This paper presents a methodology to optimally design a multi-energy microgrid with thermal and electric loads considering \(N-1\) and probabilistic regulation reserves. This methodology consists of a chance-constrained optimization that determines the optimal sizing of the microgrid. Microgrid operations are rigorously considered by modelling hourly thermal and electric demand patterns as well as technology production schedules over a year. Such schedules include both electric and thermal power balances, ramp constraints, \(N-1\) and regulation reserves, among others. To ensure a reliable microgrid design and operation, reserve constraints have been proposed to deal with both \(N-1\) generation contingencies, and forecast errors. \(N-1\) reserves guarantee that a sudden outage of any of the electric generators is strategically covered by the remaining generators in order to avoid load shedding. Additionally, nonzero-mean random forecast errors of electric load and solar production are addressed by a set of chance constraints able to schedule asymmetric up and down regulation reserves. Their levels are high enough to cover hourly random forecast errors (or intermittencies) with a high threshold probability. The proposed methodology results in a mixed integer second-order cone program. Results of microgrid designs with and without reliability reserves are carefully analysed and compared. Neglecting reliability constraints leads to a lower-cost design at the expense of exposing the microgrid to unsafe operation. Finally, sensitivity analysis to study the optimal portfolio sizing with respect to electric BESS investment cost, solar production forecast mean error, and random intermittency threshold probability are performed.
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Abkürzungen
\(\mathcal {G}\)
Fuel-fired conventional generator: ICE, MT and DSL. Subscript g.
All the available technologies in the MG \(\left( \mathcal {J}\cup \mathcal {S}\cup N\right)\): PV, ICE, DSL, MT, EC, AC, TESS and BESS. Subscript k
\(\mathcal {N}\)
Thermal energy technologies: Electric Chiller (EC), Absorption Chiller (AC). Subscript n
\(\mathcal {S}\)
Energy Storage Systems: Battery (BESS), Thermal Storage (TESS). Subscript s
\(\mathcal {T}\)
Time-step Set. Subscript t
Y
Lifespan for microgrid. Subscript y
1 Introduction
Renewable energies have become essential sources of electricity when it comes to the construction of microgrids around the world [1]. Several factors like increasing fuel prices of conventional generation, decreasing installation costs of renewable energy [2], and environmental concerns have promoted their use. Even with such drivers, in order to support microgrid deployment, several challenges such as optimal sizing and operation, optimal control strategies, and reliability supply, need to be addressed [3].
1.1 Motivation
Although the microgrid planning should consider estimates of both energy resources availability and demand, it is determined by the size of its elements (generators and storage systems). The sizing process, in turn, relies on the adopted operational strategy of the microgrid and data assumptions. Operational aspects are considered during the microgrid sizing phase to account for the operational conditions that will be employed during operation. Although the exact operational condition is not perfectly known during the sizing phase, considering operational constraints ensures that the microgrid is properly sized and thus the operational strategies can be feasible. In general, determining the “appropriate” size of components for a given set of electrical and/or thermal load profiles, remains a challenge. If components are oversized, the system capital cost and energy curtailment will be unnecessarily high. Also, generator may reach lower efficiency and higher maintenance costs which may lead to low generator reliability as fuel-fired diesel generators can develop “wet-stacking” at low power output w.r.t to rated capacity [4, 5]. If components are undersized continuous load shedding, unsatisfied demand and high non-renewable energy generation may occur leading to unsuccessful operation [6]. Thus, in order to guarantee a continuous and reliable energy supply to the users, an optimal sizing methodology requires a rigorous modeling of operational aspects in short time frames. Sudden outages of technologies supplying energy to the microgrid as well as intermittence and uncertainties in both solar photovoltaic energy production and demands are essential elements to consider to guarantee a reliable and economic microgrid performance.
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In this paper, a Mixed Integer Second-Order Cone Programming (MISOCP) model for jointly optimizing the sizing and operation of an islanded multi-energy microgrid is presented. The type of microgrids subject to study in this work is one that is not connected to the power grid and considers both thermal and electric loads on an hourly basis. The microgrid is designed considering a portfolio of distributed energy resources to deliver thermal energy and electricity to a set of thermal and electric loads commonly observed in hotel facilities. This portfolio includes conventional generation resources as internal combustion engines, microturbines and diesel-fueled generator, renewable generation resources as solar PV modules, energy storage systems such as batteries and thermal storage, and cooling supplying technologies such as both absorption and electric chillers. Additionally, in order to ensure a reliability-based microgrid design, the \(N-1\) security criterion has also been included in the proposed optimization model. Finally, with the aim of guaranteeing continuous supply–demand balance, even during random fluctuations of demand and/or intermittent generation, a set of probabilistic chance constraints are imposed to schedule an appropriate level of hourly regulation reserves.
1.2 Literature review
The reliability-based optimal sizing of multi-energy microgrids is investigated in this paper. Although scientific literature related to microgrid planning has traditionally focused on its operation and sizing, reliability criteria are not usually contemplated. In [7] authors present an optimal islanded microgrid planning with BESS. The BESS is scheduled as reserve provider. The BESS, defined by both, energy capacity and rated power, is defined as the maximum difference between pre-defined regulation reserves and power margin of committed units. Notwithstanding, reserve requirements are not estimated as a function of microgrid size, but they are considered as an input model parameter. In [8], authors present a computational framework to evaluate both economic and resilience in microgrid planning without considering reserve estimation. This work is extended in [9] to evaluate cooling and heating load, along with CHP systems, notwithstanding, reserves are not scheduled. Authors in [10] present a model to optimally size an electric microgrid using a multiyear approach. The BESS degradation is considered as a function of the hourly power-to-energy ratio. Detailed multi-year simulations spanning the entire lifetime of the project at an hourly basis are considered and the model is solved using an iterative MILP-solving strategy. Regulation reserves are computed as a portion of the load and expected renewable production to account for the uncertainties. However, neither cooling loads nor \(N-1\) reserves are modeled. In [11] a stochastic optimisation model for isolated multi-energy systems sizing is presented. Hydrogen and heat loads are considered. A tractable representation of a fully linear AC power flow equations were presented in the planning model. Authors do not model \(N-1\) and regulation reserves requirements. Authors of [12] presented a two-step Mixed-Integer Linear Programming (MILP) approach to determine the optimal sizing of a rural islanded microgrid in which four representative weeks are employed to describe one operating year; neither thermal demand nor reliability constraints were considered. In the work [13], a hybrid genetic algorithm-MILP approach is employed to find the optimal sizing of components of an islanded microgrid that delivers power to a hotel. The microgrid sizing is determined by the genetic algorithm; whereas the operational problem, that schedules technology production, is handled by a MILP model. The work [14] considers thermal and hydrogen loads. In order to reduce computational burden, twelve representative days (out of 365 days per year) are considered; \(N-1\) reserves are out of the scope of this work. In [15, 16] and [17] authors aim to find the optimal size of an islanded microgrid operating. Uncertainties in electric and thermal demand are neglected. The sizing problem is tackled by a grid-search heuristic algorithm; and the operational problem is solved employing a MILP model over a 12-hour control horizon. These two problems are solved independently, i.e., there is no joint optimization between the sizing and operation problems, and operational reliability aspects are not addressed. In [18] a MILP model is presented to design either grid-tied or islanded microgrids. Thermal and electrical loads are assumed as known. The sizing and operating strategies are based on minimization of capital, operation, and maintenance costs of the selected technologies. This work has considered the existence of a thermal and an electric network; nevertheless, power reserves to cover component outages have not been considered. Reference [19] presents a MILP model to jointly optimize sizing and operation of an islanded electric microgrid via minimization of net present cost; Operational reliability aspects have not studied.
In terms of microgrid operations, several investigations have employed chance-constraints to define optimal regulation reserve levels. These reserves allow the microgrid to have enough available energy to deal with random generation-demand balance fluctuations due to forecast errors or intermittency in renewable generation. For instance, the authors of [20] present a probabilistic approach to estimate regulation reserves based on the forecast error distribution of the PV and WT system outputs and demand. Reserves are estimated to deal with random events as generation intermittency, forecasting errors, and generation outages, while minimizing the expected energy not supplied of the microgrid. This study assumes the microgrid is already designed. In [21] authors present the concept of probability of successful islanding as the probability to cover a sudden islanding with microgrid scheduled reserves. Chance constraints based on the three-sigma rule impose conditions to estimate an amount of reserves that cover at least 99% of possible forecast deviations. A normal probability distribution of forecast errors of wind, PV and load is modeled; however, the optimization model does not find optimal microgrid sizing, but it does optimize the operation without considering the \(N-1\) security criterion. In [22], an operational scheduling model using probabilistic chance constraints is proposed to cover the uncertainty of demand and renewable resources. A method for allocating reserves to a microgrid is presented in order to deal with prediction errors and potential failures in both distributed generation and battery systems, maintaining power balance with a high probability depending on the parameter \(\text {L}^{\text {req}}\). Estimated reserves are allocated to avoid load shedding when events such as materialization of forecast errors, BESS outages or unexpected islanding occur. Given that microgrid sizing is not considered, the availability of reserves is fixed. This in turn makes difficult to maintain power balance with high probabilities. In contrast, our work focuses on properly sizing the microgrid as a function of the probability of maintaining power balance.
The work presented in [23] is similar to the work [22] in the sense that it assumes that the microgrid has already been sized. This constitutes the first key difference w.r.t our work. Second, the work [23] does not allocate reserves to deal with potential generator or storage system outages. And third, authors [23] did not take into account the amount of energy available in the storage system when defining reserves. This could lead the storage system being allocated with upward (downward) reserves that could not be implemented if its current state of charge is low (high).
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A variety of investigations on microgrid sizing guaranteeing \(N-1\) security criteria have also been reported. Generator outages are modeled to avoid power not served during post-contingency states. Remaining online generators are forced to adjust their output to support the power of the faulted generator. This can be done only if reserves have been properly scheduled. Authors in [24] developed a mixed-integer quadratically constrained quadratic programming to jointly optimize sizing and operation of an isolated microgrid introducing \(N-1\) security constraints; however, they do not schedule regulation reserves to handle renewable intermittencies and demand uncertainties. The work [25] is an extension of the work from [18] and their results ensure a secure optimal microgrid design. The generator contingencies modeled in [24] and [25] consider the loss of a single generating unit within a power plant. But, in order to consider a more extreme situation and thus achieve a robust operation, this work has modeled the outage of the entire generation plant. Also, none of these investigations have addressed the challenge of random fluctuations in demand and renewable generation in short time frame during microgrid operations.
Table 1 shows a summary of the discussed approaches in the scientific literature in the microgrids field. The table focuses on highlighting the key differences among them and displays the research gaps addressed in this manuscript.
Table 1
Summary of the most relevant formulation in the literature
To the best of our understanding, the existing literature does not provide a comprehensive optimization model for both sizing and operating a multi-energy microgrid that, at the same time, addresses the reserves provisions against both (demand and solar PV) forecast deviations and sudden outage of any generation resource in isolated microgrids. To fill these research gaps, this paper presents a methodology to optimally design a hybrid multi-energy microgrid (with thermal and electric loads) that considers both \(N-1\) and regulation reserves and minimizes the net present CAPEX and OPEX. The \(N-1\) criterion, traditionally employed to guarantee that demand is always met, even under the outage of any generation resource, is modeled to determine enough reserves to avoid load shedding under sudden generator outages. Up and down regulation reserves are estimated using chance constraints to cover joint random fluctuations of demand and solar PV with respect to their forecasted values.
The main contributions from this work are as follows:
The proposal of a MISOCP model to jointly optimize reliability-based design, sizing and operation of an islanded multi-energy microgrid with both thermal and electric load.
The design and inclusion of a set of constraints to ensure a reliable thermal-electric microgrid operation by optimally combining two types of reserves. One type of reserves is in charge of maintaining continuous enough energy supply against \(N-1\) electric generator contingencies. And the regulation reserves are computed by employing a set of hourly probabilistic chance constraints to guarantee enough asymmetric power and energy reserves to deal with nonzero-mean random forecast errors of both demand and solar PV generation. Positive and negative errors are rigorously differentiated to schedule asymmetric up and down regulation reserves respectively.
The remainder of this work is organized as follows: Sect. 2 presents the mathematical proposed model for reliability-constrained design and operation of microgrid. Section 3 presents the case study and its results are discussed in Sect. 4. Finally, Sect. 5 presents conclusions for this paper.
2 Proposed model
In this section we introduce the proposed mathematical model for selecting, sizing and operating the microgrid architecture displayed in Fig. 1, which is supposed to satisfy both electricity and thermal demands. To achieve this, we have considered a generic architecture in which relations between thermal and electrical systems is also modeled. The proposed sizing model is in charge of determining the required components and its size that guarantee demands satisfaction in a secure manner at minimum cost. Additionally, a set of reserve requirements is considered. The resulting problem is a MISOCP.
Fig. 1
Base microgrid architecture
×
2.1 Objective function
The proposed microgrid sizing model is aimed to minimize the total investment cost (CAPEX) of components and discounted operational cost (OPEX) as represented in Eq. (1).
where factor \(\gamma\) is the ratio of the number of hours per year (8760) to the number of hours employed to represent the operations of the microgrid \(|\mathcal {T}|\), i.e., \(\gamma = \frac{8760}{|\mathcal {T} |}\).
2.1.1 Capital expenditures
Capital costs are considered at year zero. The capital cost of generation technologies is computed as the product of generator size \(p^{\text {nom}}_{j}\) (in kW) and the corresponding unit investment \(I_{j}^{\text {gen}}\) (in $/kW). Likewise, investment cost of chillers is determined as the product between nominal capacity \(q^{\text {nom}}_{n}\) (in kW) and unit cost \(I_{n}^{\text {chi}}\) in ($/kW). Investment cost of energy storage systems depends on both rated power output \(p^{\text {nom}}_{s}\) (in kW) and nominal storage capacity \(e^{\text {nom}}_s\) (in kWh). Total CAPEX is presented in Eq. (2):
where \(F_{g,t}\) represents fuel cost (Eq. (4)). Fuel price \(\pi _{g}\) of fuel-fired technology g is given in \(\left[ \frac{\$}{\text {MBTU}}\right]\). Thus, the affine function \(a_{g}p_{g,t} + b_{g}z_{g,t}\) denotes fuel consumption. Parameters \(a_{g}\) and \(b_{g}\) in \(\left[ \frac{{\text{MBTU}}}{{\text{kWh}}}\right]\) and \(\left[ \frac{{\text{MBTU}}}{{\text{h}}}\right]\), respectively are employed to estimate fuel consumption from the fuel rate curve for fuel-fired power plants with power output \(p_{g,t}\) and scheduled operation defined by variable \(z_{g,t}\). Operation and maintenance cost \(\text {OM}_{k,t}\) is shown in Eq. (5). OM\(^{\text {var}}_{k}\) and OM\(^{\text {fix}}_{k}\) are the variable and fixed operational cost rates, respectively, scaled by power output (\(p_{j,t}\) or \(p^{\text {dis}}_{s,t}\)).
Carbon emissions tax \(\text {CO}_{g,t}\), displayed in Eq. (6), require emissions tax rate \(\omega _{g}\) imposed by the market. Parameter \(\text {ER}^{CO_2}_{g}\) in \(\left[ \frac{kg}{\text {kWh}}\right]\) represents the emissions factor for each technology [18, 26, 27]. Electric battery degradation cost, denoted by \(w_{\text {BESS},t}\), is computed according to Eq. (7). The operational cost is considered to account for its degradation in storage capacity. It is assumed that the BESS will be replaced once its overall energy throughput \(E^{lt}_{\text {BESS}}\) is reached [28, 29]. Energy throughput can be parameterized as a number of average“effective” cycles to failure, i.e., \(E^{\text {lt}}_{\text {BESS}} = N^{\text {lt}}_{\text {BESS}} e^{\text {nom}}_{s}\). Reference [29] shows that \(N^{\text {lt}}_{\text {BESS}}\) depends on the average product between depth of discharge and cycles to failure. Then, degradation cost per unit of cycled energy, computed as \(I^{\text {sto,E}}e^{\text {nom}}_{s}/E^{lt}_{\text {BESS}}\) needs to be multiplied by BESS discharge power \(p^{\text {dis}}_{\text {BESS},t}\) (affected by efficiency) to obtain the dollar amount of total degradation cost \(w_{\text {BESS},t}\) as in Eq. (7). Reserve cost \(\text {RC}_t\) is computed as the product of the sum of all reserves each time-step —down regulation reserves, up regulation reserves and \(N-1\) reserves— by the reserve price \(\beta ^{R}\) as shown in (8).
$$\begin{aligned} F_{g,t}&= \pi _{g} \left( \, a_{g}p_{g,t} + b_{g}z_{g,t}\right) \, \Delta t ,&\forall g \in \mathcal {G}: \mathcal {G} \subseteq \mathcal {J}, t\in T \end{aligned}$$
The model that captures dependence of solar PV output power on both solar irradiance and ambient temperature is adapted from [30]. It also affects available power as long as cell temperature increases. Available solar photovoltaic power production is computed using the global horizontal irradiance (GHI) and assummed to be totally dispatched as PV power output as in Eq. (9) and cell temperature is computed with Eq. (10):
where \(G_t\) represents global horizontal irradiance (GHI); and \(G^{\text {stc}}\) and \(T^{\text {stc}}\) describe irradiance and temperature under standard conditions respectively. \(C_{T} < 0\) is the PV temperature coefficient.
2.2.2 Chillers
Both absorption and electric chillers require an input energy source to generate cooling. An absorption chiller converts heat into cooling whereas an electric chiller turns electric power into cooling. Both cases are modeled in (11) and (12) respectively through performance coefficients as suggested in [31‐33].
The state of charge (SOC) accounts for the amount of energy stored (kWh) in the system at time t. It is the result of continuous charging and discharging processes. SOC is shown in (13):
where \(\eta ^{\text {rt}}_{s}\) is round-trip efficiency for storage system s. Also, \(p^{\text {dis}}_{s,t}, p^{\text {ch}}_{s,t}\) is the discharge and charge power (in kW) for storage system s at time t, respectively. Parameter \(\delta _{s}\) is the self-discharge rate and represents stored energy losses. The model presented in Eq. (13) applies for battery and thermal energy storage systems.
2.3 Operational constraints
2.3.1 Power balances
In each time step, electric power generated by generators needs to be either consumed by microgrid load or stored. The electric power balance is shown in constraint (14).
The heat required by the heat load and the absorption chiller needs to be balanced by heat coming from TESS and thermal power plants as described in constraint (16). If any heat excess is generated, it might be wasted.
Equations (17) and (18) ensure generation dispatch does not exceeds its nominal or minimum capacity. Binary variable \(z_{j,t}\) decides whether or not to dispatch the power plant j at time t. Parameter M is a large scalar and guarantees that power plant j is limited by its nominal output only when it is dispatched, i.e., when \(z_ {j, t} = 1\).
$$\begin{aligned} p_{j,t}&\le \min \left( p^{\text {nom}}_{j}, z_{j,t} \, M \right) ,&\forall j\in \mathcal {J}, t\in T \end{aligned}$$
(18)
2.3.3 Chillers output constraints
Constraints (19)-(20) ensure that chiller output is always between minimum and maximum operation point. Binary variable \(z_{n,t}\) and M limit power output to its nominal value or zero if cooling resource is dispatched or not, respectively:
$$\begin{aligned} \underline{\mu }_{n}q^{nom}_{n} - (1-z_{n,t}) \, M&\le q_{n,t} ,&\forall n \in N, t\in T \end{aligned}$$
(19)
$$\begin{aligned} \min \left( q^{\text {nom}}_{n}, z_{n,t} \, M\right)&\ge q_{n,t},&\forall n \in N, t\in T \end{aligned}$$
(20)
2.3.4 Storage constraints
Constraint (21) forces electricity and heat storage to be within specified minimum and maximum state-of-charge percentages. Constraints (22) and (23) bound instantaneous charge and discharge power output to nominal power output when scheduled; otherwise, charge/discharge is forced to be zero. Since \(p^{\text {nom}}_{s}\) is a decision variable, Big-M constraint with binary decision is used. Binary decisions \(z^{\text {ch}}_{s,t}\) and \(z^{\text {dis}}_{s,t}\) prevent storage systems to charge and discharge simultaneously as constrained in Eq. (24):
While some thermal generators, such as diesel generators, can respond from zero to full load in a much shorter time, ramp constraints may still apply to other types of generators with slower dynamics. In general, the proposed formulation allows for the possibility of including ramp constraints for any fuel-based generator. However, each case study will determine whether or not it is necessary to implement ramp constraints, which can be skipped setting the parameters \(r_{j}^{\text {up}}\) and \(r_{j}^{\text {down}}\) to 1. In the case of generators subject to ramp constraints, these constraints also limit the reserves allocation, ensuring that committed reserves can be depleted in the given time frame \(\Delta t\). Rate of change of power output from fuel-fired generators is limited by maximum ramp to be loaded or downloaded. Equation (25) limits the rate of change from previous time-step to be within down and up ramp rates \(r^{\text {down}}_{j}\), \(r^{\text {up}}_{j}\), respectively [34]:
Our approach guarantees reliable operation of the microgrid, i.e., microgrid components can “react” immediately to avoid load shedding once a contingency occurs. In this work, only electrical contingencies are taken into account. Thermal contingencies are not considered. To generalize the presentation of the model equations, all storage systems, including electric and thermal, were grouped together in a set called \(\mathcal {S}\). The main operational constraints that limit stored energy and power output apply to every storage s, in these cases the general representation using the subscript s (e.g., \(p^{\text {dis}}_{s,t}\)) was used; however for exclusively electrical issues such as outages, only electrical storage (i.e., BESS) is considered. Therefore, the subscript \(\left( \cdot \right) _{\text {BESS},\dots }\) for such constraints was used \(\left( e.g., \Delta p_{\text {BESS},t}^{\text {s}}\right)\) considering that \(\text {BESS} \in \mathcal {S}\).
2.4.1 \(N-1\) reliability constraints
The proposed model forces a reliable operation under electric generation \(N-1\) contingencies. When the contingency of the i-th element occurs, its power injection \(p_{i,t}\) (or \(p^{\text {dis}}_{\text {BESS},t}\) when the BESS fails) is lost. Except for the solar PV system, the remaining online generation must take over of lost power with the scheduled \(N-1\) security reserve \(\Delta p_{j,t}^{\text {S}}\) -\(\Delta p_{\text {BESS},t}^{\text {S}}\) for BESS- supplying power to meet the demand and avoid load shedding. Therefore, generators and/or BESS power output should be ramped-up. For each time-step, a potential outage in each generator and electric storage is considered (i.e., a total of five different outages per hour are considered) and reliability reserves are scheduled. Each outage compromises both current power output and scheduled regulation reserve \(\Delta p_{i,t}^{\text {SR+}}\), which need to be considered in the scheduling of reliability reserves. All of these aspects are carefully considered in constraints (26) and (27):
where \(\Delta p_{j,t}^{\text {S}}\) and \(\Delta p^{\text {S}}_{\text {BESS},t}\) are the allocated power reliability reserves in time-step t for generator j and BESS, respectively. \(p_{i,t}\) and \(\Delta p_{i,t}^{\text {SR+}}\) are lost power output from the outaged generator i and its allocated regulation reserve, respectively, that need to be supplied by all the remaining generators j-th.
2.4.2 Regulation reserves
Cloud cover, changes on irradiance patterns, and power demand fluctuations encourage the use of forecasting techniques for microgrid management. In addition to estimated \(N-1\) reserves to face generation outages under \(N-1\) criterion, the microgrid still needs to deal with both, solar PV and demand forecasting errors. The inherent random nature of these errors increase system regulation reserve requirements in order to balance generation and demand. Regulation reserves and (i.e., spinning and non spinning reserves) are in charge to balance the system by adjusting the power output of generating units [35, 36]. In this work, a chance-constraint approach is proposed to allocate the regulation reserve to each generation based in the estimated regulation reserve requirements.
Several authors have explored the concept of defining a constraint to be met with a high probability, rather than as a conventional “hard” constraint with 100% certainty of satisfaction. However, our work adapts the foundations of chance-constrained optimization to the microgrid operational model. This adaptation is driven by the inherent uncertainties associated with solar generation and demand fluctuations in maintaining power balance. This has motivated us to propose a probability-based reserves allocation on top of the microgrid dispatch to maintain the generation-demand balance with a high probability. The interested reader can refer to [37] and [38] for additional material on chance constrained optimization.
Assuming the existence of hourly forecasts for electric power demand, \(D^{\text {el}}_{t}\) and solar PV generation, \(p^{\text {av}}_{\text {pv},t}\), both actual random demand \(\tilde{D}^{\text {el}}_{t}\) and random solar generation schedule \(\tilde{p}_{\text {pv},t}\) can be expressed as (28) and (29) respectively:
where \(\varepsilon _{D,t}\) and \(\varepsilon _{\text {pv},t}\) represent hourly demand and solar PV estimation and forecast errors respectively, which are considered as independent random variables. The model so far has assumed that expected (or forecasted) demand can be met by both, forecasted solar PV and fuel-fired generation. However, random fluctuations in either demand or solar PV generation are not properly balanced in the model. Thus, it is essential to consider such random fluctuations on an hourly basis to properly scheduling regulation reserves.
Our approach is based on the idea of guaranteeing that electric demand can be satisfied with high probability. To do so, an amount of reserve is necessary to cover fluctuations of the net forecasting error \(\varepsilon _{t}\) defined as:
Net forecasting error can be either positive or negative, depending on the quality of forecasts. Net positive error requires additional generation; whereas net negative error requires a decrease in the generation to maintain demand-generation balance. As a result, in this approach, we distinguish the positive and negative parts of \(\varepsilon _{t}\) in order to schedule positive and negative regulation reserves. That is,
where \(\varepsilon _{t}^+ = \max \left( \varepsilon _{t},0\right)\) and \(\varepsilon _{t}^- = \max \left( -\varepsilon _{t},0\right)\). Therefore, we need to compute positive reserves \(\text {SR}^+_{t}\) and negative reserves \(\text {SR}^-_{t}\) such that:
According to the probabilistic criteria exposed in (30) and (31), deterministic regulation reserves \(\text {SR}^+_{t}\) and \(\text {SR}^-_{t}\) guarantee that both, positive and negative imbalances will be covered with probability \(1-\eta ^+\) and \(1-\eta ^-\) or higher, respectively. In this sense, reserve amounts \(\text {SR}^+_{t}\) need to be at least the \(\left( 1-\eta _t^+\right)\)-th quantile \(q^+_{t,1-\eta ^+}\) of the probability distribution of \(\varepsilon _t^+\). Likewise, and \(\text {SR}^-_{t}\) should be at least the \(\left( 1-\eta _t^-\right)\)-th quantile \(q^-_{t,1-\eta ^-}\) of the probability distribution of \(\varepsilon _t^-\):
$$\begin{aligned} \text {SR}^+_{t}&\ge q^+_{t,1-\eta ^+}, \quad \forall t \in T \\ \text {SR}^-_{t}&\ge q^-_{t,1-\eta ^-}, \quad \forall t \in T. \end{aligned}$$
where \(q_{n}^{\varepsilon } = F^{-1}_\varepsilon \left( n\right) , \ n \in \left[ 0, 1\right]\) refers to the n-th quantile of the distribution of the net forecast error \(\varepsilon\); and \(F^{-1}_\varepsilon \left( \cdot \right)\) is its corresponding inverse cumulative distribution function. An important feature of this formulation is that it can be employ with any probability distribution. Both \(q^+_{t,1-\eta ^+}\) and \(q^-_{t,1-\eta ^-}\) result in non-negative quantities.
When both errors are independent and normally distributed, i.e., \(\varepsilon _{\text {pv},t} \sim \mathcal {N}\left( \mu _{\text {pv,t}}, \sigma ^2_{\text {pv},t}\right)\) and \(\varepsilon _{D,t} \sim \mathcal {N}\left( \mu _{D,t}, \sigma ^2_{D,t}\right)\), it is known for a fact that net error is also normally distributed. That is,
In the proposed approach, we assume that solar forecast error uncertainty is larger as long as solar production increases. In this sense, the hourly power output of the solar PV system ends up affecting parameters of the solar forecast error probability distribution. Solar PV mean forecast error \(\mu _{\text {pv,t}}\) and standard deviation \(\sigma _{\text {pv},t}\) are parameterized as percentages \(\alpha _{\text {pv}}\) and \(\vartheta _{\text {pv}}\) respectively of hourly scheduled PV power output \(p_{\text {pv},t}\). That is,
$$\begin{aligned} \mu _{\text {pv,t}}&= \alpha _{\text {pv}} p_{\text {pv},t},\quad \forall t \in T \\ \sigma _{\text {pv},t}&= \vartheta _{\text {pv}} p_{\text {pv},t}, \quad \forall t \in T \end{aligned}$$
Quantiles \(q_{t,1-\eta ^{+}}^{+}\) and \(q_{t,1-\eta ^{-}}^{-}\) that define the amount of positive and negative regulation reserves respectively are computed as follows:
Quantiles defined by Eqs. (35) and (36) represent the minimum amount of regulation reserves to schedule on an hourly basis. This formulation leads to the second-order cone nature of the proposed model. Additionally we do not impose that \(q_{t,1-\eta ^{+}}^{+},q_{t,1-\eta ^{-}}^{-} \ge 0\) in the model given the non-convexity of such constraints. Rather we verify that the optimal solution satisfies the non-negativity constraints on both quantiles. Finally, the positive and negative regulation reserve requirements are shown in (37) and (38) respectively, resulting in a set of conic constraints:
Both \(N-1\) and positive regulation reserve requirements need to be high enough so as to meet the reliability constraints. However, reserves need to be tuned such that power output cannot exceed the nominal power of any generator as in (39); and are also bounded by maximum ramp-up power as in (40). For BESS, \(N-1\) security reserve and up regulation reserves are limited by the current available energy stored and by its nominal power output as in (41).
where \(\Delta p_{j,t}^{\text {SR}+} \ge 0\), \(\Delta p^{S}_{j,t} \ge 0\), \(\Delta p^{\text {SR}+}_{\text {BESS},t} \ge 0\), and \(\Delta p^{\text {S}}_{\text {BESS},t} \ge 0\) correspond to positive generator regulation reserve, generators \(N-1\) security reserve, BESS up regulation reserves, and BESS \(N-1\) security reserve, respectively.
Likewise regulation reserve constraints of power generation and BESS required to balance negative shifts in net forecasting error \(\varepsilon _t\) are shown in constraints (42)–(44):
where \(\Delta p_{j,t}^{\text {SR}-} \ge 0\) and \(\Delta p^{\text {SR}-}_{\text {BESS},t} \ge 0\) correspond to negative generator and BESS regulation reserves, respectively. Down regulation reserves are bounded by both the minimum operational limit and the ramp-down limit for the generators. For the BESS, down regulation reserves are bounded by the available storage capacity without exceeding its maximum SOC limit, and by its nominal power input.
This regulation reserve approach allows to differentiate probabilities according to the needs of the microgrid. Violation probabilities \(\eta ^+\) and \(\eta ^-\) must be small but can be different. Thus, it is possible to assign different small violation probabilities to either positive or negative imbalances. Additionally, probabilities can be different throughout the day. Also, any resulting probability distribution of forecast errors can be employed. In fact, even dependent random errors can be easily adapted to this formulation. Mitigating the proposed model conservativeness can be done by adjusting the maximum constraint violation probability \(\eta _t^{(\cdot )}\). The larger this probability, the less conservative the solution. Constraints (37) and (38) allow to scheduling enough generation to face potential fluctuations (forecast errors) in demand and renewable generation as a function of the probability \(\eta _t^{(\cdot )}\).
When implementing the explicit formulation of the scenarios, i.e. discretization of the model uncertainties, it results in a large-scale optimization problem with a reasonable number (maybe hundreds) of scenarios [38, 39]. Instead, the proposed computationally tractable chance-constrained has allowed to consider the continuous probability distribution of the combined (solar and demand) forecast error, which in turn permits the model user to set her/his own level of operational risk through the constraint violation probability. Also, despite the fact that the resulting second-order cone model is nonlinear (as opposed to the linear stochastic optimization version), it is still convex. Furthermore, commercial solvers achieve global optimality as long as convexity is maintained in the model.
3 Case study
Fig. 2
Electric and thermal load patterns
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This section shows a case study to test the performance of the proposed model. We assume the resulting islanded microgrid will serve the facilities of a hotel in Cartagena, Colombia. The daily electric and thermal load profiles, on an hourly basis, are obtained from a large hotel facility in DER-CAM software [40] and are shown in Fig. 2. These patterns are split in weekdays and weekends. Peak electric demand is around 630 kW, whereas thermal demand is around 216 kW. Daily electric and thermal demands are 8900 kWh and 1700 kWh respectively. Since the hotel facility is not expected to expand, energy demand levels are assumed as constant in the long term. Investment costs and emission rates are shown in Table 2. Emission rates for each technology are taken from [41] and [42]. The proposed formulation allows to account for the GHG emission for each kwh provided by the sized technology through the parameters \(\text {ER}_{k}^{\text {CO}_{2}}\) and the emission tax cost \(\omega _{k}\). We have decided to account for the emissions of several technologies. Emissions from solar PV are also considered as indicated in [42], in which authors account for the life cycle emissions from the manufacture of solar cells. The \(\text {CO}_{2}\) tax rate is 0.07 \(\left[ \frac{\$}{\text {kg}}\right]\) as suggested in [27].
The project lifespan is 15 years. This case study considers \(|\mathcal {T}|=8760\) hourly periods in order to represent the entire microgrid operations in the sizing problem. This implies \(\gamma = 1\). An interest rate of 6% is employed to discount future cash flows. Up and down regulation reserve cost \(\beta ^{R}\) is set as 40% of the degradation cost of battery storage as in [21]. The average cycles to failure \(N^{\text {lt}}_{\text {BESS}}\) is set to 995 (computed using data from [28]). Solar and electric load estimation error \(\tilde{\varepsilon }_{pv,t}\), \(\tilde{\varepsilon }_{D,t}\) are assumed to be normally distributed \(\mathcal {N}_{\varepsilon _{pv}}\left( \mu _{pv,t}=\alpha _{pv}p_{pv,t}, \sigma _{pv,t}=\vartheta _{pv} p_{pv,t}\right)\), with \(\alpha _{pv}=0\) and \(\vartheta _{pv}=0.1\) and \(\mathcal {N}_{\varepsilon _{D}}\left( \mu _{D,t}=0, \sigma _{D,t}=0.02D^{el}_{t}\right)\), respectively. The allowable violation probabilities of regulation reserves are set as \(\eta ^{-}=\eta ^{+}=0.05\). Meteorological parameters of Cartagena are displayed in Fig. 3. Temperature data have been taken from IDEAM [43] and solar irradiance from Solcast [44]. The model is implemented in Pyomo 5.7.3 [45] and solved using Gurobi 9.5.0 [46] executed on an intel Core i7-8700 with 16GB RAM.
Fig. 3
Temperature and solar irradiance for the hotel
×
4 Results
4.1 Base case
In the base case, only up and down regulation reserves are considered. Security reserves due to sudden outages are ignored in this case. The optimal sizing, CAPEX, annual OPEX and annual CO\(_2\) emissions for each technology in the base case are shown in Table 3. Diesel-generator, PV and BESS are selected for supplying electric power, while an electric chiller (EC) supplies the thermal energy demand. The objective function adds up to \(\$10.3\) million. Although the operational commitment is considered during 8760 hly periods, only the first four days are displayed in Fig. 4. Despite the relative high OPEX of diesel generation, it is heavily scheduled as the main generator to meet electric load. During peak sunlight hours, PV supplies a significant share of load and BESS charge. The remaining load is supplied mainly by the diesel generator. BESS discharge also provides support during the peak-hours.
Power output for microgrid technologies in base case
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The state-of-charge (SOC) of BESS is shown in Fig. 5. BESS is scheduled to be charged (upward SOC) mainly during peak sunlight hours with PV system. Discharge (downward SOC) is scheduled during both, the mornings and peak load hours under absence of solar PV production.
Fig. 5
State-of-charge for ESS in base case
×
Base case optimal up and down regulation reserves allocation and requirements on the base case are depicted in Fig. 6. This case ignores \(N-1\) reserves and only regulation reserves are imposed. The size of the requirements depends on the probability distribution of net forecast error (difference between both electric load and PV system forecast errors). For the PV system, the larger the scheduled PV output, the larger reserve would be needed. DSL and BESS are the only dispatchable resources installed in this case (MT and ICE are not installed). As a result, regulation reserve requirements are covered mainly by BESS. That is, BESS would face potential intermittencies given the non-dispatchability and potential high forecast error of the PV-system production. Stored energy acts as an up-regulation reserve to cover sudden net load increments or any potential renewable intermittency; down-regulation reserve is employed to store a sudden surge of energy. Considering degradation wear costs leads to the use of BESS mainly as a reserve provider. Scheduling BESS for energy provision accelerates discharge cycles, which is represented in capacity loss and higher replacement costs.
Fig. 6
Up and down reserves requirements due intermittency in basecase
×
In this case, thermal load is always supplied by EC. MT and ICE, providers of recovered heat, are of no interest to the model given their high operational and investment cost. This, in turn, yields the model to not install any AC. As a result, TESS is not necessary in the optimal microgrid design.
The Reliability-based design includes all constraints presented in the proposed model in Sect. 2. Sizing, CAPEX, annual OPEX, and annual CO\(_2\) emissions for each technology in the microgrid are shown in Table 4. The total objective function is \(\$16.5\) millions. The energy mix is more diversified than the base case to avoid high dependency on a specific technology. PV system is the largest generator, closely followed by DSL, MT and ICE. The BESS capacity is sized to be discharged in approximately an hour and a half at its nominal power output. Although conventional fuel-fired generation represents a larger share than renewable resources, CO\(_2\) emissions are reduced compared to the base case.
Fig. 7
Reserve schedules for \(N-1\) contingencies in the reliability-based case
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The provision reserves against \(N-1\) contingencies on for the reliability-based design case are shown in Fig. 7. \(N-1\) reserves requirements avoid any generator to supply the entire electric load; rather, all generators are scheduled to a moderate power output. The model allocates reserves for each dispatchable technology such that, for any sudden generator loss at any hour, the microgrid has enough mix of \(N-1\) reserves. Once a generator is faulted, \(N-1\) reserves, provided by remaining generators, should be enough to continuously satisfy electric demand. The proposed approach in this work diversifies both energy provision and reserve allocation by avoiding a single generator providing all of the reserve and energy requirements. Required reserves are determined by the largest generator’s output plus the largest allocated regulation reserve per period. Thus, this balance between operating efficiency and reserve capacity entails a trade-off: increasing generation capacity enhances efficiency; however, it also increases the need for \(N-1\) reserve, which in turn imposes generators to operate at lower power output levels
As shown in the operational scheduling in Fig. 8, MT and ICE are continuously dispatched at its minimum operative output, whereas DSL supplies the remaining load. During sunlight hours, solar PV power is employed to charge the BESS and supply a fraction of load, which eventually reduces DSL production. Thus, MT and BESS are the main reserve providers against contingencies. When PV production is high, DSL is enabled to provide reserves in case of an outage. As a result of the reliable operation of the microgrid, operational cost tends to be high given the significant use of conventional fuel-fired generation.
Fig. 8
Power output for microgrid technologies with reliability-based design
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Figure 9 displays the schedule of thermal systems. Unlike the base case results, the AC takes advantage of the heat recovered from the burnt fuel of MT and ICE to provide cooling. Remaining thermal demand is supplied by the EC, which in turn imposes an additional electric load to the microgrid. Several hours before the peak-cooling load condition, recovered heat from MT and ICE is not employed by the AC, but stored in the TESS. Once in peak-cooling load condition, the TESS provides heat to the AC, which delivers energy to thermal loads in order to reduce the EC participation.
Fig. 9
Thermal load and chillers output
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The state-of-charge for each storage system is shown in Fig. 10. BESS is charged by the PV-system and by the fuel-fired generator when solar production vanishes and electric load is the lowest. BESS is discharged during peak-load hours and during the night-hours when PV is zero. Since BESS is also scheduled as reserve provider, it is rarely fully discharged.
Fig. 10
State-of-charge for ESS in reliability-based design
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Reserves for intermittency are splitted in up/down regulation reserves as displayed by dashed lines in Fig. 11. Up regulation reserves are covered by almost all technologies online, which are available to be ramped-up to deal with either random intermittencies in PV output or changes in electric load. Down reserves are intended to decrease power production (or store any energy surplus in the BESS) given an increase in PV-power output or decrease in load demand. Down regulation reserve requirements are covered mainly by DSL, which is scheduled close to its rated power. BESS also provides down reserves when its SOC is low. MT and ICE do not offer down reserves since they are scheduled at their minimum levels.
Fig. 11
Up and down regulation reserves requirements in reliability-based design
×
For both the base case and the reliability-based case, the info of CPU time, number of constraints, number of conic constraints, number of variables, number of binary variables and objective function values are shown in Table 5. The CPU time for the full reliability-based case is close to 42 min.
Table 5
Optimization problem characteristics
Base case
Reliability-based case
Time [min]
25.98
42.6
Constraints
623,666
676,370
Conic constraints
17,568
17,568
Variables
351,372
351,372
Binary Vars
87,840
87,840
Obj. Func. [$]
10,303,246
16,556,278
4.2.2 Reserve requirements
Regulation and \(N-1\) security reserves, impose requirements to power output in generators and BESS. \(N-1\) and up regulation reserve require the generator power output and BESS discharging to be ready to increase if needed; on the other hand, down regulation reserves require the generator and BESS discharging to be reduce if needed. For the BESS, reserves are scheduled to take advantage of its ability to absorb energy in the charging mode. Figures 12 and 13 depict the scheduled power output, up/down regulation reserves allocation and \(N-1\) security reserves for DSL and BESS respectively. Both allocated reserves are scheduled based on the excess generator capacity, without exceeding their nominal or ramp limits. In the case of DSL, the output limits are imposed by the nominal power output and maximum up/down ramps. DSL schedules down regulation reserves without exceeding the minimum operational value. Likewise, \(N-1\) and up regulation reserves are rigorously scheduled avoiding producing power above nominal values of generation resources.
Fig. 12
Diesel generator scheduled output and reserves
Fig. 13
BESS Stored energy and scheduled reserves
×
×
Despite the fact that BESS reserves are limited by available stored energy and nominal charge/discharge power output, the optimal sizing shows that BESS is a significant reserve provider. For \(N-1\) reserves, BESS is scheduled to increase its power output (discharging mode). The \(N-1\) reserve provision is high when either its SOC is at medium/high levels or BESS is already discharging due to the absence of sunlight and PV output. For down regulation reserves, BESS reserves are scheduled when sunlight is at peak. If any deviation occurs, the BESS may reduce its power output and even absorb any energy excess (charging mode) to cover the demand/PV forecasting deviation. Equation (8) offers a general formulation that allows reserve costs to be allocated as proposed in the literature [21]. However, the reserve cost could be set to zero if needed. After running the model setting reserve cost to zero, the impact on the optimal microgrid sizing is negligible; but, the scheduling of regulation reserves increases. The model schedules up(down) regulation reserves beyond the computed requirements due to lack of reserve cost. Remaining available power (scheduled output above minimum) is allocated as up(down) regulation reserves. This in turn implies additional \(N-1\) reserves that need to be provided to deal with potential outages
4.3 Sensitivity analysis
4.3.1 Sensitivity analysis for investment cost
Fig. 14
Optimal microgrid sizing vs BESS investment cost
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To comprehensively evaluate the performance of the proposed model under different circumstances, a sensitivity analysis in BESS capacity investment cost is carried out. To analyze the impact of BESS investment cost \(\left( I^{\text {sto,E}}_{\text {BESS}}\right)\) on technology sizing, variations between 70% and 130% of the its nominal value (580 $/kWh) were considered. Figure 14 shows that higher investment cost per kW does not cause significant changes in the BESS installed capacity. The main change in the analysis is the amount of energy discharged from the BESS. For the lowest value (319 $/kWh) 996,961 kWh are discharged; whereas the case with the highest value (841 $/kWh), only 423,975 kWh are discharged annually from the BESS. This condition stems from the BESS investment cost, which heavily affects the wear cost. If it is low, then BESS participation is high. On the contrary, the wear cost tends to be excessively high and the BESS only participates as reserve provider. Also, MT and DSL nominal capacities barely change; but are constantly scheduled to operate, which would recover additional heat. As a result, the sizing of AC and TESS is such that it can take advantage of the increased heat and thus reduce the EC participation. The simultaneous occurrence of peaks in both electric and thermal load forces the model to employ more AC (which is fed by heat stored in the TESS and recovered heat from MT and ICE) than EC. The objective function ranges from $16.5 millions (319 $/kWh) to $16.8 millions (841 $/kWh).
4.3.2 Sensitivity analysis with respect to solar forecast error
Fig. 15
Up and down regulation reserve requirements under changes in \(\alpha _{\text {pv}}\)
×
The net error distribution of solar production and electric load directly affects the regulation reserve requirements. Figure 15 depicts the reserve requirements for up and down regulation reserves under changes in the error mean of the solar production forecast \(\mu _{pv,t}\), while Fig. 16 shows the optimal microgrid sizing in each case. Negative solar mean forecast error (forecast overestimates actual production) increases the up regulation reserves and reduce down reserves. For large negative mean errors, PV system size increases and fuel-fired production decreases, which causes a decrease in the objective function. Large positive mean error forecast decreases up regulation reserves and increases down regulation reserve requirements. Unlike up regulation reserves, down reserves are supported mainly by BESS and DSL (MT and ICE are dispatched at their operational minimum.) To face increasing down regulation reserve requirements, the model reduces the PV system size and increases DSL schedule to extend the amount of down regulation reserve available in DSL. Non-negativity conditions for quantiles in Eqs. (35) and (36) are verified on an hourly basis. Even for extreme values, \(\alpha _{\text {pv}}=\pm 10\%\) quantiles are still non negative.
Fig. 16
Optimal microgrid sizing vs changes in the the solar mean forecast error \(\alpha _{\text {pv}}\)
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4.3.3 Sensitivity analysis with respect to regulation reserve violation probabilities
According to the probabilistic criteria exposed in Eqs. (30) and (31), the model imposes that scheduled regulation reserve requirements should cover positive and negative imbalances in demand-generation with probabilities at least \(1-\eta ^+\) and \(1-\eta ^-\) respectively. The reserve commitment of the proposed model and its implications is analyzed in this part via the following three cases:
Reference case: \(\eta ^{+} = \eta ^{-} = 0.05\).
High probability case: \(\eta ^{+} = \eta ^{-} = 0.10\).
Low probability case: \(\eta ^{+} = \eta ^{-} = 0.01\).
Figure 17 shows variation of up and down regulation reserve requirements in a short time span when forecast mean errors are zero (\(\mu _{\text {D},t} = \mu _{\text {pv},t} = 0\)). And Fig. 18 shows the optimal microgrid design in each case. As expected, the lower probabilities \(\eta ^{+}\) and \(\eta ^{-}\), the higher up and down regulation reserve requirements. To schedule high regulation reserves, conventional fuel-fired generators and BESS are high-sized. As the probabilities \(\eta ^{+}\) and \(\eta ^{-}\) increase, solar PV participation increases, which reduces operational expenditures. Also, non-negativity conditions stated in Eqs. (35) and (36) always hold.
Fig. 17
Up and down regulation reserve requirements under changes in \(\eta\)
Fig. 18
Optimal microgrid sizing vs probabilities \(\eta ^{+}\) and \(\eta ^{-}\)
×
×
4.3.4 Fuel-consumption curve parameters
In this section, we present a comprehensive analysis of the fuel-consumption parameters \(a_g\) and \(b_g\) in the proposed formulation and their impact on both the sizing and operational aspects of the reliability-based study case under consideration. A new case based in the reliability case in Sect. 4.2 is proposed, where parameters \(a_{g}\) and \(b_{g}\) for each generator type were approximated from industrial data. Parameters \(a_{g}\) and \(b_{g}\) for each generator type were approximated from industrial data. Specifically, for Diesel Fuel Consumption [47], Natural Gas Consumption [48] and, ICE fuel consumption [12]. the data was fitted to a straight line as shown in Fig. 19 and converted to MBTU using average heat content from [49]. Results are presented in Table 6.
Fig. 19
Linear approximation of fuel consumption
Table 6
Sizing results for industrial parameters \(a_g\) and \(b_g\)
The DSL generator is not selected as part of the energy mix due to its high operating costs. Fuel consumption and the high price of diesel make it uneconomical to use the generator. To replace the power supplied by DSL, both ICE and the MT increase their rated power and are scheduled with a higher output at each timestep. Due to the amount of heat available to be recovered from the ICE and MT, the HAC increases its capacity while the participation of the EC decreases. Because now only two fuel-fired generators provide most of the required power with a higher output, the \(N-1\) reserve requirements are higher and provided mainly by the BESS. BESS cycling policy is now set as reserve provider to allows the reserve requirement to be met and avoid MT to run at low power output. This behaviour is depicted in Fig. 20. In contrast to the reliability-based case, in this case the consumption and cost of diesel fuel is much higher. Therefore, the model decided not to use the DSL generator, not even as reserves provider, as it is very expensive to keep it running to provide reserves due to fuel consumption.
This section presents an analysis of the BESS wear cost \(\omega _{\text {BESS},t}\) related to microgrid sizing. This cost of accounts for the BESS degradation due to cycling energy through the storage system. It quantifies the cumulative degradation incurred by each kWh provided by the battery until its lifetime energy throughput is depleted. Equation (7) incorporates the BESS wear cost approach in the objective function, which affects BESS cycling. As a result, the optimal cycling policy is based on a trade-off among generator fuel cost, generators fuel consumption, battery wear cost and reserve requirements. A full reliability-based case without BESS wear cost is performed and the corresponding results are presented in Table 7.
The main changes comparing this case with respect to the reliability-based case presented in Sect. 4.2 is the lack of ICE generator in the sized mix. The ICE power is replaced with a higher rated power to DSL, MT and BESS. Since no heat is recovered from ICE, HAC and TESS are significantly reduced. The power scheduling is depicted in Fig. 21. In this case, BESS is cycled more often at lowest SOC w.r.t. the reference case. The BESS power output is higher and more recurrent while still providing regulation and \(N-1\) reserves because no degradation is considered. Since the remaining generators provide higher power output and regulation reserves to replace the ICE, the \(N-1\) reserve requirement are increased to cover any potential outage in the sized generators.
Table 7
Sizing results for reliability-based study with no wear cost
Power output for technologies in the reliability-based case with no BESS wear cost
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5 Conclusions
This paper presents a novel reliability-based methodology to optimally design an isolated microgrid with both electric and thermal demands. The optimal design covers technology selection and sizing from a distributed energy resources portfolio. The interplay between thermal and electricity flows with the microgrid were effectively considered via linear constraints. To provide a reliable design and operation, a set of constraints against sudden \(N-1\) generation contingencies was proposed. Also, a high percentage of random fluctuations of demand and/or intermittent generation were handled by a set of probabilistic chance-constraints that determine minimum amounts of hourly regulation reserves. This type of constraints determined the second-order conic nature of the mathematical model.
When reliability constraints are included in the model, the participation of dispatchable resources in the optimal design of the microgrid increases as a result of higher reserve requirements; but, renewable technologies participation decreases. As a result, not only larger nominal power capacities are installed, but also higher operational expenditures are observed given the significant costly production of fuel-fired technologies.
Reliance on few generation projects leads to high \(N-1\) reserve requirements to mitigate the risk of generator outages. Meeting these reserve requirements with conventional generators increases fuel costs as the generators must reduce their power output to provide reserves, which leads to partially loaded fuel-fired generator and increased fuel consumption
BESS-capacity investment cost is a key parameter in the BESS operations. As long as such a cost is reduced, BESS is heavily scheduled; otherwise, BESS capacity barely changes and is mainly employed for reserves provision. This in turn increases the participation of the fuel-fired technologies, absorption chiller, and thermal storage system.
The wear cost \(\omega _{\text {BESS},t}\), accounting for energy cycling through the BESS is used to effectively assess battery capacity degradation. The BESS cycling policy is optimally determined based on a trade-off between fuel costs and wear cost in order to reduce CAPEX and OPEX.
Finally, in order to avoid the typical zero-mean assumptions of random forecast errors, our chance-constraints carefully differentiate positive from negative errors. Forecast errors are not necessarily unbiased, as in real life. While a positive error mean increases down-regulation reserves (and reduces up reserves), a negative error mean increases up-regulation reserves (and reduces down reserves). However, the effect of imposing low violation probability of handling random forecast errors always translates into high up- and down-regulation reserves.
Acknowledgements
This research was funded by Celsia Colombia S.A. E.S.P. and was awarded by Minciencias under Grant 786-2019 (code 8101-786-65958).
Declarations
Conflict of interest
The authors declare that they have no known competing financial or personal relationships that could have appeared to the work reported in this paper.
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