The energy hub system facilitates the conversion of energy carriers. The energy hub connects various energy sources such as electricity and natural gas, and different loads (e.g., electricity and heating) to one another. The hub comprises a transformer, boiler and simultaneous electricity and heat generator, receives natural gas and electricity from the upstream grid as the input and supplies the consumers’ demanded electricity and heating as the output.

Herein, electric and thermal loads are modeled as follows Eq. (1) by using a normal distribution function with a mean equal to the baseline load in each period and a standard deviation (SD) of 5% of the baseline load [20].

(1)f(Lt,l)=1σl2πexp(−(Lt,l−μl)22σl2)

The statistical analysis shows that the general distribution logic is the most appropriate model and confirms the proposed strategy in the analysis. Thus, the log-normal distribution function with the mean equal to the baseline electricity price in a certain interval and the SD of 5% of the baseline price is used for modeling the electricity market in Eq. (2) [25]:

(2)f(Ct,e)=1Ct,cσc2πexp(−(ln⁡(Ct,e)−μc)22σc2)

Two distribution functions are applied to model the uncertainties in the wind and solar units. The output power of the wind generator depends on the wind speed with zero fuel costs. The wind speed changes in real-time, which highlights the importance of a probabilistic model. The literature usually uses the Weibull normal distribution function for wind speed (Eq. (3)).

(3)fv(vt)={(βα).(vtα)β−1.exp(−(vtα)β)vt≥00otherwise

The output power from the wind farm in any interval is calculated via the wind power-wind speed curve that expresses in Eq. (4) [20,21]:

(4)RPwt(vt)={00≤vt≤vcior≤vt≥vcoPrwt.vt2−vci2vr2−vci2vci≤vt≤vrPrwtvr≤vt≤vco

The power generated from a solar unit differs with solar irradiation that depends on various factors such as environmental conditions, time, day, month, season and orientation of the solar plant towards solar irradiation. Herein, solar irradiation is modeled via a beta distribution function, expressed in Eq. (5).

(5)f(Rt)=Γ(φ,ξ)Γ(φ).Γ(ξ)⋅Rt(φ−1)⋅(1−Rt)ξ

The solar generator of the output power belongs to solar irradiation; therefore, output power modeling needs solar irradiation modeling. The output power of the solar station as a function of irradiation in any interval of the irradiation-power curve is calculated in Eq. (6) as:

(6)RPpv(Rt)={Prpv⋅(Rt2RSTAD⋅Rce)0≤Rt≤RcePrpv⋅(RtRSTAD)Rce≤Rt≤RSTADPrpvRSTAD≤Rt

There are six sources of uncertainty in the proposed multi-carrier MG: previous-day and real-time prices, solar and wind generations, as well as electric and thermal loads. The proposed grid presents energy sales/purchases in a multi-carrier MG as a two-stage model. In the first stage, the MG presents the hourly energy purchase and sales to the distribution system operator without considering uncertainties. After settling the market, the MG programs generation for day-ahead MG operation is based on deterministic parameters. In the second stage (real-time operation), by considering the confirmed previous-day sales and purchase bids, the MG operator uses the resources to balance the generation and consumption due to uncertainties. In fact, this stage precedes the real-time market settlement in each hour; therefore, the units’ level of generation, ESS charge/discharge and energy sales/purchase in the real-time market are optimized. This model is a mixed-integer linear problem, in which the objective function and constraints of the proposed multi-carrier MG are modeled for 24 h in each scenario. In this model, the objective function is to minimize the energy sales and purchase costs as well as maintenance costs of the equipment. The minimization objective function is modeled as Eq. (7) [20]:

(7)OF:∑sΩt,s⋅∑t[ρDA(t,s)⋅PeDA(t,s)]+ρg(t,s)⋅Pg(t,s)+Co&m(t,s)

where the first term is the energy purchase costs or sales profit in the previous-day market per hour. The second term is the natural gas purchase cost to supply the gas fuel-consuming units that are in charge of supplying electricity and heating to the MG. The final term is the equipment maintenance costs modeled in Eq. (8) as follows:

(8)Co&m(t,s)=PeCHP(t,s)Ko&mCHP+PhAB(t,s)Ko&mAB+PeWT(t,s)Ko&mWT+PeNet(t,s)Ko&mtran+Es(t,s)Ko&mES

In Eq. (8), each units’ maintenance cost is calculated by multiplying the generation of each unit by the maintenance cost coefficient per hour.

The following equations show the MG equipment conversion/generation equation. In this MG, energy conversion is performed by a transformer modeled in Eq. (9). The generated energy by the simultaneous generation converter and the boiler is presented in Eqs. (10) and (11), and depends on the purchased gas and equipment efficiency. The heating system is formulated in Eq. (12) [20]:

(9)Petrans=PeNet(t,s)ηtrans

(10)PlCHP(t,s)=Pg(t,s)ηlCHPv(t,s)l∈{e,h}

(11)PAB(t,s)=Pg(t,s)ηAB(1−v(t,s))

(12)PEHP(t,s)=PiEHP(t,s)ηEHPCo

Battery modeling is formulated in Eq. (13) such that Eq. (14) prevents simultaneous battery charge/discharge. In Eq. (12), for sustainable use of the battery, the amount of energy at the first and last moments of the day must be equal, as shown by Eq. (15):

(13)M(t,s)=[E(t,s)−E(t−1,s)]×(ηchIch(t,s)+ηdisIdis(t,s))

(14)Ich(t,s)+Idis(t,s)=1

(15)El(1,m)=El(24,m)

Eq. (16) presents the electric and thermal load balance that must hold for each scenario in each hour. (16) shows that the electric load and electricity required for supplying the heating system can be supplied by the upstream grid, CHP generator, wind and solar power stations and battery. Based on Eq. (17), the thermal load can be supplied by the CHP generator, absorption chiller and heating system.

(16)Loade(t,s)+PiEHP(t,s)=PeNet(t,s)α(t)+PeCHP(t,s)+PWT(t,s)+PPV(t,s)+M(t,s)

(17)Loadh(t,s)≤PhCHP(t,s)+PhAB+PhEHP

The upper and lower bounds of each unit, e.g., transformer capacity, CHP generator, boiler, heating system, ESS and the battery charged/discharged energy, are in Eqs. (18)–(23):

(18)Pmintrans<Petrans(t,s)<Pmaxtrans

(19)PminCHP<PCHP(t,s)<PmaxCHP

(20)PminAB<PAB(t,s)<PmaxAB

(21)PminEHP<PEHP(t,s)<PmaxEHP

(22)Emin<E(t,s)<Emax

(23)|MES(t,s)|<MES

Eq. (24) models the input gas coefficient to the CHP generator and the boiler.

(24)0≤v(t,s)≤1

The high sensitivity of the previous-day and real-time electricity prices to gas supply was demonstrated vividly in the winter of 2014. Although this electricity price follows the natural gas price model, their relative dependence must be the starting point for creating proper interdependence models; still, the real-time market electricity price depends on many other factors, a few of which can be measured by available data; e.g., natural gas price in the real-time market, the effect of gas pipe capacity, load model deviations and load responsiveness, RES penetrability and previous-day market electricity price, each somehow affecting the real-time market electricity price. Finally, to determine how various factors affect future and real-time electricity prices in the mentioned periods, the correlation coefficient of electricity prices and different factors are calculated in each local hub. The carriers’ price dependence is demonstrated with Eq. (25), in which the y-axis shows the previous-day and real-time market prices. The x-axis shows the load prediction error, wind prediction error, natural gas price or previous-day electricity price.

(25)y=β1x+β2

Coefficients 1β and 2β denote the best line for carrier dependencies.

In this stage, the first-stage variable (the forward purchase/sales of electricity) is placed in the second-stage objective function as a constant parameter (first term), and the electricity shortage/surplus is traded on the real-time market (second term). In this stage, all the uncertainties are taken into account and the MG operator can sell the forward purchased surplus electricity on the market in real-time. Eq. (26) shows the second-stage objective function.

(26)∑sΩt,s⋅∑t[ρDA(t,s)⋅PeDA(t,s)]+ρgRT(t,s)⋅PgRT(t,s)+∑tρeRT(t,s)⋅PeRT(t,s)+Co&mRT(t,s)

All the constraints mentioned above, except Eqs. (27) and (28) which have been changed, hold. Thus, in this stage, the electricity purchased in Eq. (27) changes, as the result of which the electricity load balance is altered based on Eq. (28).

(27)PeRT(t,s)=PeRT(t,s)+PeRT(t,s)

(28)Loade(t,s)+PiEHP(t,s)=PeNet(t,s)+PeCHP(t,s)+PWT(t,s)+PPV(t,s)+M(t,s)

Based on Eq. (27), the total electricity purchased from the grid is the sum of the electricity purchased from the previous-day market and the real-time market.

To create knowledgeable choices withinside the existence of uncertainties, threat control issues of electricity utilities can be simulated through multistage stochastic packages. These packages utilize a hard and fast bevy of situations (or potential realizations), as well as taking into consideration the corresponding changes to the version of the multivariate random statistics procedure, e.g., electric demand, stream flows to hydro units, the output of production in intermittent clean sources, fuel and electricity prices. To show the involved uncertainty, the number of scenarios is demanded while, reduction scenario approaches are often used because of computational complexity and time limitations. This work suggests a novel model for to recursive backward scenario reduction is possible regarding next-day scenarios to generate wind power. Therefore, a set of the initial scenario set is determined by the following algorithm and assigns new probabilities to the stored scenarios. The output is planned to assist production scheduling of power systems employing intermittent type clean energy sources.

Backward scenario reduction is a positive method for reducing the number of scenarios. This technique provides a set of scenarios to estimate their primary set.

Considering ξs(S=1,2,…,Ns) as different N_s scenarios with the probability, probabilities and DTs,s⋅ as the paired distance of the scenarios, the following steps should be followed:

Step 1: Take S as the primary set of scenarios and DS is the set of scenarios that should be eliminated. The primary DS is zero. Calculate the distance of all the scenario pairs: DTs,s⋅=DT(ξs,ξs),S,S′=1,2,…,Ns while; DTs,s⋅=∑(Vis−Vis)2⋅Vis, the vector of unspecified parameters in scenario s.

Step 2: For any scenario k, DTk(r)=minDTk,s,S′,k∈s and S′≠k,r are the scenario criteria with minimum distance from scenario k.

Step 3: Calculate PDk(r)=prob(k)×DTk(r),k∈s and select d in it PDd=minPDk,k∈S .

Step 4: S=S−{d},DS=DS+{d},prob(r)=prob(r)+prob(d).

Step 5: Repeat Steps 2–4 to eliminate the intended number.

A multi-carrier MG is a system that boosts energy flexibility. To respond to different forms of demands, the multi-carrier MG is equipped with energy converters, generators, and ESS. Fig. 1 illustrates the structure of a multi-carrier MG. Each multi-carrier MG comprises small-scale energy sources, several demands and a battery. The small-scale energy sources include PV and wind power stations, a boiler, CHP generator, transformers and a heating system. Table 1 lists the technical specifications of the equipment in this MG.

In this simulation, the number of scenarios for each parameter is reduced to three scenarios by using the introduced method. The data of electric and thermal loads, wind and solar generations, previous-day electricity price, and real-time electricity price for the three scenarios are presented in Figs. 3 and 4.

The electricity and natural gas prices for the three scenarios are given in Fig. 5.

Solar and wind units’ generation for each scenario is depicted in Figs. 6 and 7.

The simulation results are presented and discussed in this section. The bidding strategy problem in the multi-carrier MG is applied to implemented on the proposed grid. Seven modes are considered to evaluate the model.

Mode 1. Purchase and sales strategy with previous-day and real-time electricity price uncertainty

Mode 2. Purchase and sales strategy with all the uncertainties in previous-day and real-time electricity price, electric and thermal loads as well as RES generation

Mode 3. Effect of previous-day electricity price variation on the purchasing and sales strategies with all the uncertainties in the paper

Mode 4. Effect of RES power station (wind and solar) on the purchasing and sales strategy with all the uncertainties in the paper

Mode 5. Effect of battery capacity on the purchasing and sales strategy with all the uncertainties in the paper

Mode 6. Effect of gas price on the purchasing and sales strategies, with all the uncertainties in the paper

Mode 7. Effect of uncertainties on the cost of MG operation

Mode 1: Here, the proposed strategy is applied to a multi-carrier MG. Fig. 8 compares the cost of MG operation in the previous-day and real-time market with previous-day and real-time market electricity price uncertainty. Based on Fig. 8, the cost of operation is considerably reduced for the dual participation model compared to the model, in which it participates only in the previous-day or real-time markets. The cost of MG operation is severely increased if it participates in the real-time market due to the high electricity price, compared to participating in the previous-day market. The comparison of optimal MG operation with/without participation in the previous-day market shows that the cost of MG operation is reduced by about 5.2% with dual participation, compared to participation only in the real-time market.

Mode 2: To demonstrate the importance of the proposed strategy for a multi-carrier MG and using the proposed strategy for MG operation by considering previous-day and real-time market electricity price, the electric and thermal load and RES generation (solar and wind) are simulated in mode 2.

Fig. 9 compares the cost of MG operation in dual participation mode with the uncertainty in all previous-day and real-time market parameters. Based on Fig. 9, each cost of operation is increased in the dual participation mode compared to participating only in the previous-day market, due to a rise in electric and thermal loads and a reduction in DG prediction. Compared to the single participation in the real-time market, it has been reduced due to the expensive electricity price in the real-time market. This mode reveals that the cost of operation has increased during dual participation compared to Mode 1 due to having a more realistic model with different uncertainties. This difference in costs could seem unreasonable at first; however, due to uncertainties in DG and energy price, the MG tends to purchase energy from the previous-day market due to the low electricity price, while it tends to sell energy at a higher price on the real-time market.

Figs. 10 and 11 display the average purchased/sold energy on the previous-day market and the energy exchanges on the real-time market to compensate for the imbalance. In Fig. 11, the image of a clock is used to display the forward sales/purchases on the previous-day market, as well as the energy sales/purchase on the real-time market to compensate for load imbalance. A binary variable is used to see whether the forward sales/purchase of energy in the first stage was cost-effective or not, denoted by 0 (no) and 1 (yes). Based on the following two figures, in the first stage, the MG tended to sell energy at 1, 2, 3, 5, 6, and 23–24 due to the low consumption and high generation, but generation or economic profit from the real-time purchase was impossible for the MG. This could be because, appraising and considering maximum generation for RES units on the previous day, less energy was generated on the real-time market due to uncertainty.

After second-stage analysis and examining the defined binary variable (confirmation of sales or purchase on the previous-day market), it is concluded that the MG should sell the surplus electricity in the real-time market due to the high electricity price. Evidently, the MG operator had decided to sell energy in the forward sales and purchase strategy at 4, while it needed to purchase a small amount of electricity from the upstream grid, even at that hour in the real-time market. At 7–9, 11–15, 17, 19, and 20–22, the MG operator decided to purchase electricity from the upstream grid, which is acceptable at most hours, and even purchase a small amount of electricity from the grid to supply the surplus load on the real-time market. The rejection of the forward electricity purchase at 10 and 18 is notable; the operator has realized in the real-time stage that the electricity price has been reduced on the real-time market and found it cost-effective to purchase all the electricity on the real-time market.

Finally, it is concluded that the MG operator should forward purchase electricity at peak hours due to the uncertainties; however, it should not forward sell any electricity and, instead, sell the surplus electricity on the real-time market. Forward purchase of surplus electricity is more optimal than a little forward purchase to overcome the uncertainties because it is possible to sell the forward-purchased surplus electricity on the real-time market.

Mode 3: Fig. 12 and Table 2 show the effect of variations (increases/decreases) in the previous-day electricity price on the cost of MG operation. Evidently, if the previous-day market electricity price has a declining trend, the cost of MG operation is reduced when participating only in the previous-day market; in this case, it is cost-effective for the MG operator to forward purchase on the previous-day market.

Interestingly, in this mode, the final cost of participation in the real-time market alone is more than the other two modes. Overall, if the electricity price remains unchanged, the costs of operation in the previous-day and real-time markets also remains unchanged. If the electricity price increases by 10%, then the cost of operation only in the previous-day participation model or dual participation slightly increases. Nevertheless, if the electricity price gradually increases by 20% and above, the cost of operation in the previous-day participation model rises, while it decreases during dual participation. By raising the electricity price up to 30% on the previous-day market, the cost of operation increases only when participating in the previous-day market; however, in the dual participation model, the changes were not significant; the cost of operation is still higher when participating only in the real-time market. The declining trend of the operation cost with a >30% rise in the previous-day electricity price in the dual participation model is notable, which is due to the sale of electricity on the previous-day market, only to then purchase it in the real-time market.

Based on Fig. 12, the minimum 50% rise in the electricity price in the previous-day market has led to an exponential elevation in the cost of operation in the previous-day market, but a reduction in the cost of dual participation. In this model, the operation cost of dual participation is less than participation only in the real-time market. It is shown that when the electricity price is increased by 100% in the previous-day market, to reduce the cost of MG operation, a very little amount of energy should be purchased from the previous-day market and the maximum amount of energy from the real-time market.

Mode 4: Fig. 13 depicts the effects of RES (wind and solar) power station capacity on the cost of operation; the more the capacity of the RES power station increases, the more the cost of operation would decrease. The notable point in this scenario is that, by increasing the RES units’ capacity by 10 times, the cost of MG operation will greatly decrease. Based on Table 3, with this extent of RES power station installation, there is no need to participate in the previous-day market and the consumer can trade all the required energy on the real-time market.

Mode 5: Fig. 14 and Table 4 present the effect of battery capacity on the cost of operation. Based on this scenario, by increasing the battery capacity, the system operator will have more flexibility in selling/purchasing energy in all three modes (participation in previous-day market, participation in day-ahead market and both), which eventually reduces the cost of MG operation.

As an example, the ESS performance in the main model is illustrated in Fig. 15. The battery performance shows that the MG tends to charge in non-peak hours, while energy discharge is done to supply loads in peak hours when the electricity price is high. Also, the effect of increasing natural gas price is presented in Table 5 for the operation cost.

Mode 6: In Table 6 and Fig. 16, it is observed that the electricity price depends on the gas price on the previous day and in real-time; by increasing the gas price, the electricity prices also rise, thereby increasing the total cost of MG operation.

Mode 7: In this model, the effect of uncertainty on the MG cost in the dual participation model is studied based on Table 6. Evidently, the absence of uncertainty in RES generation has unrealistically decreased the costs.

In Scenario 1, the importance of RES and battery uncertainty on the cost of MG operation is greater than in the other modes. If uncertainty is not included, the real cost of MG is underestimated unrealistically due to assuming maximum RES generation. Therefore, considering RES uncertainty even with low capacity in MG operation is inevitable. However, in the model that does not consider load uncertainty, the cost of operation is markedly increased compared to Scenario 1, which could partly be due to considering uncertainty in RES generation. In Scenario 3, the effect of disregarding uncertainty on price has increased the cost of operation compared to Scenario 2. Comparison of Scenarios 2 and 3 reveals it is due to the greater importance of considering load uncertainty and, to a lesser extent, disregarding price uncertainty, which has further increased the costs in the MG. Uncertainty is considered in all cases in Scenario 4, in which compared to Scenario 3, the cost of operation is slightly reduced. The important conclusion is that the simultaneous operation of several energy carriers, e.g., electricity and gas, greatly mitigates the effects of electricity price variations on the cost of MG operation; in fact, the effects of electricity price uncertainty have diminished, whether regarding the previous-day market or the real-time market.