Cleaner energy microgrids under market power and limited regulation in developing countries

Cleaner energy microgrids under mark et p o w er and limited regulation in dev eloping coun tries Elsa Bou Gebrael a , Ma jd Olleik a , Sebastian Zwic kl-Bernhard b a A meric an University of Beirut, Mar oun Semaan F aculty of Engine ering and A r chite ctur e, Industrial Engine ering and Management Dep artment, L eb anon b Vienna University of T e chnolo gy, Institute of Ener gy Systems and Ele ctric al Drives, Ener gy Ec onomics Gr oup (EEG), Austria, Norwe gian University of Scienc e and T e chnolo gy, Norway Abstract In many lo w-income coun tries, neighborho o d diesel generators are widely used to comp ensate for unreliable or una v ailable national electricit y grids. These diesel-based microgrids are t ypically characterized b y market pow er, significan t pollution, and w eak regulatory o versigh t. In parallel, households increasingly deplo y off-grid solar photov oltaic (PV) systems to gain control o ver electricity supply . Ho wev er, these systems suffer from curtailed ex- cess generation during p eak solar hours and unreliable access at other times. While prior studies hav e optimized microgrids in developing contexts from a tec hno-economic p ersp ectiv e, they largely neglect the mark et p ow er exerted b y monop olistic priv ate generators. This pap er addresses this gap b y devel- oping a bi-lev el game-theoretic mo del that enables household-generated elec- tricit y to b e fed in to the microgrid while explicitly accounting for the market p o w er of a neighborho o d diesel generator compan y (DGC). The regulator sets price and feed-in-tariff caps to maximize household economic surplus (HES), while the DGC acts as a profit-maximizing agen t con trolling access and sup- ply . The mo del is applied to a Lebanese case study using high-resolution empirical data collected via logging devices. Results show that: (i) price and feed-in-tariff caps substantially increase HES and consisten tly induce signif- ican t household PV feed-in to the microgrid; (ii) higher DGC budgets or greater PV-owner p enetration lead to pronounced gains in HES; and (iii) the renew able energy share reaches 60% under base conditions and approaches 100% at sufficien tly high budgets or PV-o wner p enetration lev els, compared to 0% under the status quo. Keywor ds: Microgrids, renew able energy, developing coun tries, market p o w er, bi-level mo del 1. Introduction Coun tries under geop olitical stress or other crises face multiple c hallenges in reliable and sustainable electricity generation and distribution. W ars tak e their toll on national grids, damaging infrastructure and preven ting grid up- grades and extensions [ 1 , 2 , 3 ]. Dysfunctional gov ernments suffering from corruption hinder the dev elopment and implemen tation of adequate p olicies for asset management, and inefficiently allo cate resources in self-profiting se- tups [ 4 , 5 , 6 ]. As of 2022, 64% of the p opulation lacking electricit y access liv e in fragile, conflict-affected or vulnerable regions [ 7 ]. Consequen tly , these coun tries observ e a rise in consumer-led generation initiatives, commonly resorting to individual or neigh b orho o d diesel generators and solar home systems (SHS) [ 8 , 9 ]. Ho w ever, these solutions often come at the expense of energy efficiency or clean generation. The literature around electrification in the Global South addresses the tec hno-economic asp ects of these arrangemen ts, but abstracts the p olitical 2 or regulatory con texts that were at the origin of the grid issues [ 10 , 11 ]. At the same time, existing studies pay limited attention to the emerging market dynamics b et w een different pla yers, particularly around neighborho o d diesel generators. Neigh b orho od diesel generators hav e b een the fallback solution of many coun tries suffering from wars, crises or corruption. Lebanon, Iraq and Nigeria are notable examples, where geop olitical instability , sanctions and systemic corruption hav e resulted in an unreliable national electricit y supply and pro- longed main grid outages of up to 20 hours p er day [ 12 , 13 , 14 ]. The long- term reliance on diesel generators in these countries, as well as in Y emen, Afghanistan, Keny a and other African sub-Saharan countries [ 8 ] comes with p olitical, economic, and en vironmental challenges. When a diesel generator serv es a neighborho o d or communit y , a microgrid develops in parallel to the national grid [ 8 , 13 ]. The diesel generator op erators b enefit from a natural monop oly ov er this microgrid, assuming b oth roles of generator and distrib- utor, and p oten tially using their p osition to control access and maximize profits [ 8 , 13 , 15 ]. In the absence of a reliable national grid, the gov ernmen t has little lev erage in regulating diesel-based microgrids, as they address the national grid’s shortcoming [ 16 ]. Moreo ver, and besides the relatively high lev elized cost of electricit y (LCOE) of diesel generators, compared to other sources [ 11 ], the almost exclusive dep endence of the microgrid on fossil fuels mak es generation more sensitive to the frequen t disturbances in fuel supply . While the final electricit y price in the microgrid reflects the high generation cost and prev ailing mark et p o wer, it often ov erlo oks emission-related exter- nalities incurred by consumers. The degraded air quality in affected coun tries 3 p oses imp ortan t health risks, particularly around the generation sites gen- erally em b edded in densely p opulated neighborho o ds [ 17 ]. Greenhouse gas emissions resulting from excessive diesel-based generation are also kno wn to significan tly contribute to climate c hange [ 17 ]. T o coun ter the monopoly p o wer of the diesel generator company (DGC) o ver the microgrid, and to enhance consumer autonomy in generation and pricing, households in dev eloping coun tries are increasingly relying on off-grid household photov oltaic (PV) systems [ 9 ]. In Y emen, for example, war-ridden areas are primarily relying on ro oftop solar PV systems for electricity gener- ation, with 50% and 75% of households in rural and urban areas resp ectiv ely ha ving inv ested in an off-grid solar PV system [ 18 ]. As of 2024, 4% of the sub-Saharan p opulation relies solely on solar home systems (SHS), with an additional 2.5% using such systems as bac kup sources. Although SHS allow for cleaner generation, and are often praised as sustainable “last-mile” elec- trification solutions [ 19 ], their in termittent nature and ad-ho c deploymen t at the household level introduce sev eral inefficiencies. During hours of p eak sun- ligh t, lev els as high as 75% of excess generation potential are w asted, reducing the economic attractiveness [ 20 , 21 ]. Conv ersely , in hours of low sunligh t, the supply from PV is unreliable, and could fail to meet the h ousehold’s demand [ 20 ]. Multiple studies hav e discussed the hybrid setups in tegrating SHS to diesel-based microgrids from tec hno-economic p erspectives. This arrange- men t allo ws PV-o wners to sell their excess supply during da y-time and satisfy their excess demand in other times of the day [ 9 , 22 ]. In doing so, the share of distributed renewable energy in the microgrid increases while the DGC 4 acts as a reliable backup source. How ever, this mo dalit y must b e designed in a wa y that considers the en trenc hed p o w er that the DGC currently ex- ercises and the limited regulatory o v ersight of go vernmen ts. Otherwise, an y theoretically optimized system will not b e adopted in practice as the DGC remains the primary decision maker on access and supply (or lack thereof ) within the microgrid [ 15 , 23 ]. Therefore, the purp ose of this pap er is to answer the following researc h question: How c an a r e gulator, with limite d r e gulatory oversight and who c annot chal lenge the pr ofitability of the entr enche d DGC, design a p olicy for cle aner ener gy micr o grids? Particularly, these micr o grids, although c ontr ol le d by a DGC, should incr e ase the utilization of the existing household PV assets, r e duc e r elianc e on diesel-b ase d gener ation, limit unmet demand and impr ove the affor dability of ele ctricity. T o answ er this question, w e dev elop a bi-level game theoretical mo del where the regulator, constrained by the entrenc hed profitability of the DGC, acts first then the DGC resp onds. A t the first level, w e represen t the reg- ulator as an en tity interested in maximizing the economic surplus of the households, b oth PV-owners and non-PV owners, while maintaining at least the same lev el of profitability of the DGC op erations compared to the status quo. Building on existing regulatory practices on price caps in diesel-based microgrids [ 24 ], we foresee the regulator able to design a p olicy comp osed of t wo price caps: (i) an upper cap on the price of electricit y sold b y the DGC to the households, and (ii) a low er cap on a feed-in tariff (FiT) for the electricit y sold from household PV o wners to the DGC. At the second level, the DGC acts as a profit maximizer while con trolling access to the micro- 5 grid, generation from its assets, and electricity purc hases from PV o wners with excess generation. Through logging devices, we collect actual demand profiles for households in a microgrid in Lebanon, a coun try that witnessed con tinuous electricity challenges. W e then apply our mo deling framew ork to the Lebanese case generating k ey techno-economic and regulatory insights while properly accoun ting for the existing p olitical realities of the market participan ts. The rest of this paper pro ceeds as follo ws: Section 2 reviews the literature on microgrids and FiT policies for developing countries, Section 3 details the prop osed bi-lev el game theoretical framew ork along with the solution approac h. Section 4 presen ts our case study and the data used. Section 5 discusses the results while Section 6 summarizes the main findings. 2. Related works The literature on the managemen t of crisis-driv en microgrids is scarce. The most related existing literature cov ers the tec hno-economic ev aluation of swarm grids, where in vestmen ts are the result of the organic addition of capacit y to the microgrid b y individual pla yers [ 22 ]. In this case, energy man- agemen t is often an afterthought, as opp osed to microgrids that are originally designed to efficiently meet a forecasted demand. This t yp e of grid, similar to the ones dev eloping in the Global South, suffer from a lac k of regulatory framew orks to efficiently manage the existing assets. Another av en ue of w ork tac kles feed-in tariff p olicies as incentiv es for the installation of ro oftop PV systems and trading with the national grid, but they are rarely designed at the microgrid lev el. In either case, the issue of microgrid o wnership is ov er- 6 lo ok ed, making b oth techno-economic and regulatory analyses only partially represen tative of realit y . 2.1. Swarm grids and micr o grids in developing c ountries With microgrids and sw arm electrification emerging as a natural con- sumer resp onse in crisis settings, the Global South has observ ed the dev el- opmen t of renew able energy comm unities, with household o wnership and participation b eing a k ey success factor in implemen tation [ 25 , 26 ]. The existing literature has thoroughly discussed the techno-economic asp ects of sw arm grids and microgrids, fo cusing on the reduction of ro oftop PV systems generation inefficiencies and the optimal design and pricing under multiple ob jectives [ 22 , 27 ]. F or example, in the Philippines, a case study shows im- pro vemen ts in microgrid efficiency and black out reduction when connecting all solar home systems [ 28 ]. Similarly , a Y emeni case study compares three setups, including standalone priv ate systems, a swarm grid, and a formally organized microgrid, sho wing that when PV-o wners are allow ed to participate in either the sw arm grid or the microgrid, electricity costs were reduced and demand realization w as improv ed [ 29 ]. The results of a study from Mada- gascar further show that the trading of excess energy b et ween individual PV o wners (prosumers) increased consumer economic surplus through a more ef- ficien t mark et-clearing approach relying on a p eer-to-p eer algorithm, leading to lo w er electricit y prices [ 30 ]. Applications of game theory in comm unit y grids highlight as w ell the b enefit of p eer-to-p eer trading, citing b etter uti- lization of renewable energy , reduced carb on emissions, and increased so cial w elfare, under the assumption of a neutral grid op erator [ 31 , 32 ]. In practice, how ever, the prev alence of diesel-based microgrids in fragile 7 con texts restricts PV-owner participation [ 8 ]. In such microgrids, regulatory o versigh t is limited, and the DGC o wns the grid, obstructing the adequate p olicy design and fair tariff setting needed for effective renewable integration [ 11 , 33 ]. In fragile states, authorities often lac k b oth the capacit y and the mandate to regulate or formalize existing microgrids [ 10 , 34 ], lea ving the DGCs’ con trol ov er them largely unaddressed. 2.2. FiT p olicies The general consensus around the role of feed-in tariff p olicies at the con- sumer level is that they incen tivize in vestmen ts in renewable sources [ 35 ], and enhance the efficiency of already-installed assets [ 18 ]. Dev elop ed coun tries ha ve led in establishing p olicies and mechanisms to manage prosumer en- ergy exc hange with the grid, including at the microgrid level, with particular consideration for greenhouse emission reduction [ 36 ]. Extensive framew orks deriving the optimal price of electricity fed hav e b een developed. F or exam- ple, a sequential mo del-based optimization is prop osed for flexible FiT design in microgrids, based on historical and pro jected data, while keeping in line with Australian regulations [ 37 ]. When implementing FiT p olicies, developing countries aim to tackle addi- tional concerns, such as electricit y access, av ailabilit y and affordability . The connection of ro oftop PV systems to the grid often corresp onds to the cost- optimal configuration, and PV-o wners are generally willing to participate in FiT programs [ 38 ]. It is therefore not surprising that m ultiple developing coun tries ha ve set national FiT p olicies, with considerations for the a v ail- abilit y of renewable energy technologies [ 39 ]. Hence, sp ecific frameworks for the financial mo deling of FiT s hav e b een dev elop ed in accordance with lo- 8 cal market complexities, and country-specific applications, suc h as in Ken ya and Mala wi [ 40 ]. How ever, suc h national p olicies are not adequate when national grids are intermitten t, as feed-in can only happ en when the grid is energized. Accoun ting for this limitation, a decision analysis framew ork has b een prop osed to plan hybrid renew able energy systems under uncertain grid interconnection in Lebanon, explicitly considering the unreliabilit y of the coun try’s national utility in the discussion of future FiT s [ 41 ]. T anzania is one of the only countries where considerations for lo cal c hal- lenges and microgrid-lev el FiT p olicies o v erlap. It explores ho w FiT s can b e adapted sp ecifically for remote mini-grids, determining a tariff reflecting rural op erating conditions, and comp eting with diesel generator economics [ 39 , 40 ]. Y et, despite addressing the gap of microgrid-sp ecific FiT p olicies in dev eloping economies, the prop osed framework shares a lac k of consider- ation for mark et pow er with the ab o ve-men tioned works, treating op erators as neutral entities. The ov erall scarcity of regulations in informal microgrids is not unexp ected, as their v ery emergence is the result of w eak regulatory p o w er [ 10 ]. 3. Mo deling framew ork The prop osed mo deling framework considers that the regulatory entit y , acting as a leader in a bi-lev el game, is interested in achieving the follow- ing four goals: (i) increasing the utilization of the existing household PV assets, (ii) reducing reliance on diesel-based generation, (iii) limiting the un- met demand, and (iv) reducing the electricit y price in the DGC-controlled microgrid. This should be done without reducing the profits of the DGC. 9 Figure 1: Diagram of the bi-lev el game A chieving these desired regulatory goals contributes to increasing the house- hold economic surplus comp osed of the com bined v alue of (i) the household satisfied demand, and (ii) the household PV electricity fed into the microgrid. The regulatory entit y is able to set an upp er b ound on the price of electricit y sold to the households 𝑃 𝑚 𝑎 𝑥 , and a low er b ound on the price of electricit y sold to the DGC b y the household PV owners 𝐹 𝑖𝑇 𝑚𝑖 𝑛 (Figure 1 ). The DGC acts as a follo wer in terested in maximizing its profits in re- sp onse to the p olicy adopted by the regulatory entit y comp osed of the tuple ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) . T raditionally , drawing on practical exp eriences in develop- ing countries with diesel-based microgrids, the DGC con trols the generation sc hedule (and accordingly the outage sc hedule) and the household access to the microgrid [ 8 ] (Figure 2a ). It sells the electricity at a price 𝑃 of its c hoice suc h that 𝑃 ≤ 𝑃 𝑚 𝑎 𝑥 . In our prop osed framework, the DGC is offered the additional flexibility of purc hasing electricit y from households with PV sys- tems at a price 𝐹 𝑖 𝑇 of its choice as long as 𝐹 𝑖 𝑇 ≥ 𝐹 𝑖𝑇 𝑚𝑖 𝑛 . The DGC is also free to c ho ose to inv est in its own PV and battery storage assets if such an 10 (a) Status-quo microgrid (b) Prop osed microgrid Figure 2: Representativ e diagram of the microgrid b efore and after change in vestmen t is economically attractive (Figure 2b ). F ormally sp eaking, in this Stac kelberg game, the leader sets a p olicy affecting the feasible set of the follo wer, and ev aluates its own pa yoff by an ticipating the follow er’s reaction. A ccordingly , the full game theoretical mo del can b e written as: max 𝑃 𝑚𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 household economic surplus (leader ob j) subje ct to: ensuring: profits 𝐷 𝐺 𝐶 ≥ base profits 𝐷 𝐺 𝐶 (leader constrain t) and: max S profits 𝐷 𝐺 𝐶 (follo wer ob j) subje ct to: economic and tec hnical constraints (follo wer constrain ts) The first lev el decisions are comp osed of the tuple ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) . T a- 11 ble 1 presen ts the decision v ariables used in the second-lev el mo del, whic h constitute set S . The parameters and relev an t set names are shown in Ap- p endix A , in T ables 5 and 6 resp ectiv ely . T o preserv e the tractability of the mo del, w e emplo y representativ e days, eac h w eighed by 𝜔 𝑑 , c haracterizing the en tire year. F or all v ariables and parameters, the indices 𝑖 , 𝑔, 𝑦 , 𝑑 , ℎ refer to the household t yp e, the tec hnology , the year, the represen tative day , and the hour, resp ectiv ely . Moreo ver, all decision v ariables are nonnegativ e, and represen ted b y Latin letters, while parameters are represen ted b y Greek ones. The t wo lev els of the game are further detailed in Sections 3.1 and 3.2 . 3.1. First-level The regulatory en tity’s pa y off, equiv alent to the discoun ted household economic surplus (HES), is giv en by Equation 1 : max 𝑃 𝑚𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛  𝑦 ∈Y " 𝑄 𝑦 ( 𝜉 − 𝑃 ) +  𝑑 ∈ D 𝜔 𝑑  ℎ ∈H 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ × 𝐹 𝑖𝑇 #  ( 1 + 𝛾 𝑅 𝐸 ) − 𝑦  (1) It is made up of tw o comp onen ts: (i) the v alue of the demand met, which can b e computed as the pro duct of the met demand 𝑄 𝑦 (detailed in Equation 2 ) and the difference b et w een the v alue of lost load 𝜉 and the price 𝑃 , and (ii) the v alue from feeding in household-generated electricit y to the microgrid, computed as the pro duct of the fed-in capacity 𝐹 𝑖 𝑦 ,𝑑 , ℎ and the feed-in tariff 𝐹 𝑖𝑇 . The sym b ol 𝛾 𝑅 𝐸 denotes the discoun t rate of the regulatory entit y . The total demand in the microgrid is expressed as − Í 𝑖 ∈I min  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  where the symbols Ω 𝑖 = 0 , 𝑦 and Ω 𝑖 = 1 , 𝑦 resp ectiv ely represent the n umber of non PV-o wner households and PV-owner households in the microgrid. The sym- b ol 𝜎 𝑖 , 𝑦, 𝑑 , ℎ denotes the excess electricity supply for household category 𝑖 . A 12 Sym b ol V ariable name Unit 𝐴 𝑔 , 𝑦 A dded DGC capacity k W 𝐵 + 𝑦 , 𝑑 , ℎ Battery charge during hour ℎ k W 𝐵 − 𝑦 , 𝑑 , ℎ Battery discharge during hour ℎ k W 𝑏 𝑘 , 𝑦 , 𝑑 , ℎ Binary v ariable enforcing the 𝑘 𝑡 ℎ constrain t on the heat rate curv e binary 𝐶 𝑔 , 𝑦 Capacit y installed by the DGC k W 𝐷 𝑔 , 𝑦 , 𝑑 , ℎ Dispatc hed p o wer k W 𝐹 𝑖 𝑖 , 𝑦 , 𝑑 , ℎ F ed-in capacity from PV-owner k W 𝐹 𝑖𝑇 F eed-in T ariff USD/k Wh 𝑃 Price of electricity set by the DGC USD/k Wh 𝑄 𝑦 Demand served by the DGC k Wh 𝑅 𝑒 𝑡 𝑔 , 𝑦 Retired capacity k W 𝑆 𝑜𝐶 𝑦 , 𝑑 , ℎ State of charge at the end of ev ery hour ℎ k Wh 𝑆 𝑜𝐶 0 𝑦 , 𝑑 State of charge at the b eginning of the day k Wh 𝑈 𝑦 , 𝑑 , ℎ Unmet demand across all households k Wh 𝑅 𝑦 , 𝑑 , ℎ Diesel consumption as a function of the diesel generator heat rate and the dispatch L/k Wh 𝑆 𝑔 Salv age v alue of the installed capacity at the end of the planning horizon USD 𝑇 𝑅 𝑦 T otal reven ues earned b y the DGC USD 𝑇 𝐶 𝐶 𝑦 T otal capital costs incurred by the DGC USD 𝑇 𝑂𝑉 𝐶 𝑦 T otal op eration v ariable costs incurred b y the DGC USD 𝑇 𝑂 𝐹 𝐶 𝑦 T otal op eration fixed costs incurred by the DGC USD 𝑇 𝑈 𝐷 𝐶 𝑦 T otal unmet demand costs incurred by the DGC USD T able 1: Second-level (DGC) decision v ariables 13 p ositiv e 𝜎 𝑖 , 𝑦, 𝑑 , ℎ means that household generation p otential from PV exceeds its instantaneous demand, and that PV-owners can feed at most 𝜎 𝑖 , 𝑦, 𝑑 , ℎ in to the microgrid. A negativ e 𝜎 𝑖 , 𝑦, 𝑑 , ℎ indicates a net demand. The served demand 𝑄 𝑦 is defined in Equation 2 . It is derived from the total demand in the microgrid, reduced b y the unserv ed demand 𝑈 𝑦 ,𝑑 , ℎ de- termined b y the DGC: 𝑄 𝑦 =  𝑑 ∈ D 𝜔 𝑑  ℎ ∈H −  𝑖 ∈I min  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  ! − 𝑈 𝑦 ,𝑑 , ℎ ∀ 𝑦 (2) The PV-o wner’s generation excess is the difference b et ween the household PV total generation and the household demand, whereas the non-PV owner’s excess is simply the negativ e demand, as defined b elo w: 𝜎 𝑖 = 0 , 𝑦, 𝑑 , ℎ = − 𝜇 𝑖 = 0 , 𝑦, 𝑑 , ℎ ∀ 𝑦 , 𝑑 , ℎ , 𝑖 𝜎 𝑖 = 1 , 𝑦, 𝑑 , ℎ = 𝜙 𝑦 ,𝑑 , ℎ × 𝜃 𝑃𝑉 𝑖 = 1 − 𝜇 𝑖 , 𝑦, 𝑑 , ℎ ∀ 𝑦 , 𝑑 , ℎ , 𝑖 where 𝜇 𝑖 , 𝑦, 𝑑 , ℎ is the household demand, 𝜙 𝑦 ,𝑑 , ℎ is the solar capacit y factor, and 𝜃 𝑃𝑉 𝑖 = 1 is the a verage size of the household PV-owner PV system. The decision v ariables of the regulator are only b ounded by 0 on the lo wer end. An y nonnegative feed-in tariff is acceptable to PV-owners, as their excess generation p oten tial w ould hav e b een otherwise wasted. How- ev er, ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) m ust k eep the DGC economically incen tivized, i.e., the maximized net present v alue (NPV) of profits of the DGC under an y p ol- icy ( 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ) should b e greater than or equal to the one under the current mark et conditions ( 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 0 ) (Equation leader constraint ), establishing the constrain t for the DGC’s participation in the game. The DGC’s profit max- imizing mo del, presented in Section 3.2 , is implicit to this constraint. 14 3.2. Se c ond-level The b elo w mo del describing the microgrid under DGC control is inspired b y the a v ailable literature around determining the optimal generation asset sizing in a microgrid under a profit-maximization ob jectiv e [ 42 , 43 ]. The DGC desires to maximize its NPV. W e assume that, if multiple sets of deci- sions lead to the same maximized NPV, the DGC prefers the one with the lo west level of unmet demand (Equation 3 ). This preference is illustrated b y a v ery small unmet demand p enalt y w eigh t 𝜀 introduced to the DGC’s ob jective function: max S 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 − 𝜀  𝑦  𝑑 𝜔 𝑑  ℎ 𝑈 𝑦 ,𝑑 , ℎ · ( 1 + 𝛾 𝐷 𝐺 𝐶 ) − 𝑦 ! (3) The NPV form ulation is detailed in Equation 4 : 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 =  𝑦 ∈Y   𝑇 𝑅 𝑦 − 𝑇 𝐶 𝐶 𝑦 − 𝑇 𝑂 𝑉 𝐶 𝑦 − 𝑇 𝑂 𝐹 𝐶 𝑦  ( 1 + 𝛾 𝐷 𝐺 𝐶 ) − 𝑦  +  𝑔 ∈ G 𝑆 𝑔 ( 1 + 𝛾 𝐷 𝐺 𝐶 ) − Υ (4) The symbol 𝛾 𝐷 𝐺 𝐶 represen ts the discount rate. F or eac h year 𝑦 , 𝑇 𝑅 𝑦 is the total reven ues, 𝑇 𝐶 𝐶 𝑦 the total capital costs, 𝑇 𝑂𝑉 𝐶 𝑦 the total op eration v ariable costs, 𝑇 𝑂 𝐹 𝐶 𝑦 the total op eration fixed costs, and 𝑆 𝑔 the salv age v alue of the technology 𝑔 . Eac h of these terms are detailed in Equations 5 through 8 ∀ 𝑦 , and 9 ∀ 𝑔 , where the summations indexed b y 𝑦 , 𝑑 , ℎ , 𝑔 , 𝑖 are 15 o ver the en tire sets Y , D , H , G , I , unless otherwise sp ecified. 𝑇 𝐶 𝐶 𝑦 =  𝑔  𝜆 𝐶 𝑔 , 𝑦 · 𝐴 𝑔 , 𝑦  ∀ 𝑦 (5) 𝑇 𝑂 𝑉 𝐶 𝑦 =  𝑑 𝜔 𝑑  ℎ ©  « 𝜆 𝑂𝑉 𝑔 = 𝐵 · 𝐵 + 𝑦 ,𝑑 , ℎ +  𝑔 ∈ G 𝑔 𝜆 𝑂𝑉 𝑔 · 𝐷 𝑔 , 𝑦, 𝑑 , ℎ + 𝑅 𝑦 ,𝑑 , ℎ · 𝜋 +  𝑖 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ · 𝐹 𝑖𝑇 ª ® ¬ ∀ 𝑦 (6) 𝑇 𝑂 𝐹 𝐶 𝑦 =  𝑔  𝜆 𝑂 𝐹 𝑔 · 𝐶 𝑔 , 𝑦  ∀ 𝑦 (7) 𝑇 𝑅 𝑦 = 𝑃  𝑑 𝜔 𝑑  ℎ ©  « 𝐵 − 𝑦 ,𝑑 , ℎ − 𝐵 + 𝑦 ,𝑑 , ℎ +  𝑔 ∈ G 𝑔 𝐷 𝑔 , 𝑦, 𝑑 , ℎ +  𝑖 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ ª ® ¬ ∀ 𝑦 (8) 𝑆 𝑔 = Υ − 1  𝑦 =Υ − 𝜈 𝑔  𝐴 𝑔 , 𝑦 · 𝜆 𝐶 𝑔 , Υ ·  1 − Υ − 𝑦 𝜈 𝑔   ∀ 𝑔 (9) The change from the status quo to the prop osed microgrid is represented b y the binary input 𝛽 , with 𝛽 = 1 indicating, from the p ersp ectiv e of the DGC, the p ossibilit y of feeding in electricity from households and installing PV and battery capacities. Equations 10 and 11 limit the installation of PV and batteries resp ectively to 0 in the curren t situation, or to a large enough n umber 𝑀 otherwise. Equation 12 sets an upp er b ound on the PV-owners’ feed-in when 𝛽 = 1 : 𝐶 𝑔 = 𝑃𝑉 , 𝑦 ≤ 𝛽 × 𝑀 ∀ 𝑦 (10) 𝐶 𝑔 = 𝐵 , 𝑦 ≤ 𝛽 × 𝑀 ∀ 𝑦 (11) 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ ≤ max  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  × 𝛽 ∀ 𝑦 , 𝑑 , ℎ , 𝑖 (12) The DGC’s ob jective function is sub ject to the following constrain ts: • The supply-demand balance constraint: 16 𝑈 𝑦 ,𝑑 , ℎ + 𝐵 − 𝑦 ,𝑑 , ℎ +  𝑔 ∈ G 𝑔 𝐷 𝑔 , 𝑦, 𝑑 , ℎ +  𝑖 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ = 𝐵 + 𝑦 ,𝑑 , ℎ −  𝑖 min  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  ∀ 𝑦 , 𝑑 , ℎ (13) Equation 13 ensures that the dispatc hed energy 𝐷 𝑔 , 𝑦, 𝑑 , ℎ , the disc harging of batteries 𝐵 − 𝑦 ,𝑑 , ℎ and the fed-in capacit y 𝐹 𝑖 are enough to satisfy the household demand − Í 𝑖 min  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  and the c harging of bat- teries 𝐵 + 𝑦 ,𝑑 , ℎ , while allo wing unmet demand 𝑈 𝑦 ,𝑑 , ℎ . F urthermore, at any hour, the unmet demand must not exceed the total demand (Equation 14 ): 𝑈 𝑦 ,𝑑 , ℎ ≤ −  𝑖 min  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  ∀ 𝑦 , 𝑑 , ℎ (14) • The budget constraint: While the DGC can c ho ose which capacities to ins tall, its decision is b ounded by a total budget Π . This budget is av ailable at year 𝑦 = 0 , and all subsequen t discounted cash flows due to capacit y installation exp enses should amount to it, as expressed in Equation 15 .  𝑔 ∈ G 𝑔  𝑦 𝐴 𝑔 , 𝑦 × 𝜆 𝐶 𝑔 , 𝑦 × ( 1 + 𝛾 𝐷 𝐺 𝐶 ) − 𝑦 ≤ Π (15) • The regulator p olicy constraints: The electricity price and the FiT set by the DGC are constrained by the regulator’s decisions on 𝑃 𝑚 𝑎 𝑥 and 𝐹 𝑖𝑇 𝑚 𝑎 𝑥 (Equations 16 and 17 ). 𝑃 ≤ 𝑃 𝑚 𝑎 𝑥 (16) 𝐹 𝑖𝑇 ≥ 𝐹 𝑖𝑇 𝑚𝑖 𝑛 (17) • The technologies’ capacity and retiremen t constraints: 17 App endix B.1 details the constraints on installation and retirement of capacit y for the different tec hnologies. • The dispatch constraints: The dispatc h (or in the case of batteries, c harge and disc harge) for eac h tec hnology is limited b y its corresp onding installed capacit y , and a corresp onding capacity factor when applicable. The equations for these constrain ts are in app endix B.2 . The diesel generator should also ob ey a set of constrain ts p ertaining to its heat rate. T raditionally , the heat rate function is a U-shap ed curve showing the fuel consumed p er generated k Wh against the utilization rate of a diesel generator. T o preserv e the linearity of constrain ts, w e approximate the heat rate curve by a piecewise constant function [ 44 ]. In the following, we mo del the curv e as three piecewise constant functions where the diesel consumption 𝑅 𝑦 ,𝑑 , ℎ is defined as: 𝑅 𝑦 ,𝑑 , ℎ = 𝜌 1 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ if 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ ≤ 0 . 30 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 (HR.1) 𝑅 𝑦 ,𝑑 , ℎ = 𝜌 2 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ if          0 . 30 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 < 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ ≤ 0 . 60 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 (HR.2) 𝑅 𝑦 ,𝑑 , ℎ = 𝜌 3 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ if 0 . 60 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 < 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ (HR.3) ∀ 𝑦 , 𝑑 , ℎ . • The storage technologies constraints: App endix B.3 presen ts the constraints related to the trac king of the 18 batteries state of c harge. 3.3. Solution appr o ach The second-lev el mo del prop osed in this pap er is non-linear in the equa- tions defining 𝑇 𝑂 𝑉 𝐶 𝑦 and 𝑇 𝑅 𝑦 (Equations 6 and 8 resp ectiv ely). The non- linearit y is caused b y the decision v ariables 𝑃 and 𝐹 𝑖𝑇 . How ev er, lemma 1 sho ws that 𝑃 and 𝐹 𝑖𝑇 are optimized at 𝑃 𝑚 𝑎 𝑥 and 𝐹 𝑖 𝑇 𝑚𝑖 𝑛 resp ectiv ely , and therefore can b e treated as fixed parameters. Lemma 1. L et 𝑁 𝑃𝑉 ∗ ( 𝑃, 𝐹 𝑖𝑇 ) b e the maximize d NPV for a given 𝑃 , 𝐹 𝑖𝑇 under c onstant demand. Then, 𝑁 𝑃𝑉 ∗ ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) ≥ 𝑁 𝑃𝑉 ∗ ( 𝑃, 𝐹 𝑖𝑇 ) ∀ 𝑃 ≤ 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 ≥ 𝐹 𝑖𝑇 𝑚𝑖 𝑛 . Pr o of. Refer to app endix C . □ Therefore, the bi-lev el mo del is solv ed b y iterating o ver a grid of the tuple ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) set by the regulatory en tity , and tak en as equal to 𝑃 and 𝐹 𝑖𝑇 resp ectiv ely by the DGC in its profit-maximizing mo del. At every step, the regulator’s ob jectiv e function is ev aluated, ultimately determining the optimal p olicy tuple. 4. Case study and data W e apply the game theoretical framework describ ed in Section 3 to the case of a real microgrid in Deir Qanoun Ennaher, in Lebanon. Lebanon’s residential electricity market is c haracterized b y a highly inter- mitten t utility , Electricite du Liban (EDL), and a proliferation of diesel-based microgrids [ 16 ]. Since the 2019 economic crisis, the country has observed an 19 organic increase in distributed renewable energy , namely , household solar PV systems [ 45 ]. As of 2023, the installed solar PV capacit y is estimated at 1000 MW [ 46 ]. The microgrid considered in our case study is centered around a 400 k W diesel generator. Around 40% of the households connected to the micro- grid are PV-owners. The num b er of households, their av erage PV capacity and characteristics of the diesel generator are summarized in T able 2 . The lifetimes of all considered tec hnologies are sho wn in T able 3 . T o obtain the represen tative household demand profiles, sho wn in Figure 3 , w e installed log- gers in three households and collected hourly load data from August 2024 to April 2025. The ra w logs are av ailable on a public database [ 47 ]. As for solar capacit y factor data, w e relied on the widely cited w ebsite renew ables.ninja [ 48 , 49 ]. The load and solar capacity factor data obtained w ere a v eraged (using k-means clustering) to generate three representativ e daily load pro- files, for summer, winter and spring/fall, as shown in Figures 3 and 4 . The inputs and their sources are a v ailable in the pro ject’s Gith ub repository 1 , and summarized in App endix E . 5. Results and discussion W e first lo ok at the status-quo microgrid, where the DGC only op erates a diesel generator without feed-in from PV-o wners’ excess, nor installing PV nor batteries. This case constitutes the b enchmark against which our prop osed mo del is compared. W e compute the current NPV of the DGC’s 1 link: https://github.com/molleik/microgrid_MP 20 Num b er of PV-o wner households 400 Num b er of non-PV-o wner households 250 A v erage installed rooftop PV capacity p er PV-o wner household (k W) 4 Installed diesel generator capacit y (k W) 400 Remaining lifetime of installed diesel generator (y) 3 T able 2: Data on Deir Qanoun Ennaher T echnology Lifetime Solar PV 20 y ears Batteries 8 y ears Diesel generator 5 y ears T able 3: T ec hnology lifetimes (a) All households owning PV (b) All households not owning PV Figure 3: Representativ e microgrid demand profiles 21 Figure 4: Representativ e profiles of solar capacity factors profits ( 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 0 ), which is used as a threshold for the leader’s constraint. W e also iden tify the budget needed to supp ort current op erations ov er the planning horizon. This budget, denoted Π 0 , is used as the default budget in the mo dified mo del. 5.1. Change fr om status-quo Holding the budget at Π 0 , and allowing renew able energy in the microgrid, i.e. , PV-owner feed-in and DGC-owned PV-battery system, The optimal ( 𝑃, 𝐹 𝑖𝑇 ) p olicy can b e found by v arying 𝑃 b et w een 0 and 0.4 USD (the status-quo price) and 𝐹 𝑖𝑇 b et ween 0 USD and 𝑃 . T able 4 compares b oth the status quo and the new mo del along the following metrics: • The p ercen tage of unmet demand, defined as the ratio of unmet demand to the total demand, • The p ercentage of wasted excess household PV generation p oten tial, defined as the ratio of wasted excess household PV generation to total household PV generation p oten tial, 22 Status quo Prop osed mo del 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 (USD) 2.25 M 2.46 M Budget (USD) 420 k 420 k HES (USD) 6.99 M 8.26 M Unmet demand (%) 1.9 12.2 W asted excess PV generation p oten tial (%) 58.8 6.2 Price (USD/k Wh) 0.4 0.36 F eed-in T ariff (USD/k Wh) - 0.12 Renew able energy p enetration (%) 0 60.1 T able 4: Comparison b et ween status-quo and prop osed mo del, where M refers to millions and k to thousands. • The renew able energy p enetration, defined as the DGC-owned PV gen- eration and household PV fed-in capacity , as a p ercentage of the serv ed demand. The prop osed case shows a 18% impro vemen t in HES. It is driven by a 10% decrease in electricity price, and the sale of household PV-owner excess gen- eration at 0.12 USD/k Wh. The corresponding fed-in capacity results in a decrease of 52.6 p ercen tage p oin ts of w asted excess generation p oten tial. Fi- nally , the increased unmet demand in the prop osed mo del is further discussed in Section 5.4 . Figure 5a highlights the feasible p olicy tuples, for which 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ≥ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 0 . The dra wn curve maps, for ev ery 𝑃 , the maximum 𝐹 𝑖 𝑇 that k eeps the DGC satisfied with profits equal to the ones in the current case. As the 𝐹 𝑖𝑇 on the b oundary increases, it b ecomes high enough for the DGC 23 (a) F easibility region (b) HES (maximum inside red b o x) Figure 5: Prop osed mo del to drastically reduce feed-in, and rely on owned PV generation instead. This case is represented by the sharp increase in the maximum feasible 𝐹 𝑖𝑇 af- ter 𝑃 = 0 . 37 USD/k Wh. The hatched area u nder this p ortion of the curv e sho ws the p olicy tuples where the fed-in household PV excess is limited and mak es up less than 5% of the total supply to the microgrid. The remaining highligh ted region represen ts the set of ( 𝑃 , 𝐹 𝑖 𝑇 ) tuples for which significantly feeding household PV excess in to the system is economically viable for the DGC. F or every feasible ( 𝑃 , 𝐹 𝑖𝑇 ) tuple, the HES is sho wn in Figure 5b . Con- sidering Figures 5a and 5b , the optimal and near-optimal tuples include significan t PV-owner feed-in. The HES resulting from the prop osed mo del is a reflection of the c hange in the capacity and generation p ortfolio in the microgrid. Figure 6a shows the yearly in-place capacities. During the first three y ears of op erations, 24 (a) Installed capacities (b) Y early generation profile Figure 6: Capacity and generation p ortfolio in the prop osed mo del the DGC uses the existing diesel capacity , holding off on an y additional in vestmen t in a PV-battery system un til the end of the diesel generator’s lifetime. Then, the low er LCOE of PV-battery is more attractive than the relativ ely exp ensiv e diesel generator. With the installation of batteries, the DGC also feeds-in more PV-owner excess (Figure 6b ). The addition of a PV-battery system leads to unmet demand during some hours in win ter. The av ailable PV capacity and battery charge cannot meet the demand, and the expansion of any capacity for the purp ose of serving these few hours is prohibitiv ely exp ensiv e. 5.2. Sensitivity analysis on the budget c onstr aint Financing generation capacit y expansion is a key concern when deter- mining the optimal size of a microgrid, particularly in dev eloping coun tries. As in vestmen t budgets often constitute a barrier to achieving grid reliability , the literature commonly ackno wledges this constraint [ 50 ]. The regulatory en tity must therefore find the set of p olicies that do not violate the DGC’s 25 Figure 7: F easible regions of regulatory entit y p olicies preference under an assumed DGC budget, then establish the ( 𝑃, 𝐹 𝑖 𝑇 ) tuple that maximizes the HES. The impact of v arying this budget is imp ortant on the DGC’s decisions, and therefore, on the b est ( 𝑃 , 𝐹 𝑖𝑇 ) p olicy . Similarly to Figure 5a , Figure 7 sho ws the feasible regions under dif- feren t budgets. F or the budget of 1.75 M USD, we find that the budget constrain t becomes non-binding for all ( 𝑃, 𝐹 𝑖𝑇 ) tuples, and the DGC sig- nifican tly reduces its reliance on household PV excess as soon as the 𝐹 𝑖 𝑇 exceeds the LCOE of in vesting in additional PV capacity , corresp onding to 0.08 USD/k Wh. F or low er budgets the tipping 𝐹 𝑖𝑇 betw een feeding-in elec- tricit y from households and almost fully relying on DGC owned PV system increases to 0.17, 0.18, and ov er 0.40 USD/k Wh for the budgets of 0.75 M, 0.4 M and 0.25 M USD resp ectiv ely . Figure 8 shows the HES for all feasible ( 𝑃, 𝐹 𝑖𝑇 ) tuple for ev ery considered budget. As it is the case under Π 0 , the optimal solution still allows for significan t feed-in from PV-owner excess, and 26 (a) Budget = $250k (b) Budget = $420k (c) Budget = $750k (d) Budget = $1.75M Figure 8: HES of feasible p olicies, with red b o xes indicating the maximum do es not entirely rely on DGC-o wned generation. Higher budgets generally yield higher HES (Figure 9 ). Under the pro- p osed mo del, the status-quo HES can b e reco vered for a budget lo wer than Π 0 b y 58.3%. Main taining the b enchmark b udget Π 0 , the HES would increase b y 18% with the prop osed mo del. F or every optimal HES, at eac h budget, we lo ok at the four v alues driving the regulator’s ob jectiv e function: 𝑃 , 𝐹 𝑖𝑇 , serv ed demand, and household PV-o wner excess utilization. 27 Figure 9: Optimal HES at differen t budgets Figure 10: Optimal price and feed-in tariff at different budgets 28 Figure 10 shows the optimal prices and feed-in tariffs for every consid- ered case. As the budget increases, the DGC replaces exp ensiv e diesel-based capacit y with cheaper PV-battery systems, reducing the ov erall costs and allo wing for a price decrease. The 𝐹 𝑖 𝑇 decreases as well: with tighter bud- gets, the DGC is constrained to use more household PV-owner excess as a c heap er alternative to the diesel generator. As the budget is relaxed, the p ossibilit y to add abundant DGC-owned PV capacities results in reducing the 𝐹 𝑖𝑇 b elo w the PV LCOE to ensure that the DGC prefers feed-in ov er PV self-generation. Budgets of 1.25 M USD or ab o v e are non-binding to the DGC and result in stable 𝑃 and 𝐹 𝑖𝑇 . These findings are further reflected in Figure 11a . With tigh t budgets, additional diesel capacity is needed. Batteries are first introduced to increase the utilization of the a v ailable prosumer feed-in. When the budget reaches 0.75 M USD or more, sizable PV in vestmen ts are undertak en. At budgets of 1 M USD and ab o ve, diesel generator capacities are entirely phased out follo wing the retiremen t of existing units. Figure 11b highlights the consid- erable share of total generation that the fed-in electricit y constitutes. At a budget of 0.75 M USD, the large installed capacity of battery storage results in maximizing the contribution of fed-in electricity to the microgrid d emand to 39% and in minimizing the wasted household PV excess generation p oten- tial to 5%. F or higher budgets, the increased PV capacit y then reduces the utilization of prosumer feed-in resulting in the wasted household PV excess generation p oten tial to reac h 10%. F or the tightest budget, the unmet de- mand share reac hes 15%. As the budget is further relaxed, this share drops to a stable 4%. 29 (a) T otal added/retired capacity (b) T otal energy generation Figure 11: T otal added capacities and total energy for differen t budgets 5.3. Sensitivity on numb er of household PV-owners PV-o wner participation is a k ey factor for HES impro vemen t. It is there- fore imp erative to understand the effects of v arying the share of household PV-o wners in the microgrid on the HES. F or ev ery household PV-o wner p en- etration case, corresp onding b enc hmark NPV and budget are computed. The c hange in HES b et ween each microgrid’s status-quo and the corresp onding prop osed mo del is illustrated in Figure 12 . At 0% household PV-o wners, in tro ducing a PV capacit y up on retirement of the diesel generator low ers op erating costs and increases the DGC’s NPV, ev en though unmet demand rises (Figure 13 ). Coupled with the absence of feed-in, this results in a de- crease in the HES from the status quo. With higher shares of PV-o wners, this undesired outcome is mitigated. The HES increases with the increasing p enetration of PV-o wners until b eing maximized at a 90% p enetration rate. Understandably , the p ercen tage of w asted household PV generation p oten- tial increases with the share of PV-owners, due to b oth the decrease in net 30 Figure 12: Difference in HES b et w een the prop osed mo del and status-quo (a) T otal added/retired capacity (b) T otal energy generation Figure 13: Added capacities and energy for different shares of PV-owners demand on the microgrid, and the increase in the total generation capacity of households (Figure 13b ). It is also w orth noting that, as the PV-o wner share reac hes 75% or higher, the DGC’s op erations are almost completely fo cused on battery storage. 5.4. Constr aining the unmet demand As sho wn in previous sections, the prop osed microgrid mo del results in limited unmet demand except when the budget is v ery tight or when the 31 (a) Unmet demand under different budgets (b) Unmet demand under different PV-owner shares Figure 14: Unmet demand for prop osed mo del and status quo share of PV-o wners in the microgrid is lo w. T o further shed ligh t on this imp ortan t measure, Figure 14 compares the unmet demand of the proposed mo del to that of the status quo for different budgets (Figure 14a ) and shares of PV-o wners in the microgrid (Figure 14b ). It is noticeable that the prop osed microgrid mo del results in higher unmet demand compared to the status quo for almost all the cases considered. Reg- ulating the ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) tuple alone, while p ermitting a profit-maximizing DGC to access household PV feed-in and inv est in PV-battery systems, gen- erally results in a low unmet demand but is insufficient to k eep it at levels ac hiev able in diesel-only microgrids. This observ ation reflects the regula- tor’s limited o versigh t ov er supply qualit y in the microgrid, which cannot b e addressed through price caps alone. T o quan tify the effects of the limited regulatory ov ersigh t o ver the qualit y of supply , and therefore the HES, w e in tro duce a theoretical regulator with the extended p o w er to imp ose, in addition to ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) , the status-quo unmet demand levels. Let 𝜒 0 represen t the unmet demand in the status-quo microgrid. The following constrain t guaran tees that the unmet demand in 32 (a) HES under different budgets (b) HES under different PV-owner shares Figure 15: HES for different microgrid cases the prop osed mo del do es not exceed 𝜒 0 ( 18 ):  𝑦  𝑑 𝜔 𝑑  ℎ 𝑈 𝑦 ,𝑑 , ℎ ≤ 𝜒 0 (18) Figure 15 compares the HES for three cases: i) the status-quo, ii) the prop osed mo del where the regulator only controls the tuple ( 𝑃 𝑚 𝑎 𝑥 , 𝐹 𝑖𝑇 𝑚𝑖 𝑛 ) , and iii) the proposed model with extended regulation imp osing constraint 18 under differen t budgets and shares of PV-owners in the microgrid. The follo wing observ ations can b e made. As revealed in previous results, the pro- p osed mo del strongly outp erforms the status-quo except for cases where the PV-o wner share in the microgrid is 10% or low er. The extended regulation is alw ays p erforming b est. The difference b et ween limited and extended regula- tions is less than 5% for budgets at or ab o ve 0.75 M USD and less than 7.6% for shares of PV-owners at or ab o v e 40%. This highligh ts that, despite the limitations of only relying on price caps in ac hieving low er unmet demand lev els, the loss in HES in the prop osed mo del compared to a regulator with the p o w er to imp ose a certain qualit y of supply is small. 33 6. Conclusion In man y dev eloping coun tries affected by political and economic insta- bilit y , central go vernmen ts often fail to meet so ciet y’s electricity needs. In resp onse, neighborho o d diesel generators hav e emerged to fill this gap by forming informal, diesel-based microgrids that supply p ow er to nearby house- holds and institutions. With the increasing adoption of household-level pho- to voltaic (PV) systems, new questions arise regarding the optimal utilization of existing assets to form microgrids capable of deliv ering more affordable and renew able electricity . Addressing these questions requires not only a tec hno- economic p ersp ectiv e but also an examination of the market p o wer exercised b y incum b en t actors, specifically diesel generator companies (DGC), within en vironments c haracterized by limited regulatory ov ersight. In this pap er, tw o types of contributions are made: • In terms of energy system mo deling. W e prop ose a bi-lev el game theo- retical mo del represen ting a regulator with limited ov ersight at the first- lev el, and a monop olizing diesel generator company at the second. The regulator decides on the maxim um electricity price and minim um feed- in-tariff in the microgrid in order to maximize the household economic surplus (HES). The regulator promotes the in tegration of renewable energy in the microgrid through allo wing household PV-o wner feed-in and incentivizing DGC-owned PV and battery installation, while ac- coun ting for the economic interests of the DGC reflecting its mark et p o w er. The DGC controls access to the microgrid, and decides on capacit y additions, dispatc h, and the amoun t of fed-in capacity from household PV o wners. 34 • In terms of p olicy insights, the mo del is applied to the case of a real microgrid in a village in Lebanon, relying on lo cal data collected ov er six mon ths. The following main conclusions are reached: Under DGC market p ow er, microgrid-level price and feed-in-tariff caps are generally successful in considerably increasing the HES. The p olicies that maximize the HES alw ays en tail a large p ortion of household PV- o wner fed-in electricit y into the microgrid satisfying up to 39% of the demand in the base case. Increasing the DGC’s budget leads to substan tial gains in HES com- pared to the status quo. The HES impro vemen ts amounting to 18% at the base budget increase to 41% under a non-binding budget. When the household PV-owners’ p enetration in the microgrid exceeds 10%, considerable HES impro vemen ts compared to a diesel-based mi- crogrid are observ ed. The improv ements p eak at a p enetration level of around 90%, where the household PV-owners’ generation surplus al- lo ws for cheaper electricit y , and the demand serv ed b y the grid is still significan t. The renew able energy p enetration in the microgrid reaches 60% at the base case conditions compared to 0% in the status quo diesel-based microgrid. This p enetration rate approaches 100% for budgets starting 1 million USD or for PV-owning households constituting 75% of all households in the microgrid. The limited regulator’s control o ver unmet demand en tails measurable HES losses under tigh t budget conditions and low household PV-o wner 35 shares in the microgrid. A t the status-quo budget and PV-owner share, extending price-cap regulation to also control unmet demand leads to a 10% increase in HES. This study assumes a microgrid that op erates indep endently of the na- tional grid. F uture research could inv estigate cases of partial interconnection with the national grid, as w ell as the inclusi on of time-v arying electricity prices and feed-in tariffs o ver the planning horizon. A dditionally , consider- ing a dynamic PV ownership p enetration rate represen ts another in teresting extension. 7. Ac knowledgmen ts W e thank Prof. Georges Zaccour and Prof. Anne Neumann for providing commen ts and review. This publication is based on research supp orted by the T empleton W orld Charit y F oundation, Inc. (funder DOI 501100011730) under the gran t https://doi.org/10.54224/32645 . 36 References [1] Akram Almohamadi, Neil McCullo c h, Jo ev as Asare, and Moussa Saab. Impro ving electricit y services in yemen. IGC r e- p ort https://www. theigc. or g/sites/default/files/2021/11/Impr oving- ele ctricity-servic es-in-Y emen_Final-r ep ort. p df , 2021. [2] Siham Matallah, Khadidja Zerigui, and Amal Matallah. Renewable en- ergy solutions to the lack of access to electricity in conflict-ridden coun- tries: A case study of yemen. Ener gy , 296:131233, 2024. [3] In ternational Energy Agency. Ukraine’s energy security and the coming winter. https://www.iea.org/reports/ ukraines- energy- security- and- the- coming- winter , 2024. IEA, P aris. Licence: CC BY 4.0. [4] T etiana Kurbatov a, Roman Sidortsov, Galyna T ryp olsk a, Daniil Hulak, and Iryna Sotn yk. Maintaining ukraine’s grid reliabilit y under rapid gro wth of renew able electricity share: Challenges in the pre-war, war- time, and p ost-w ar p eriods. 2024. [5] Jie Liu, Hao qi Qian, Qian Zhang, Zhiy an Lin, and Pierluigi Siano. Cor- ruption induced energy inefficiencies: Evidence from c hina’s energy in- v estment pro jects. Ener gy Policy , 183:113825, 2023. [6] Laszlo Lov ei, Alastair McKechnie, et al. The Costs of Corruption for the Po or- The Ener gy Se ctor . W orld Bank, 2012. [7] Energy Sector Management Assistance Program (ESMAP). Off-grid solar mark et trends rep ort 2024: Outlook, 2024. 37 [8] Charles Lawrie and Camillo Stub en b erg. F riend or fo e? diesel generators and the global energy transition. Ener gy R ese ar ch & So cial Scienc e , 126:104124, 2025. [9] In ternational Energy Agency (IEA). T racking SDG7: The Energy Progress Rep ort, 2025, 2025. Licence: CC BY NC 3.0 IGO. [10] T emilade Sesan, Unico Uduk a, Lucy Baker, Ok ech ukwu Ugwu, Ewah Eleri, and Subhes Bhattac haryy a. Exploring the connections b et w een mini-grid mark et regulation and energy access expansion: The case of nigeria. Ener gy Policy , 184:113891, 2024. [11] Emília Inês Come Zebra, Henn y J v an der Windt, Geraldo Nh umaio, and André PC F aaij. A review of hybrid renewable energy systems in mini-grids for off-grid electrification in developing countries. R enewable and Sustainable Ener gy R eviews , 144:111036, 2021. [12] AFP. Lebanon in blac kout as p o wer stations run out of fuel. https://www.france24.com/en/live- news/ 20211009- lebanon- in- blackout- as- power- stations- run- out- of- fuel , 2021. Last accessed: 19 April 2022. [13] Ali Al-W ak eel. Lo cal energy systems in iraq: neighbourh oo d diesel gen- erators and solar photo v oltaic generation. In Micr o grids and L o c al En- er gy Systems . In techOpen, 2021. [14] Eric Akp o viroro Obar, Ab delw ahed T ouati, Oluw aseun Simon Adek anle, Benjamin Aga jelu, Laince Pierre Mouleb e, and Nabila Rabbah. Na vi- 38 gating the prev ailing c hallenges of the nigerian p ow er sector. WSEAS T r ansactions on Power Systems , 17(2):234–243, 2022. [15] Sebastian Zwickl-Bernhard, Anne F Neumann, Ma jd Olleik, and Ha ytham M Db ouk. Mark et organization in lo w-income countries’ mi- crogrids: Insigh ts from electricity demand elasticit y and game-theory optimization. case study: Lebanon. IEEE T r ansactions on Ener gy Mar- kets, Policy and R e gulation , 2025. [16] Dana Abi Ghanem. Energy , the city and ev eryda y life: Living with p o w er outages in p ost-w ar lebanon. Ener gy r ese ar ch & so cial scienc e , 36:36–43, 2018. [17] Sherv an Babamohammadi, Amy R Birss, Hamid P ouran, Jagro op Pand- hal, and T ohid N Borhani. Emission control and carb on capture from diesel generators and engines: A decade-long p ersp ectiv e. Carb on Cap- tur e Scienc e & T e chnolo gy , 14:100379, 2025. [18] Ali Q Al-Shet wi, MA Hannan, Ma jid A Ab dullah, MSA Rahman, Pin Jern Ker, Ammar A Alk ah tani, TM Indra Mahlia, and Kashem M Muttaqi. Utilization of renew able energy for p o w er sector in y emen: curren t status and p otential capabilities. IEEE A c c ess , 9:79278–79292, 2021. [19] Alwin Long, Mazlin Bin Mokh tar, Minhaz F arid Ahmed, and Chen Kim Lim. Enhancing sustainable dev elopment via lo w carb on energy transi- tion approac hes. Journal of Cle aner Pr o duction , 379:134678, 2022. 39 [20] Mohammad Amin V aziri Rad, Alibakhsh Kasaeian, Xiaofeng Niu, Kai Zhang, and Omid Mahian. Excess electricity problem in off-grid hybrid renew able energy systems: A comprehensive review from challenges to prev alent solutions. R enewable Ener gy , 212:538–560, 2023. [21] Ida F uchs, Sergio Balderrama, Sylv ain Quoilin, P edro Cresp o del Granado, and Ja y aprak ash Ra jasekharan. Swarm electrification: Har- nessing surplus energy in off-grid solar home systems for univ ersal elec- tricit y access. Ener gy for Sustainable Development , 77:101342, 2023. [22] Stev e Sheridan, Keith Sunderland, and Jane Courtney . Sw arm electri- fication: A comprehensive literature review. R enewable and sustainable ener gy r eviews , 175:113157, 2023. [23] Jak ob Sv olba, Sebastian Zwickl-Bernhard, Ma jd Olleik, and Elsa Bou Gebrael. Renewable integration in fossil-fuel-p o w ered microgrids under mark et p o w er conditions. In 2025 21st International Confer enc e on the Eur op e an Ener gy Market (EEM) , pages 1–5. IEEE, 2025. [24] Ali Ahmad, Neil McCullo c h, Muzna Al-Masri, and Marc A youb. F rom dysfunctional to functional corruption: the p olitics of decentralized electricit y provision in lebanon. Ener gy R ese ar ch & So cial Scienc e , 86:102399, 2022. [25] Hannes Kirchhoff, Noara Kebir, Kirsten Neumann, P eter W Heller, and Kai Strunz. Dev eloping m utual success factors and their application to sw arm electrification: microgrids with 100% renew able energies in the 40 global south and germany . Journal of Cle aner Pr o duction , 128:190–200, 2016. [26] Hannes Kirchhoff and Kai Strunz. Key driv ers for successful dev elop- men t of p eer-to-peer microgrids for sw arm electrification. Applie d En- er gy , 244:46–62, 2019. [27] Munish Manas, Shivi Sharma, K Shashidhar Reddy , and Abhina v Sriv as- ta v a. A critical review on tec hno-economic analysis of h ybrid renew able energy resources-based microgrids. Journal of Engine ering and Applie d Scienc e , 70(1):148, 2023. [28] Giulio Prevedello and Annette W erth. The b enefits of sharing in off-grid microgrids: A case study in the philippines. Applie d Ener gy , 303:117605, 2021. [29] Martha M Hoffmann and Da wud Ansari. Simulating the potential of sw arm grids for pre-electrified communities–a case study from y emen. R enewable and Sustainable Ener gy R eviews , 108:289–302, 2019. [30] Lea Bertram, Ida F uchs, Victor Ban uls Ramirez, P edro Cresp o del Granado, and Sergio Balderrama. Lo cal electricit y market designs for in terconnected nanogrids: Impact on rural electrification in madagascar. Journal of Cle aner Pr o duction , 449:141786, 2024. [31] Xi Luo, W ence Shi, Y usen Jiang, Y anfeng Liu, and Jinw en Xia. Dis- tributed p eer-to-peer energy trading based on game theory in a commu- nit y microgrid considering ownership complexity of distributed energy resources. Journal of Cle aner Pr o duction , 351:131573, 2022. 41 [32] Edstan F ernandez, MJ Hossain, Khizir Mahm ud, Mohammad Sohrab Hasan Nizami, and Muhammad Kashif. A bi-level optimization- based communit y energy management system for optimal energy sharing and trading among p eers. Journal of Cle aner Pr o duction , 279:123254, 2021. [33] Jussi V alta, Saku Mäkinen, Kirsi K otilainen, P ertti Järven tausta, and Gonçalo Mendes. Comparison of regulatory challenges faced by different microgrid o wnership mo dels. In 2018 IEEE PES Innovative Smart Grid T e chnolo gies Confer enc e Eur op e (ISGT-Eur op e) , pages 1–9. IEEE, 2018. [34] Subhes C Bhattacharyy a. Mini-grids for the base of the pyramid market: A critical review. Ener gies , 11(4):813, 2018. [35] Gob ong Choi, Sung-Y o on Huh, Eunn yeong Heo, and Chul-Y ong Lee. Prices versus quantities: Comparing economic efficiency of feed-in tar- iff and renew able p ortfolio standard in promoting renewable electricity generation. Ener gy Policy , 113:239–248, 2018. [36] Hanaa F eleafel and Mic hel Leseure. F eed-in tariff p olicy for microgrids: past and future a case study of the option v alue of complementary energy assets in turbulent conditions. International Journal of Ener gy Se ctor Management , 2025. [37] Md Ahasan Habib and MJ Hossain. Smart grid, smart fit: A data- driv en approac h to optimize microgrid energy market. Ener gy Policy , 203:114618, 2025. 42 [38] Hari Kumar Sub eri, Muhammad Asif, and T alha Bin Nadeem. Rooftop solar pv in bh utan: A systemic analysis of feed-in-tariff program. Ener gy for Sustainable Development , 83:101591, 2024. [39] Subhes C Bhattacharyy a. T o regulate or not to regulate off-grid elec- tricit y access in developing coun tries. Ener gy p olicy , 63:494–503, 2013. [40] Magda Moner-Girona, R Ghanadan, M Solano-P eralta, I Kougias, K Bódis, T Huld, and S Szab ó. A daptation of feed-in tariff for remote mini-grids: T anzania as an illustrative case. R enewable and Sustainable Ener gy R eviews , 53:306–318, 2016. [41] Ma jd Olleik, Amir Boushahine, and Kareem Abou Jalad. Planning h ybrid renewable energy systems under uncertain grid in terconnection conditions. Sustainable Ener gy, Grids and Networks , page 102131, 2026. [42] Carlos Villa and F elip e Henao. Oversizing grid-connected microgrids as a business mo del—an optimisation assessment approac h. Ener gy R ep orts , 8:2100–2118, 2022. [43] Hamid Karimi and Ehsan Heydarian-F orushani. A multi-ob jective bi- lev el framework to mo del distribution system op erator’s b eha vior in the wholesale and lo cal transactive mark ets. Ener gy , 312:133695, 2024. [44] Rogier Hans W uijts, Marjan v an den Akker, and Mac hteld v an den Bro ek. Effect of mo delling choices in the unit commitment problem. Ener gy Systems , 15(1):1–63, 2024. [45] Leila Dagher, Sara Diab, Razan Zw ein, et al. F rom crisis to opp ortunity: A dv ancing solar energy in lebanon through effectiv e p olicymaking. 2025. 43 [46] Mohamad El Ha jj, Hadi Ab ou Moussa, and Grace Mikhael. Solar pho- to voltaic status rep ort for lebanon 2023. T echnical report, Lebanese Cen ter for Energy Conserv ation (LCEC), Beirut, Lebanon, April 2025. Sev enth edition. [47] Ha ytham Db ouk, Ma jd Olleik, and Els a Bou Gebrael. Elec- tricit y demand data in a lebanese village: Households, in- dustrial sites and a municipalit y . [Data set], 2025. doi: h ttps://doi.org/10.5281/zeno do.15514295. [48] Stefan Pfenninger and Iain Staffell. Long-term patterns of europ ean p v output using 30 y ears of v alidated hourly reanalysis and satellite data. Ener gy , 114:1251–1265, 2016. [49] Iain Staffell and Stefan Pfenninger. Using bias-corrected reanalysis to sim ulate curren t and future wind p ow er output. Ener gy , 114:1224–1239, 2016. [50] An thony Afful-Dadzie, Eric Afful-Dadzie, Iddrisu A wudu, and Joseph K waku Banuro. P ow er generation capacit y planning under bud- get constrain t in developing countries. Applie d ener gy , 188:71–82, 2017. 44 App endices A. Nomenclature The parameters and sets used in the description of the bi-lev el framew ork are sho wn in this app endix. 45 Symbol Input name Unit 𝛼 Minimum state of charge 𝛽 Binary input allowing the use of renewable sources binary 𝛾 𝐷 𝐺𝐶 Discount rate of the DGC 𝛾 𝑅 𝐸 Discount rate of the regulatory entit y 𝜀 Unit p enalt y of unmet demand USD/k Wh 𝜂 Charging and discharging efficiency 𝜃 𝑃 𝑉 𝑖 A verage PV capacity of households k W 𝜅 𝑔 Initial installed DGC capacity k W 𝜆 𝐶 𝑔 , 𝑦 Unit capital cost of technology USD/k W 𝜆 𝑂 𝐹 𝑔 Unit fixed op eration cost of technology USD/k W/ 𝑦 𝜆 𝑂𝑉 𝑔 Unit v ariable op eration cost of technology USD/k Wh 𝜇 𝑖 , 𝑦 , 𝑑 , ℎ T otal electricity demand of households k W 𝜈 𝑔 Lifetime of technology years 𝜈 0 𝑔 Remaining lifetime of installed technology at year 0 years 𝜉 V alue of lost load USD/k Wh Π Budget av ailable for installing new capacity USD 𝜋 Price of diesel USD/L 𝜌 𝑗 Heat rate of the diesel generator on the 𝑗 𝑡 ℎ portion of the heat rate curv e L/k Wh 𝜎 𝑖 , 𝑦 , 𝑑 , ℎ Surplus electricity p er household ∗ k W 𝜏 𝑦 Maximum electricity price set by the ministry USD/k Wh 𝜐 Minimum required level of satisfied demand % Υ Planning horizon 𝜙 𝑦 , 𝑑 , ℎ Capacity factor of PV Ω 𝑖 , 𝑦 T otal num b er of households av ailable to the microgrid 𝜔 𝑑 W eigh t of representativ e day 𝑑 T able 5: parameters ∗ A negative surplus indicates a net demand 46 Sym b ol Set name D Represen tative p eriods (days) within a year ( { 0 , 1 , 2 } ) G All technologies (diesel generator 𝐷 𝐺 , DGC photo voltaic cells 𝑃𝑉 , DGC batteries 𝐵 ) G 𝑔 Non-storage technologies ( 𝐷 𝐺 , 𝑃𝑉 ) H Hours in a representativ e p erio d ( { 0 , . . ., 23 } ) I Household types (Non-PV-owner household, PV-owner household) R Renewable sources ( 𝑃𝑉 , fed-in PV-owner surplus 𝐹 𝑖 ) S All decision v ariables Y Y ears ( { 0 , . . ., 14 } ) T able 6: Derived sets B. Second-level constrain ts B.1. Cap acity and r etir ement of te chnolo gy c onstr aints Equation 19 keeps track of the installed capacit y of the technologies 𝐶 𝑔 , 𝑦 , as a function of the previously installed 𝐶 𝑔 , 𝑦 − 1 , the added 𝐴 𝑔 , 𝑦 , and the retired capacities 𝑅 𝑒𝑡 𝑔 , 𝑦 , where the initial capacities are detailed in 20 . 𝐶 𝑔 , 𝑦 = 𝐶 𝑔 , 𝑦 − 1 + 𝐴 𝑔 , 𝑦 − 𝑅 𝑒 𝑡 𝑔 , 𝑦 ∀ 𝑔, 𝑦 − 1 ≥ 0 , (19) 𝐶 𝑔 , 0 = 𝜅 𝑔 + 𝐴 𝑔 , 0 − 𝑅 𝑒 𝑡 𝑔 , 0 ∀ 𝑔 (20) The retirement of the DGC’s capacities is mo deled in Equations 21 through 23 . When the remaining life of the initial capacity ends, the retired capacit y is equal to the initial capacity ( 21 ). Before that, no capacit y gets retired ( 22 ). After the initial capacity is retired, the capacit y of tec hnology 𝑔 added 47 𝜈 𝑐 𝑔 y ears ago (i.e.: a lifetime ago) is retired ( 23 ). 𝑅 𝑒 𝑡 𝑔 , 𝑦 = 𝜅 𝑔 ∀ 𝑔, 𝑦 = 𝜈 0 𝑔 (21) 𝑅 𝑒 𝑡 𝑔 , 𝑦 = 0 ∀ 𝑔 , 𝑦 < 𝜈 0 𝑔 (22) 𝑅 𝑒 𝑡 𝑔 , 𝑦 = 𝐴 𝑔 , 𝑦 − 𝜈 𝑔 ∀ , 𝑦 > 𝜈 𝑔 (23) B.2. Disp atch c onstr aints Equations 24 through 27 ensure that the dispatch 𝐷 𝑔 , 𝑦, 𝑑 , ℎ from neither the diesel generator, the DGC’s PV nor the batteries charging or disc harging exceed their resp ectiv e capacities. 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ ≤ 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 ∀ 𝑦 , 𝑑 , ℎ (24) 𝐷 𝑔 = 𝑃𝑉 , 𝑦, 𝑑 , ℎ ≤ 𝜙 𝑦 ,𝑑 , ℎ × 𝐶 𝑔 = 𝑃𝑉 ∀ 𝑦 , 𝑑 , ℎ (25) 𝐵 − 𝑦 ,𝑑 , ℎ ≤ 𝐶 𝑔 = 𝐵 , 𝑦 ∀ 𝑦 , 𝑑 , ℎ (26) 𝐵 + 𝑦 ,𝑑 , ℎ ≤ 𝐶 𝑔 = 𝐵 , 𝑦 ∀ 𝑦 , 𝑑 , ℎ (27) B.3. Stor age te chnolo gy c onstr aints Equation 29 follows the hourly state of charge of the batteries as a func- tion of the previous state of charge, c harge, discharge, and efficiency 𝜂 , while the initial condition on the state of c harge is set in 30 . Equation 31 ensures that the state of c harge nev er drops b elo w a given minim um (as a p ercentage of the energy capacit y), and do esn’t exceed the energy capacit y . 𝑆 𝑜 𝐶 𝑦 ,𝑑 , ℎ = 𝑆 𝑜𝐶 𝑦 ,𝑑 , ℎ − 1 + 𝜂 × 𝐵 + 𝑦 ,𝑑 , ℎ − 𝐵 − 𝑦 ,𝑑 , ℎ / 𝜂 ∀ 𝑦 , 𝑑 , ℎ − 1 ≥ 0 , (28) 𝑆 𝑜 𝐶 𝑦 ,𝑑 , 0 = 𝑆 𝑜𝐶 0 𝑦 ,𝑑 + 𝜂 × 𝐵 + 𝑦 ,𝑑 , 0 − 𝐵 − 𝑦 ,𝑑 , 0 / 𝜂 ∀ 𝑦 , 𝑑 , ℎ − 1 ≥ 0 , (29) 𝑆 𝑜 𝐶 0 𝑦 ,𝑑 = 𝑆 𝑜𝐶 𝑦 ,𝑑 , 23 + 𝜂 × 𝐵 + 𝑦 ,𝑑 , 23 − 𝐵 − 𝑦 ,𝑑 , 23 / 𝜂 ∀ 𝑦 , 𝑑 (30) 𝛼 × 4 ℎ × 𝐶 𝑔 = 𝐵 , 𝑦 ≤ 𝑆 𝑜𝐶 𝑦 ,𝑑 , ℎ ≤ 4 ℎ × 𝐶 𝑔 = 𝐵 , 𝑦 ∀ 𝑦 , 𝑑 , ℎ (31) 48 C. Pro of of Lemma 1 Assuming 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ , 𝑃, 𝐹 𝑖 𝑇 ) is the NPV for any fixed set of v ariables S ′ = S − { 𝑃, 𝐹 𝑖 𝑇 } , price 𝑃 and feed-in tariff 𝐹 𝑖𝑇 , and 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗ , 𝑃, 𝐹 𝑖 𝑇 ) the NPV for the optimal set of v ariables S ′ ∗ = S ∗ − { 𝑃, 𝐹 𝑖𝑇 } , price 𝑃 and feed-in tariff 𝐹 𝑖𝑇 . With c hanges of price and feed-in tariff having no effect on the mo del’s fea- sibilit y and parameters, i.e., 𝑃 and 𝐹 𝑖𝑇 are indep enden t of all constraints, and no input is a function of 𝑃 and 𝐹 𝑖𝑇 , we first pro v e that the NPV is non-decreasing in 𝑃 . 𝜕 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 𝜕 𝑃 =  𝑦  𝑑 𝜔 𝑑  ℎ ©  « 𝐵 − 𝑦 ,𝑑 , ℎ − 𝐵 + 𝑦 ,𝑑 , ℎ +  𝑔 ∈ G 𝑔 𝐷 𝑔 , 𝑦, 𝑑 , ℎ +  𝑖 ∈I 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ ª ® ¬  1 ( 1 + 𝛾 𝐷 𝐺 𝐶 ) 𝑦  (C.1) Recalling Equation 13 , where ∀ 𝑦 , 𝑑 , ℎ : 𝑈 𝑦 ,𝑑 , ℎ + 𝐵 − 𝑦 ,𝑑 , ℎ +  𝑔 ∈ G 𝑔 𝐷 𝑔 , 𝑦, 𝑑 , ℎ +  𝑖 ∈I 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ = 𝐵 + 𝑦 ,𝑑 , ℎ −  𝑖 ∈I min  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  Moreo ver, equation 14 states that that ∀ 𝑦 , 𝑑 , ℎ 𝑈 𝑦 ,𝑑 , ℎ ≤  𝑖 ∈I − min  0 , Ω 𝑖 , 𝑦 × 𝜎 𝑖 , 𝑦, 𝑑 , ℎ  (C.2) 49 Joining Equations 13 and 14 implies 𝐵 − 𝑦 ,𝑑 , ℎ +  𝑔 ∈ G 𝑔 𝐷 𝑔 , 𝑦, 𝑑 , ℎ +  𝑖 ∈I 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ ≥ 𝐵 + 𝑦 ,𝑑 , ℎ ∀ 𝑦 , 𝑑 , ℎ (C.3) ⇔  𝑑 𝜔 𝑑  ℎ ©  « 𝐵 − 𝑦 ,𝑑 , ℎ − 𝐵 + 𝑦 ,𝑑 , ℎ +  𝑔 ∈ G 𝑔 𝐷 𝑔 , 𝑦, 𝑑 , ℎ +  𝑖 ∈I 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ ª ® ¬ ≥ 0 ∀ 𝑦 (C.4) ⇔ 𝜕 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 𝜕 𝑃 ≥ 0 (C.5) ⇔ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ , 𝑃 + 𝜖 , 𝐹 𝑖𝑇 ) ≥ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ , 𝑃, 𝐹 𝑖 𝑇 ) ∀ 𝑃 , 𝐹 𝑖𝑇 , 𝜖 ≥ 0 (C.6) ∴ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ , 𝑃, 𝐹 𝑖 𝑇 ) is non-decreasing in 𝑃 , implying that: 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ ∗ , 𝑃 + 𝜖 , 𝐹 𝑖𝑇 ) ≥ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗ , 𝑃, 𝐹 𝑖 𝑡 ) ∀ 𝑃, 𝐹 𝑖𝑇 , 𝜖 ≥ 0 and, with 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗∗ , 𝑃 + 𝜖 , 𝐹 𝑖𝑇 ) the maximum NPV, for a re-optimized S ′ ∗∗ , price 𝑃 + 𝜖 , and feed-in tariff 𝐹 𝑖𝑇 , 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗∗ , 𝑃 + 𝜖 , 𝐹 𝑖𝑇 ) ≥ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ ∗ , 𝑃 + 𝜖 , 𝐹 𝑖𝑇 ) ∀ 𝑃, 𝐹 𝑖𝑇 , 𝜖 ≥ 0 ⇔ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗∗ , 𝑃 + 𝜖 , 𝐹 𝑖𝑇 ) ≥ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗ , 𝑃, 𝐹 𝑖 𝑇 ) ∀ 𝑃 , 𝐹 𝑖𝑇 , 𝜖 ≥ 0 Then, w e prov e that 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 is non-increasing in 𝐹 𝑖𝑇 . ∀ 𝑃 , 𝐹 𝑖𝑇 : 𝜕 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 𝜕 𝐹 𝑖𝑇 =  𝑦  𝑑 𝜔 𝑑  ℎ −  𝑖 𝐹 𝑖 𝑖 , 𝑦, 𝑑 , ℎ !  1 ( 1 + 𝛾 𝐷 𝐺 𝐶 ) 𝑦  (C.8) ⇔ 𝜕 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 𝜕 𝐹 𝑖𝑇 ≤ 0 (C.9) ⇔ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( 𝑃, 𝐹 𝑖𝑇 + 𝜖 ) ≤ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( 𝑃, 𝐹 𝑖𝑇 ) ∀ 𝜖 ≥ 0 (C.10) 50 ∴ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ , 𝑃, 𝐹 𝑖 𝑇 ) is non-increasing in 𝐹 𝑖𝑇 , implying that: 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗ , 𝑃, 𝐹 𝑖 𝑇 + 𝜖 ) ≤ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ ∗ , 𝑃, 𝐹 𝑖 𝑇 ) ∀ 𝑃 , 𝐹 𝑖𝑇 , 𝜖 ≥ 0 and, with 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗∗ , 𝑃, 𝐹 𝑖 𝑇 + 𝜖 ) the maximum NPV, for a re-optimized S ′ ∗∗ , price 𝑃 , and feed-in tariff 𝐹 𝑖𝑇 + 𝜖 , 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ( S ′ ∗ , 𝑃, 𝐹 𝑖 𝑇 + 𝜖 ) ≤ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗∗ , 𝑃, 𝐹 𝑖 𝑇 + 𝜖 ) ∀ 𝑃, 𝐹 𝑖𝑇 , 𝜖 ≥ 0 ⇔ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗ , 𝑃, 𝐹 𝑖 𝑇 + 𝜖 ) ≤ 𝑁 𝑃𝑉 𝐷 𝐺 𝐶 ∗ ( S ′ ∗∗ , 𝑃, 𝐹 𝑖 𝑇 ) ∀ 𝑃 , 𝐹 𝑖𝑇 , 𝜖 ≥ 0 □ 51 D. V ariable heat rate of diesel generator As first denoted in constrain ts HR.1 through HR.3 mo del the v ariable heat rate of the diesel generator. Linearly , they translates to: 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ ≤ 0 . 30 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 + ( 1 − 𝑏 1 , 𝑦 ,𝑑 , ℎ ) 𝑀 𝑅 𝑦 ,𝑑 , ℎ ≥ 𝜌 1 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ − ( 1 − 𝑏 1 , 𝑦 ,𝑑 , ℎ ) 𝑀 𝑅 𝑦 ,𝑑 , ℎ ≤ 𝜌 1 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ + ( 1 − 𝑏 1 , 𝑦 ,𝑑 , ℎ ) 𝑀 (D.1) 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ ≥ 0 . 30 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 + ( 1 − 𝑏 2 , 𝑦 ,𝑑 , ℎ ) 𝑀 + 𝜖 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ ≤ 0 . 60 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 + ( 1 − 𝑏 2 , 𝑦 ,𝑑 , ℎ ) 𝑀 𝑅 𝑦 ,𝑑 , ℎ ≥ 𝜌 2 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ − ( 1 − 𝑏 2 , 𝑦 ,𝑑 , ℎ ) 𝑀 𝑅 𝑦 ,𝑑 , ℎ ≤ 𝜌 2 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ + ( 1 − 𝑏 2 , 𝑦 ,𝑑 , ℎ ) 𝑀 (D.2) 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ ≥ 0 . 60 × 𝐶 𝑔 = 𝐷 𝐺 , 𝑦 + ( 𝑏 1 , 𝑦 ,𝑑 , ℎ + 𝑏 2 , 𝑦 ,𝑑 , ℎ ) 𝑀 + 𝜖 𝑅 𝑦 ,𝑑 , ℎ ≥ 𝜌 3 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ − ( 𝑏 1 , 𝑦 ,𝑑 , ℎ + 𝑏 2 , 𝑦 ,𝑑 , ℎ ) 𝑀 𝑅 𝑦 ,𝑑 , ℎ ≤ 𝜌 3 × 𝐷 𝑔 = 𝐷 𝐺 , 𝑦, 𝑑 , ℎ + ( 𝑏 1 , 𝑦 ,𝑑 , ℎ + 𝑏 2 , 𝑦 ,𝑑 , ℎ ) 𝑀 (D.3) where 𝑀 is a large enough num b er, and 𝜖 is a small enough num b er. In other w ords, we mo del a set of constrain ts D.1 , D.2 and D.3 for each of cases 1, 2 and 3 resp ectiv ely , such that only one p ortion of the heat rate curv ed is enforced at each time step. 52 E. Case study parameters The costs of the tec hnologies used in the case study are summarized in T able 7 , noting that the capital expenditure (Capex) of solar PV and batteries decrease along the planning horizon. The op erating exp enditures (Op ex) remain constant. P arameter V alue Unit PV Cap ex in year 0 964 USD/k W PV fixed Op ex 22 USD/k W/y ear PV v ariable Op ex 0 USD/k Wh Diesel generator Cap ex in year 0 800 USD/k W Diesel generator fixed Op ex 84 USD/k W/y ear Diesel generator non-fuel v ariable Op ex 0.014 USD/k Wh Diesel generator fuel-related v ariable Op ex 0.21 USD/k Wh Batteries Cap ex in year 0 335 USD/k W Batteries fixed Op ex 61 USD/k W/y ear Batteries v ariable Op ex 0.0006 USD/k Wh T able 7: Case study parameters 53