where it has bounded derivatives. The results of [START_REF] Bailleul | Rough differential equations driven by Banach-space valued rough paths[END_REF] show that such a rough differential equation has a unique maximal solution started from any given point, as awaited. It also follows from the results of [START_REF] Bailleul | Rough differential equations driven by Banach-space valued rough paths[END_REF], or other classical works, that the solution path x ' depends continuously on the driving signal X in the following sense. Fix T ă ζ and cover the compact support of the path px t q 0ďtďT by finitely many local chart domains pO 1 i q 1ďiďN and pO i q 1ďiďN , with O 1 i Ă O i for all 1 ď i ď N . Then, there exists a positive constant δ such that for Y δ-close to X in the Hölder rough path distance, the solution path y ' to the rough differential equation dy t " F `yt ; Ypdtq started from x 0 , will be well-defined on the time interval r0, T s and will remain in the open neighbourhood Ť N i"1 O i of the support of px t q 0ďtďT , with y t in O i whenever x t is in O 1 i . Note here that the rough path setting allows to work with any Banach manifold, without any restriction on the Banach space on which it is modelled, unlike what needs to be done in a stochastic setting where a robust theory of stochastic integration is only available for some special types of Banach spaces, the so-called M-type 2 and UMD spaces, see [START_REF] Brzezniak | Stochastic differential equations on Banach manifolds[END_REF][START_REF] Van Neerven | Stochastic integration in Banach spaces -a review[END_REF].

Examples. 1. Developping a spinning straight line. Pick 2 ď p ă 3 and let M be a Riemannian surface; denote by H 1 and H 2 the canonical horizontal vector fields on the orthonormal frame bundle OM of M , equipped with Levi-Civita connection. Denote by V 12 the unique canonical vertical vector field and denote by Ω the (scalar-valued) curvature (tensor). Let p 1 , 2 q stand for the canonical basis of R 2 , and let X be the weak geometric Hölder p-rough path with first level pt ´sq 1 and area process pt ´sq ˆ0 1 ´1 0 ˙; this path was called above the spinning straight line. Denote by

H the T OM -valued 1-form u P R 2 Þ Ñ u 1 H 1 `u2 H 2 .
Then the OM -valued solution path to the rough differential equation

de t " H `et ; Xpdtq ˘(2)
is actually the solution path to the ordinary differential equation 9 e t " `H1 `Ω V 12 ˘pe t q.

2. Finding back a geodesic. Assume for simplicity that Ω is constant, and take as a rough path in equation (2) the weak geometric Hölder q-rough path Y, with 4 ă q ă 5, whose logarithm is given by log Y ts " pt ´sq

´ 1 ' r 1 , 2 s ' " ´ 1 , " 1 , r 1 , 2 s ‰ ı¯.
Then the solution path to the rough differential equation de t " H `et ; Ypdtq ȃlso solves the ordinary differential equation

9 e t " H 1 pe t q,
that is, this path is the canonical lift to OM of a geodesic. This can be seen directly from the definition, or by appealing to the results proved in [START_REF] Friz | Rough path limits of the Wong-Zakai type with a modified drift term[END_REF] by Friz and Oberhauser.

3. Rough differential equations with values in Banach Lie groups. Let G be a Banach Lie group, with Lie algebra LiepGq. Think for instance to the loop groups, made up of C k maps from some finite dimensional manifold M 0 to some finite dimensional Lie group G 0 , with pointwise multiplication and inversion operations. Their Lie algebra is the set C k `M0 , g 0 ˘of C k maps from M 0 to the Lie algebra g 0 of G 0 , with bracket defined pointwise by the relation " u, v ‰ pxq :" " upxq, vpxq ‰ , for any u, v in C k `M0 , g 0 ˘and x P M 0 . These groups are extensively used in gauge theory or quantum field theory.

Write L g for the left translation by g in G. One defines a continuous linear map from LiepGq to the space of smooth vector fields on G setting Fpg ; uq :" ´DId L g ¯puq, for any g P G and u P LiepGq; so Fp¨; uq is for any u P LiepGq a smooth left invariant vector field on G. It is elementary to proceed as in the proof of theorem 4.20 in [START_REF] Cass | Constrained rough paths[END_REF] and see that for any weak geometric Hölder p-rough path X over LiepGq, defined on some interval r0, T q, for some 0 ď T ď 8, the solution path to equation (3) cannot explode before time T .

Rough integrators on Banach manifolds

The definition of a p-rough integrator given below is best understood in the light of the following example. Given a 1-form α on a d-dimensional manifold M equipped with a connection, how can one define the integral ş 1 0 αp˝dx t q, of α along an Mvalued continuous semimartingale px t q 0ďtď1 ? The classical answer consists in showing that x ' can always be constructed as the projection in M of a GLpM q-valued path pe t q 0ďtď1 obtained by solving a stochastic differential equation in GLpM q involving the horizontal vector fields associated with the connection and some R dvalued semimartingale pw t q 0ďtď1 -see for instance Chapter 2 in the book [START_REF] Hsu | Stochastic analysis on manifolds[END_REF] of Hsu. One then defines ş 1 0 αp˝dx t q in terms of w only, setting ż 1 0

αp˝dx t q " ż 1 0 `α ˝e´1 t ˘˝dw t .
So the datum of w and the horizontal vector fields are all we need to define x ' and use it as a control. The next definition adopts a similar point of view in the present setting.

1. Definition -Let p ě 2 be given. A weak geometric Hölder p-rough path on M is a a triple Θ " `px t q 0ďtăζ , F, X ˘, where ' X a weak geometric Hölder p-rough path over some Banach space U, defined on the time interval r0, ζq, ' F is a continuous linear map from U to the space of vector fields on M of class C rps`1 , ' the path px t q 0ďtăζ solves the rough differential equation

dx t " F `xt ; Xpdtq ˘. (3)
We also call Θ a basic p-rough integrator.

Given a p-rough integrator Θ as above, the triple `px t q 0ďtďT , F, X ˘, with T ă ζ, is also called a basic p-rough integrator. Let us insist on the fact that the Banach space U in the above definition is not fixed a priori, and depends on the basic prough integrator Θ on M . So any weak geometric Hölder p-rough path Y over some Banach space V can be canonically seen as a weak geometric Hölder p-rough path in the above sense by choosing U " V and X " Y, the identity map for F, and the first level of the rough path X t for x t . Note that we do not endow the set of basic p-rough integrators with a topology or a distance as this is somewhat subtle and we do not need it in the sequel.

The definition of a solution to the rough differential equation (3) makes it clear that if g is any sufficiently regular diffeomorphism between M and another manifold M 1 , one defines a weak geometric Hölder p-rough path on M 1 setting g ˚Θ :" ´`gpx t q ˘0ďtăζ , g ˚F, X ¯, where g ˚Fpy ; uq :" `Dg ´1pyq ˘g´F `g´1 pyq ; u ˘¯, is the push forward on M 1 by g of the vector field Fp¨; uq on M .

Definition 1 encompasses the characterization of a weak geometric Hölder p-rough path on a submanifold of R d given in [START_REF] Cass | Constrained rough paths[END_REF], only for 2 ď p ă 3, in terms of a priori extrinsic considerations. Roughly speaking, they define their class of weak geometric p-rough paths on a compact submanifold M of R d , for 2 ď p ă 3, as the class of all weak geometric rough paths Y over R d with the property that Y 0 " p1, y 0 , 0, . . . q with y 0 P M , and dY t " Qpy t q dY t (4)

for some projector-valued sufficiently regular map Q such that Qpyq has range included in T y M if y P M ; this equation ensures that 4) and the chain rule for rough integrals shows that one defines an image rough path Z over R d 1 setting dZ t " dϕpdY t q and that it satisfies the identity

y t P M . If M 1 Ă R d 1 is any other embedded compact manifold diffeomorphic to M through ϕ : M Ñ M 1 , equation (
ż 1 0 pϕ ˚αqpdZ t q " ż 1 0 αpdY t q,
for any sufficiently regular 1-form α on T M . (The expression ϕ ˚α stands for the push-forward of the 1-form α by ϕ, and both integrals make sense because of condition (4).) So the class of weak geometric p-rough paths is intrinsically defined as a class of rough integrators, independently of any particular embedding of M , seen as an abstract manifold. Note that they use general controls ωps, tq to measure the size of their rough paths, while we simply use ωps, tq " t ´s and Hölder scales. A weak geometric p-rough path in the sense of [START_REF] Cass | Constrained rough paths[END_REF] is simply described in our terms as a special kind of weak geometric p-rough path in the ambiant linear space via equation ( 4). The interest of working with the intrinsic definition 1 is clear in an infinite dimensional setting where embeddings of Banach manifolds into larger Banach spaces are less natural and rarely happen, unless M is parallelizable [START_REF] Eells | Open embeddings of certain Banach manifolds[END_REF]. Note however that the frame bundle of M is always parallelizable if T M is equipped with a connection. See section 3.

One of the nice features of the notion of rough integrator on a manifold proposed in [START_REF] Cass | Rough paths on manifolds. New trends in stochastic analysis and related topics[END_REF] and [START_REF] Cass | Constrained rough paths[END_REF] is the possibility to push forward a rough integrator by a sufficiently regular map g from M to another manifold N , giving as a result a rough integrator on N . The rough integral of a 1-form α on N against the image rough integrator is simply defined as the rough integral of α ˝dg against the rough integrator on M . A similar transport operation can be done in our framework, by seeing the push forward of a p-rough integrator Θ on M by g : M Ñ N as the p-rough integrator gΘ on M ˆN associated with pF, dg ˝Fq and X.

It may happen that the vector fields Fp¨; uq and Fp¨; vq commute for any u, v P U, and X has null first level, so the path x ' is constant. We say in that case that Θ is a pure rough path. It can still give rise to some dynamics, when seen as a rough integrator in a rough differential equation, as shown in the next section.

Weak geometric Hölder p-rough paths as integrators

Recall the construction of a line integral along a continuous semimartingale on a manifold described at the beginning of section 2.2. The next proposition shows that a weak geometric Hölder p-rough path on M can be used as an integrator in a rough differential equation, which justifies calling it a p-rough integrator.

2. Definition -Let Θ be a weak geometric Hölder p-rough path on M in the sense of definition 1. Let also π : B Þ Ñ M be a smooth fiber bundle over M , equipped with a connection, of class C rps`1 , given under the form of a smooth T B-valued 1-form H on M such that π ˚: T e B Ñ T πpeq M , is an isomorphism for every e P B. Given a rough integrator Θ " `px t q 0ďtăζ , F, X ˘in M , and e 0 P B with πpe 0 q " x 0 , a path pe t q 0ďtăζ 1 in B is said to be a lift of px t q 0ďtăζ 1 , if ζ 1 ď ζ and it is the solution path to the B-valued rough differential equation

de t " H ´F`π pe t q; Xpdtq ˘¯(5)
started from e 0 . This equation ensures in particular that πpe t q " x t . The p-rough integrator `pe t q 0ďtăζ 1 , H ˝F ˝π, X ˘is then said to be the lift of Θ to B started from e 0 .

A commonly encountered situation is to have B " M ˆN , for some other Banach manifold N , and a connection form defined for any p P T x M and y P N by the formula Hpx, yqp " `p, Gpy, x ; pq ˘, with T pM ˆN q canonically identified with T M ˆT N , and some function G : N ˆT M Ñ T N of class C rP s`1 , with Gpy, x; pq P T y N , for each p P T x M , and x P M . This definition is reminiscent of the approach chosen by Cass, Litterer and Lyons, in section 5 of [START_REF] Cass | Rough paths on manifolds. New trends in stochastic analysis and related topics[END_REF], to define in their setting rough differential equations driven by manifold-valued rough paths. Note however that their theory requires the use of the rigid and non-trivial notion of Lip ´γ manifold; no such constraint holds in our setting.

The fact that the two vector fields `H˝F˝π ˘p¨, ¨; uq and `H˝F˝π ˘p¨, ¨; vq may not commute while Fp¨; uq and Fp¨; vq may commute, for some u, v P U, explains why pure rough paths can generate dynamics. The results on rough differential equations recalled in the introduction of this section apply and show that 3. Proposition -The rough differential equation (5) has a unique maximal solution started from any given point.

Solving successively some rough differential equations of the form (5) with some basic p-rough integrators Θ p1q " ``x p1q t ˘0ďtďt 1 , F p1q , X p1q ˘, . . . , Θ pkq , with x pjq 0 " x pj´1q t j´1 , y pjq 0 " y pj´1q t j´1 and t j´1 ă ζ 1 j´1 ď ζ j´1 , for 2 ď j ď k, defines, whenever this makes sense, the concatenation Θ pkq ¨¨¨Θ p1q of the basic p-rough integrators Θ piq . We call such an object a p-rough integrator.

Canonical representation of rough integrators

We show in this section that p-rough integrators have a canonical representation when the tangent bundle of M is equipped with a connection. This representation is the analogue of the representation of a regular path γ on M by a regular path in T γ 0 M using Cartan's development map.

Cartan's moving frame method

Let M be a manifold of finite dimension d. One owes to E. Cartan the introduction in differential geometry of the moving frame method, which provides a chart for the set of M -valued paths of class C 1 based at some fixed starting point, in terms of C 1 paths in R d started from 0. Its construction requires the use of a connection on T M and of the frame bundle of M ; it can basically be described as follows. Given a C 1 path γ ' " pγ t q 0ďtď1 on M and a frame e 0 at γ 0 , its parallel transport along the path γ ' defines a section pe t q 0ďtď1 of the frame bundle above γ ' . One can use these frames to describe 9 γ t at any time in terms of its coordinates `9 u i t ˘1ďiďd in e t . The R d -valued path u ' defined by the formula u t :"

ż t 0 9 u i s ds
is called the anti-development of the path γ. One finds back the M -valued path γ ' from the R d -valued u ' and px 0 , e 0 q as the projection on M of the unique solution of the ordinary differential equation in the frame bundle ∇ 9 γt e t " 0, 9 γ t :" e t `9 u t ˘.

The interest of Cartan's moving frame method is that it somehow provides an optimal coding of all paths in M in terms of vector space-valued paths living in a space of minimal dimension. Here is a trivial illustration of this fact. Let u ' be an R 10 -valued path and γ t " ř 10 i"1 u i t be an R-valued path. The coding of γ ' by the 10 components of u ' is optimized by coding it with its real value at each time. Replacing R 10 and R by infinite dimensional Banach spaces emphasizes the importance of such parcimonious representations.

Get back to the general setting and recall that M is modelled on some Banach space E. Under some additional mild condition, one can give a parcimonious description of weak geometric Hölder p-rough paths on M similar to the above one, providing a description of these objects in terms of weak geometric Hölder p-rough paths in the tensor space T rps pEq, as opposed to the a priori unrelated Banach space U involved in the primary definition of a weak geometric Hölder p-rough path, see definition 1. As in Cartan's moving frame method, this requires the tangent bundle T M Ñ M to be equipped with a connection; this is a non-trivial assumption in an infinite dimensional setting, linked to the fact that there exists some Banach spaces that do not even admit a smooth partition of unity. No such pathology happens in finite dimension, and finite dimensional manifolds can always be endowed with an arbitrary connection; so the results of the forthcoming section 3.2 always hold for rough integrators on finite dimensional manifolds.

The frame bundle GLpM q of M will have a crucial role in that play. This is the collection of all bi-continuous isomorphisms from E to T m M , for m P M ; it has a natural manifold structure modelled on GLpEq ˆE. We shall denote by e a generic element of GLpM q, and by VGLpM q the vertical sub-bundle of T GLpM q, canonically identified with the Lie algebra glpEq " LpEq of GLpEq. The connection ∇ on the bundle T M Ñ M is naturally lifted into a connection on the bundle π : GLpM q Ñ M , still denoted by the same symbol. Remark that π ˚is a bicontinuous isomorphism between the horizontal sub-bundle H ∇ GLpM q in T GLpM q and T M . The horizontal destribution H ∇ GLpM q in T GLpM q can be used to define a continuous linear map H ∇ from E to the space of horizontal vector fields on T GLpM q, defined by the requirement that H ∇ pe ; aq P T e GLpM q is horizontal and corresponds to epaq P T M , for any a P E and e P GLpM q. This vector field valued 1-form H ∇ on E is called the canonical horizontal 1-form.

Once again, we refer the reader to the nice books [START_REF] Lang | Differential and Riemannian manifolds[END_REF] and [START_REF] Abraham | Manifolds, tensors, analysis, and applications[END_REF] for the basics of differential geometry on Banach manifolds.

Canonical representation of rough integrators

Theorem 4 below gives a canonical and parcimonious representation of a given weak geometric Hölder p-rough path, seen as a rough integrator. We assume for that purpose that the manifold M is endowed with a connection ∇. Given a 1-form F on U with values in the space of vector fields on M as in the above definition of a weak geometric Hölder p-rough path on M , we denote by F ∇ its lift to a 1-form with values in the space of ∇-horizontal vector fields on GLpM q. Parallel translation along a weak geometric Hölder p-rough path Θ " `px t q 0ďtăζ , F, X ˘is defined as the solution path to the rough differential equation in GLpM q de t " F ∇ `et , Xpdtq started from any given frame e 0 P GLpM q above x 0 . It is elementary to see that the paths e ' and x ' are defined on the same maximal interval.

Given another Banach manifold N , and a 1-form G : N ˆT M Ñ T N , as used above to write down a rough differential equation in definition 2, define G on GLpM q ˆN ˆU by the formula Gpe, y ; uq :" ´F∇ pe ; uq, G `y, πpeq ; π ˚F∇ pe ; uq ˘¯, and G ∇ on GLpM q ˆN ˆE by the formula G ∇ pe, y ; aq :" ´H∇ pe ; aq, G `y, πpeq ; π ˚H∇ pe ; aq ˘¯.

(Insist on the fact that the one-form H ∇ depends only on the connection ∇ on T M .) 4. Theorem -Let M be a Banach manifold modeled on the Banach space E, and endowed with a connection ∇. Let Θ " `px t q 0ďtăζ , F, X ˘be a basic p-rough integrator on M .

(a) Given e 0 P GLpM q, one defines a weak geometric Hölder p-rough path Z over E, on the time interval r0, ζq, by solving the rough differential equation de t " F `et ; Xpdtq ˘, dZ t " Z t b e ´1 t F `πpe t q ; Xpdtq ˘.

in GLpM q ˆT rps pEq, started from pe 0 , Idq.

(b) The solution paths pe t , y t q and pf t , z t q in GLpM qˆN to the rough differential equations dpe t , y t q " G `et , y t ; Xpdtq ˘, dpf t , z t q " G ∇ `ft , z t ; Zpdtq ˘,

coincide if they start from the same initial condition.

It is elementary to state an analogue of this theorem when M ˆN is replaced by a fiber bundle B over M , equipped with a regular connection G. There is no more difficulties in stating a similar result for any p-rough integrator Θ " Θ pkq ¨¨¨Θ p1q instead of basic p-rough integrators. The interesting point is that although the rough paths in each Θ piq may be defined on different Banach spaces, the canonical representation of the concatenation of the Θ piq only involves a rough path over E.

Proof -The first claim follows from general principles. For the second point, pick p ă p 1 ă rps `1. Using the continuity result for the Itô map recalled in section 2.1, in the topology of Hölder p 1 -rough path, together with the continuous embedding of the space W G p pUq of weak geometric Hölder p-rough paths in U into the space G p 1 pUq of geometric Hölder p 1 -rough paths in U, and the continuous embedding of G p 1 pUq into W G p 1 pUq, we see that it suffices to prove the claim when X is the weak geometric Hölder (p 1 -) rough path associated with a smooth U-valued control. The result is clear in that case as it appears as a rephrasing of Cartan's moving frame method, described above in section 3.1.

So one can always understand the solution of a rough differential equation driven by a p-rough integrator Θ as the solution to another rough differential equation driven by a p-rough integrator involving a weak geometric Hölder p-rough path over the model space E, and the canonical horizontal 1-form H ∇ . The dependence on F in the original p-rough integrator is hidden in the definition of Z in this reformulation, equation [START_REF] Li | Smoothness of Itô maps and diffusion process on path spaces[END_REF]. Given e 0 P GLpM q above x 0 , the "GLpM q-valued" p-rough integrator `pe t q 0ďtăζ , H ∇ , Z ˘is said to be the canonical representation of Θ. This result echoes one of the main results of [START_REF] Cass | Constrained rough paths[END_REF], corollary 6.19, where a one-to-one correspondence between the classes of weak geometric rough path on a given finite d-dimensional manifold, weak geometric horizontal rough paths on its frame bundle, and classical weak geometric rough paths on R d , is proved in their setting, using Cartan's development map as an essential ingredient; Theorem 4 has wider scope; applications of our setting will be developed in a forthcoming work.

1

This research was partially supported by the program ANR "Retour Post-Doctorants", under the contract ANR 11-PDOC-0025. The author thanks the U.B.O. for their hospitality.