2.1. Problem Description and Classic Model
In this section, we present the models of integrated planning of multimodal energy generation plants. We begin by developing the basic risk-neutral model, then propose the risk-averse model based on a coherent risk measure in which we consider multiple objectives (i.e., minimizing total costs and controlling a certain CO
2 emission level). In our model, different energy technologies that correspond to various clean energy generation methods are identified and included. The decisions we make are based on the amounts of energy that can be used to provide an electrical and heating supply. Notice that heating demand (e.g., industrial plants, houses, etc.), can be obtained from several sectors, either heating plants or electricity supply, or as a by-product of electricity generation (i.e., co-generation plants). We report the sources of electrical and heating supplies in
Figure 1.
Without a loss of generality, we assume that the region in this paper has access to the electrical and co-generation supplies illustrated in
Figure 1, in which we depict sets of possible demand scenarios. Each true demand value is considered to be a realization of the scenario. Under such settings, we should provide a plan to open appropriate plants and allocate energies to meet the specific demands of a region. We illustrate this in
Figure 2.
To continue to build the model, we first present the notations that will be used through the paper, shown in
Table 1.
With these notations, we then present the classic model of integrated planning of multimodal energy generation plants.
Model (A) is the basis of our paper, and it aims to minimize the single objective (i.e., total costs), while keeping the total emissions under budget. Constraint (2) enforces that electrical demands should be satisfied. Constraint (3) requires the heating demand to be served. Constraint (4) indicates that only opened plants can supply energy. Constraints (5) and (6) specify the capacity of each plant. Constraint (7) enforces that CO2 emissions should not exceed a certain level. Constraints (8) and (9) define the variables.
Although Model (A) is typically used in practice, we may achieve a more aggressive approach to reducing emissions by minimizing CO
2 emissions in the objective function. To do so, we have a bi-objective optimization Model (B):
Model (B) is a multi-objective stochastic optimization problem. One can obtain a single-objective model by introducing the coefficient of each objective; however, it is pretty hard to obtain the exact value of each coefficient. Therefore, we transform Model (B) into the following equivalent model by defining one of the multi-objectives as a chance constraint:
Model (B-1) has the same solution as that of Model (B). The in Constraint (14) is the confidence level or risk level, which can be used to denote the risk preference of a decision maker; is an auxiliary variable. Notice that Constraint (13) derives the main contributions of our risk-averse model from the chance constraint.
As supplementary to above model, we now explain the reason that we assume a case with access to electrical and co-generational sources of power. We aim to consider a more general case where various kinds of generation and demand should be covered. In particular, if the area has no access to co-generation, there is no need to specify and , which leads to the fact that is trivial to the objective function and Constraints (2)–(5). Thus, the model will be totally different without the assumption. However, with the assumption in place, we can regard the case with a single access to be a special case of the current model (i.e., . In addition, in our numerical case, the area does have access to electrical and co-generation sources in practice.
2.2. Risk-Averse Model
Before proposing our risk-averse model, we briefly review the concept of risk measurement. Coherent risk measures have been considered extensively in the literature (for more details, see [
25]). Mapping
is coherent if it satisfies the following properties for random variables
and
.
Monotonicity: If , then ;
Subadditivity: holds;
Positive homogeneity: If , then .
Translation invariance: For , we have .
In this paper, a widely used coherent risk measure (i.e., conditional value-at-risk (CVaR)) is introduced to measure risk of uncertainty. CVaR has been considered extensively in stochastic optimization. Given a risk level
, CVaR is formulated as
where
. Generally, the quantile
is used to denote the risk preference of the decision maker. If
, then the decision maker is extremely averse to the uncertainty. Here, we present Proposition 1 to for the computational consideration of a formal formulation of CVaR. Suppose
Z is bound, with a support contained in
; in this scenario, we have an equivalent reformulation:
where
. This formulation facilitates the calculation of
by shrinking the feasible region of η. Considering the outstanding merits of CVaR in leading to a computationally tractable formulation, we will model the problem based on CVaR.
Considering the definition of CVaR and Constraint (14), we are motivated to transform Constraint (14) according to CVaR as
where
. Observe that the
in Constraint (14) is exactly the
in CVaR. Then, introducing a constant
which is a user input parameter of risk budget, we have the risk-averse model of the problem:
We can linearize the
by defining that
and
Model (C) has many interesting properties compared to the classic models. We report these properties in following propositions. For ease of notation, we define the solution space as In particular, we let and denote the solution of the risk-neutral Model (B) and risk-averse Model (C), respectively.
Proposition 1. Given, we haveif.
Proof. Consider an energy allocation solution
with
. Let
denote the CO
2 emissions resulting from
, which gives us
and
.
Since
and
, we have
Therefore, if
,
which means the risk constraint is consistent with Constraint (7), (i.e., the risk-neutral model).
Thus, we have □
Note that if α = 0, then we have a special case of the above proposition, which we report that in the following corollary.
Corollary 1. For α = 0, we have
Proof. Based on the definition of CVaR and VaR, we have if α = 0. Then, according to Proposition 1, we can complete the proof. □
Proposition 1 and Corollary 1 indicate that our risk-averse model can be equivalent to the traditional risk-neutral model under certain conditions. Moreover, it implies that the risk-averse model can provide solutions that generate less CO2 than traditional models.