A Mean-Variance Optimization Approach to the ... - IEEE Xplore

A Mean-Variance Optimization Approach to the Development of Portfolios of Renewable Generation in Transmission-Constrained Systems R. S. Ferreira, L. A. Barroso, M. M. Carvalho, M. V. F. Pereira Email: {ferreira, luiz, martha, mario}@psr-inc.com PSR, Rio de Janeiro, Brazil Abstract—We propose a general modeling framework that, under a set of assumptions, allows the representation of problems involving the construction of portfolios of renewable generators, with explicit modeling of the effects of intermittency and variability of generation over the loading of transmission facilities, as mean-variance portfolio optimization problems. The proposed formulation can be solved with classical mathematical programming techniques with little computational effort and may be used for rapid assessment and screening of renewable portfolio options and of operating scenarios, and for the computation of Pareto frontiers of efficient portfolios. Keywords—renewable energy; mean-variance portfolio optimization; quadratic programming; semidefinite programming.

I.

INTRODUCTION

With the growing participation of renewables in electricity generation, the variability of their output presents significant challenges to power system engineers. One alternative to cope with these challenges is to construct portfolios of renewable generators, such that the variability of the aggregate output of the portfolio is smoothed out in comparison with that of single plants. Two mechanisms may be used for building portfolios: (i) interconnecting projects located within large distances from each other, such that the environmental characteristics of each specific location will result in different availability patterns of primary energetic resources; and/or (ii) interconnecting plants of different technologies, even if those are located in the same geographical region, to take advantage of any eventual negative correlation between their power output. References [1] and [2] are examples of studies that report statistical dependence patterns of the types alluded to in this paragraph. The emphasis on the interconnection of resources makes it clear that transmission facilities play an important role in harvesting the benefits of portfolios of renewable generation projects. In several occasions, transmission capacity constrains the output of these generators. These constraints may be notably relevant either in the short term, in occasions in which the time required for reinforcing the network is longer than the time needed for deploying generation facilities; or in the long term, e.g. when environmental constraints make reinforcements to transmission systems infeasible in practice. The role of transmission for the deployment of renewables and the need to expand the grid taking the variability of renewable generators into account have been objects extensive research – see [3]-[4] and reference therein for recent examples.

In this paper, we explore another facet of the problem of coordinating the development of renewable generation with transmission constraints –we focus on the effects of renewables on the variability of the power flows through transmission facilities, while building and analyzing portfolios of generators. Among the probabilistic approaches to portfolio synthesis and analysis is Markowitiz’s mean-variance portfolio (MVP) theory [5], in which an efficient portfolio is defined as that with the maximum expected return for a certain level of risk (indicated by the variability of returns), or the minimum risk for a given a level of expected return – a definition that alludes to the efficient frontier of portfolios. The MVP approach has been used in the electricity sector since the late 1970’s, with early applications, such as [6], focusing on portfolio of fossil fuels. A revived interest in MVP has been associated with the development of renewable generation, with the work of Awerbuch (e.g., [7]-[8]) helping consolidate the attention to the MVP approach. In the electricity industry, particular attention has been given to the problem of synthetizing and analyzing efficient portfolios of generators. Recent examples of meanvariance portfolio optimization problems (MVPOPs) applied to generation assets are found in [9]-[14]. In this paper, we develop a general modeling framework for incorporating the effects of renewables on the variability of the power flows through transmission facilities in MVPOPs for the analysis and synthesis of portfolios of renewable generators. The approach we propose is oriented towards problems in which it is important to choose projects according not only to their generation patterns, but also to their location and point of connection to the electricity grid. The formulation of those problems as MVPOPs allows the efficient use of classical mathematical programming techniques, such as quadratic and semidefinite programming (QP, SP). The proposed framework is based on the assumptions that: (i) the power output of the various candidate renewable energy plants at times of heavy loading of the transmission system can be roughly modeled as joint normally distributed random variables; and (ii) the network topology remains unchanged (i.e., transmission system expansion is not accounted for) and losses in the network can be neglected. Though these assumptions are somewhat restrictive, they lead to problem formulations that can be treated with little computational effort. The proposed approach might thus be of relevant use for rapidly assessing numerous problems associated with the location of renewable generators and evaluating impacts of their deployment on the patterns of

power flows through transmission networks. The approach has application for the rapid screening of alternatives of portfolios; the screening of network operating conditions for the purpose of selecting scenarios for further detailed analysis; and the estimation of Pareto frontiers (set of nondominated solutions [5], [15]) of portfolios of renewable generators according to their impacts on the variability of power flows through the grid. It is worth mentioning that there are various techniques for explicitly treating uncertainties, within a stochastic framework, in the power flow and optimal power flow problems. Examples of these are found in [16]-[21] and references therein. None of these references are oriented to the consideration of the effects of renewables on the loading of transmission facilities for mean-variance portfolio optimization problems. II.

PROPOSED FRAMEWORK: FORMULATION

A. General Modeling Framework Consider a transmission system with N buses (identified by n in {1…N}), K circuits (identified by k in {1…K}), G generators (identified by g in {1…G}) and D loads (identified by d in {1…D}). The set of generators connected to bus n is denoted by ΩG,n and the set of loads connected to bus n by ΩD,n. We denote the instant power injected into the network by generator g by pggen, and the power consumed by load d by pdload. The signal of each quantity is defined such that the total power injected into a bus to which only generator g and load d are connected would be pggen – pdload. Pgen and Pload are the column vectors whose elements are pggen and pdload, respectively. We use the work agent to refer to either a load or a generator in the system. Vector P is formed by concatenating Pgen and Pload and has M = G + D elements – the number of agents in the system. The elements of P are denoted by pi. We are interested in the instant loading of transmission facilities, and therefore in the instant power injections at buses. Referring to instant power quantities requires us to define the (time) period and the conditions of interest for the MVP analysis. For instance, one may be interested in analyzing system behavior specifically during hot summer afternoons, if those are the hours in which the loading of the system is the highest and we wish to evaluate to which extent the local deployment of renewables may alleviate or aggravate this critical loading. The definition of the period and the conditions of interest clearly depends on the application. Due to the focus on the variability of power injections, vector P can be described as a multivariate random variable. Specifically, we assume that P has a multivariate normal distribution with a mean vector μ and a covariance matrix C – i.e., P ~ N(μ,C). This limits the application of the proposed approach to problems for which the multivariate normal distribution is a satisfactory approximation of the actual data.1 1 The values of load of interest for MVPOPs are usually forecasts, as future scenarios are usually being evaluated. The assumption of normality of load forecast errors (and thus of load forecasts themselves) is often adopted in the electricity sector, as argued in [20], [22]. However, as also indicated in [20], the Gaussian distribution is not the most commonly employed to describe the instant output of renewable generators (e.g., wind plants). Despite

Next, we introduce the concept of activity level zi of agent i. The activity level zi equals the mean value μi of the power injected into (drawn from) the grid by the generator (load) i at the time period and conditions of interest. Hence, we have:

μ i = z i ; Ci , j = z i ⋅ z j ⋅ α i ⋅ α j ⋅ ρ i , j

(1)

where Ci,j is the element at row i and column j of C; αi is the coefficient of variation (ratio of standard deviation to μi) of pi; and ρi,j is the correlation coefficient between pi and pj. The corresponding correlation matrix is denoted by R. Clearly, all statistics should be descriptive of the behavior of the system in the time period and the conditions of interest. We now proceed to the model of active power flows. We assume that the linearized power flow model is adequate for representing the transmission network under analysis, and that ohmic losses can be neglected. The situation of interest for the analysis is that in which the network topology is known and fixed for the time period and conditions of interest – i.e., expanding the transmission system is not an option within the analysis horizon, as discussed in the introduction. Let ξk,i be the linear coefficient describing the relation between the power flow through circuit k, fk, and pi, such that:

f k = ∑i =1 ξ k ,i ⋅ pi M

(2)

The matrix equation corresponding to (2) is F = Ξ·P, where the elements of vector F are fk and Ξ is the matrix whose elements are ξk,i. We refer to active power flows simply as flows. As (2) is linear, F is a multivariate normal random vector with mean μF = Ξ·μ and covariance matrix CF = Ξ·C·(Ξ)T, where T indicates transposition. Hence, the mean value μkF and the variance Ck,kF = σk2 of the flow through circuit k, fk, are:

μ kF = ∑i =1 ξ k ,i ⋅ z i M

(3)

CkF,k = σ k2 = ∑i =1 ∑ j =1 ξ k ,i ⋅ ξ k , j ⋅ zi ⋅ z j ⋅ α i ⋅ α j ⋅ ρ i , j M

M

(4)

We may also be interested in linear functions of the flows through any subset ΩK of circuits, i.e., a function h of the form:

h( f k k ∈ Ω K ) = a + ∑k∈Ω bk ⋅ f k

(5)

K

These linear combinations of power flows can be used, e.g., for the calculation of power imports/exports of control areas. We will usually be interested in the simple sum of flows (i.e., a = 0 and bk = 1 ∀ k). Analogously to (3)-(4), the expected value μh and the variance σh2 of the linear function h are given by:

μ h = a + ∑i =1 (∑k∈Ω bk ⋅ ξ k ,i ) ⋅ zi M

(6)

K

σ h2 = ∑i =1 ∑ j =1 (∑k∈Ω bk ⋅ ξ k ,i ) ⋅ (∑k∈Ω bk ⋅ ξ k , j ) ⋅ M

M

K

K

zi ⋅ z j ⋅ α i ⋅ α j ⋅ ρ i , j

(7)

the fact that the simplifying assumption of normality of the instant power output of these generators is at times used as an approximation (see, e.g., [20],[23]), assuming normality may bring about important modeling problems – the most obvious of which is the association of a non-negligible probability to negative values of output, depending on the parameters of the distribution.

Equations (3)-(7) provide simple expressions for the mean and the variance of power flows through circuits, or of linear combinations of these flows. They can be used for formulating MVPOPs for the design of portfolios of renewable generators, when it is important to consider the influence of renewables on the variability of the loading of transmission facilities. Examples of such formulations are provided in Sections II.B and C. The activity levels of generators (and, eventually, loads) are the fundamental decision variables in these formulations. We now present the mathematical definition of the coefficients ξk,i that describe the relation among fk and pi. For systems in which there is a single slack bus, ξk,i is associated with the well-known generation-shift sensitivity factor βk,n, which relates the injection at bus n (to which i connects) to the flow in circuit k [24]. For all generators connected to n, ξk,i equals βk,n (i.e., ξk,i = βk,n ∀ g ∈ ΩG,n). For all loads connected to bus n, ξk,i equals –βk,n (that is to say, ξk,i = –βk,n ∀ d ∈ ΩD,n). A more complex definition applies when the balancing of power within the system is made by a set of distributed slack buses with pre-defined participation factors [24]. Let Ψ be the set of distributed slacks, with each slack bus s having a participation factor of ws in the active power balance. Thus, if 1 MW of generation is needed to balance the power within the system, bus s will generate ws·1 MW. The set Ψ and the participation factors ws may be defined in different ways, according to the problem at hand – e.g., they may be calculated in order to approximate the results of local sensitivity analyses of the economic dispatch of the system of interest. Obviously, we must have ∑s Ψ ws = 1. We may arbitrarily choose one of the buses in Ψ to be the leading slack m and then write the generalized generation-shift sensitivity factor of the flow through circuit k with respect to an injection at bus n, βk,nGR, as:

β kGR , n = β k , n − ∑ s∈Ψ ( ws ⋅ β k , s ) s≠m

(8)

Notice that, with this model, it is not possible to have as a slack bus any bus n to which a generator or load that is part or P connects. With the definition of the βk,nGR at hand, we can determine the coefficients ξk,i for systems with distributed slack buses. For all generators connected to n, ξk,i equals βk,nGR (that is to say, ξk,i = βk,nGR ∀ g ∈ ΩG,n). For all loads connected to bus n, ξk,i equals –βk,n (that is, ξk,i = –βk,n ∀ d ∈ ΩD,n). Before moving on, we emphasize that that agent i may have αi = 0 (i.e., no variability), if required. Furthermore, it the activity level of an agent i cannot be modified (this will normally be the case of loads and of existing generators), we can make the fixed value of pi an input parameter of the MVPOP and: (i) either add the constraint zi = pi to the formulation; (ii) or substitute the decision variable zi with the parameter pi, in all previous equations. We use the approach (i) in this paper, merely for the sake of conciseness of notation. We have now presented all foundations of our modeling framework, and shall proceed to examples of applications. B. Synthesis problems We first consider the synthesis of portfolios of renewables, in situations in which the effects of the variability of generation over the loading of transmission facilities must be considered.

We may want either to incorporate linear functions of the mean and the variance of flows through transmission facilities as constraints in synthesis problems; or to optimize an objective function that is a linear combination of the mean and/or the variance of these flows. These synthesis problems will be formulated as MVPOPs. In this paper, we formulate these MVPOPs as quadratic programs (QP). Under the assumption that P is a multivariate normal random variable, all Hessian matrices with which we deal in the QP formulations are positive semidefinite, as they are obtained by simple linear transformations of C. Therefore, the MVPOPs correspond to convex QPs, whose solution is often similar in difficulty to linear programs [25]. The ease of computation is one of the main reasons for the adoption of the assumptions listed before, as we are interest in rapidly assessing numerous problems, for instance for the purposes of screening multiple portfolio options or multiple operating conditions, or for the estimation of Pareto frontiers of efficient portfolios. The following sections contain examples of formulations of MVPOPs – the examples aim not at exhaustively enumerating all possible QP formulations, but merely at illustrating possible applications of the proposed modeling framework. 1) Meeging policy requirements: maximizing penetration of renewables while managing the variability of power flows As a first example, consider the situation of a local utility facing the requirement to foster the connection of renewable plants to its system, such that the expected amount of the yearly generation of these renewables is at least of Γ. The subtransmission grid of this utility is connected to the bulk transmission system through a set of tie-lines Φ, and the utility wishes to manage the variance of power imports through the tie lines at critical loading hours, while meeting the requirement Γ. The need to reduce the variance of imports through each tieline may be due to penalties associated with divergences of measured imports with respect to values contracted in advance. The proposed modeling framework may be used to aid the synthesis of a portfolio of renewable generators with the desired features. The following formulation may be used:

min{z} ∑k∈Φ ηk ⋅ ∑i=1 ∑ j =1 ξ k ,i ⋅ ξ k , j ⋅ zi ⋅ z j ⋅ α i ⋅ α j ⋅ ρ i , j (9) M

s.t

∑

i∈Ω REN

z

min i

μ

min k

τ i ⋅ zi ≥ Γ

≤ zi ≤ z

zi = p

M

max i

load d = ( i −G )

≤ ∑i =1 ξ k ,i ⋅ zi ≤ μ M

(9.1)

∀i ∈ {1...G}

(9.2)

∀i ∈ {(G + 1)...M }

(9.3)

∀k ∈ Φ

(9.4)

max k

where z is the vector whose elements are zi; ηk is the weight for the variance of the tie-line k in the objective function (defined to reflect some criterion from the utility); ΩREN is the set of candidate renewable generators; τi equals the ratio of the expected yearly output of plant i to its activity level (a constant, in GWh/year/MW); zimin and zimax are lower and upper bounds on activity levels of each generator (existing generators may be modeled by letting zimin = zimax); μkmin and μkmax are lower and upper bounds for the mean flows through circuits. Other constraints, such as budget constraints or requirements on the

minimal energy output per primary energetic resource, may be added to problem (9) depending on the application. Constraint (9.1) models the minimum policy requirements. The bounds on zi for each candidate generators, imposed by (9.2), may be used to represent limits on available primary energy resources or any other constraints to generation expansion. Constraints (9.3) fixate the activity levels of loads in pre-defined values. Constraint (9.4) may be used to impose lower and/or upper bounds on the expected value of flows through circuits. Notice that (9.4) merely imposes bounds on the expected value of fk. Thus, whenever the stochastic variable fk has non-zero variance, there will always be a non-zero probability that the flow through circuit k is either lower than μkmin or higher than μkmax. Therefore, this constraint cannot be used to impose operational limits on the maximum power flows through circuit k, or even on values of power flows corresponding to a given percentile of the probability distribution. In fact, none of the formulations presented in this paper account for such limits, as imposing these limits would require departing from the convex QP formulations2 to which our attention is purposefully restricted. This alludes to an important characteristic of the proposed mathematical formulation: the cost of being able to rapidly evaluate a large number of alternative portfolios and operating conditions is to employ a simplified representation of the problem, which restricts the formulation to applications such as those mentioned in the introductory section. The utility may use the QP (9) to estimate the Pareto frontier of portfolios (according to their effects on the grid). For that, it suffices to treat the parameters ηk as weights of the different objectives of a multicriteria optimization problem, in which the variability of the flow through each tie-line is a different objective. Let η be the vector whose entries are ηk. The Pareto frontier may be estimated by solving the problem for various η in Η = {η ∈ ℝk | ηk ≥ 0, Σk∈Φ ηk = 1 } [15]. 2) Trade-off between mean and variance of a weitghted sum of flows: discovering the Pareto frontier As a second example of application, consider a utility that has the task of determining a portfolio of renewable generators that results in an optimal trade-off among the objectives of minimizing the mean μh of the sum of flows through a set of circuits ΩK that connect two different regions of its network, and minimizing the variance σh2 of this sum of active power flows. One possible formulation for this problem is:

min{z}{λm ⋅ [∑i =1 (∑k∈Ω bk ⋅ ξ k ,i ) ⋅ z i ] + M

λv ⋅ [∑i=1 ∑ j =1 (∑k∈Ω bk ⋅ ξ k ,i ) ⋅ M

(10)

K

(∑k∈Ω bk ⋅ ξ k , j ) ⋅ zi ⋅ z j ⋅ α i ⋅ α j ⋅ ρ i , j ]} K

with constraints identical to (9.2)-(9.3) – and also (9.1), if desired – complementing the QP (10): they are not reproduced above for the sake of conciseness. In (10), λm and λv are weights For instance, adding constraints such as fkmin ≤ μk – δ·√(σk2) and μk + δ·√(σk2) ≤ fkmax, where fkmin and fkmax are operational limits and δ is a positive constant, would result in a nonlinear model that does not correspond to a convex quadratic program. Tough these constraints lead to relatively well behaved non-linear programs, they are beyond the scope of this paper. 2

The utility may use QP (10) to obtain the Pareto frontier for the multicriteria optimization problem involving the trade-off between mean and variance.3 It can: (i) define the parameter λ, letting λm = λ and λv = (1–λ); and (ii) solve (10) for λ varying in [0, 1]. The requirements for the solution of convex QPs such as (10) are such that the efficient frontier can be computed rapidly even for small discretizations of the interval [0, 1], as the results of case studies will show. The reader may refer to [15] for an interpretation of λm and λv associated with the marginal rate of substitution of an arbitrary utility function. C. Analysis problems In synthesis problems, we assumed that the statistics αi and ρi,j were known with certainty (for every i and j), and used this certain information to construct portfolios of generators. In this section, we introduce analysis problems, in which we wish to determine, for any given solution z* of the synthesis problem, what are the worst-case results that can be obtained (the highest variances of power flows through circuits) in case the abovementioned parameters are not known with certainty. The reader will easily verify that the parameters αi and ρi,j only appear in the MVPOPs of the previous section in the products αi ·αj ·ρi,j, which are the entries of the covariance matrix C before the multiplication by zi ·zj. The product αi ·αj ·ρi,j can be interpreted as the value of the entry i,j of the covariance matrix when all activity levels equal zi = 1. Thus, we refer to αi ·αj ·ρi,j as the entry υi,j of the unitary covariance matrix υ. Consider the case in which an elementwise description of the uncertainty in the unitary covariance matrix is possible [26] – i.e., lower and upper bounds (υi,j min and υi,jmax) are available for each element of the matrix, such that υi,j min ≤ υi,j ≤ υi,jmax. We wish to calculate the worst-case variance [26], given a portfolio described by z*. As low variances are desired in most (if not all) applications, the worst-case variances of the power flows are the highest that can occur (considering the uncertainty in υ) for any given portfolio z*. The variables of interest are those whose variances appear either in the objective function or the constraints of the associated synthesis problem. In the case of QP (9), we wish to determine the worst-case variance of the flows through each tie-line k, given the solution z*. The worst-case variance of the flow through k, σk2,W, can be calculated by solving the following semidefinite program (SP):

σ k2,W = max{υ} ∑i=1 ∑ j =1 ξ k ,i ⋅ ξ k , j ⋅ zi* ⋅ z *j ⋅ υi , j M

K

M

for the mean and the variance of the sum of flows. If we are interested in a simple sum of power flows, bk = 1 ∀ k ∈ ΩK.

s.t

min υimin , j ≤ υi ≤ υi , j υ ; 0

M

∀i ∈ {1...M }, j ∈ {1...M }

(11) (11.1) (11.2)

where (11.2) indicates that υ must be positive semidefinite. 3 Knowing the Pareto frontier, the utility may screen solutions and select a number of them for further investigation. Information on the efficient frontier may be used in other ways: e.g., the utility can indicate that the lowest achievable coefficients of variation of the power imports through pre-defined transmission corridors are still high enough to justify the renegotiation of penalties (due to divergence to contracted values) in power import contracts.

Considering a solution z* of QP (10), we may be interested in finding the worst-case variance of the sum of flows through the set of circuits ΩK. We can denote this worst-case variance by σh2,W, and calculate it by solving the following SP:

Γ = {268, 536, 804, 1072 } GWh/year. The Pareto frontiers are shown in Fig. 2. Each Pareto frontier was constructed with 1000 points, and the execution times for their computation were respectively of 7.40 s, 7.55 s, 7.89 s and 7.62 s.

σ h2,W = max{υ} ∑i =1 ∑ j =1 (∑k∈Ω bk ⋅ ξ k ,i ) ⋅ M

M

TABLE I.

CASE STUDY A: INPUT DATA

K

(∑k∈Ω bk ⋅ ξ k , j ) ⋅ zi ⋅ z j ⋅ υi , j ]}

Entries ρi,j of correlation matrix R [-]

(12)

K

Id.

with constraints identical to (11.1)-(11.2) complementing SP (12): they are not reproduced above for the sake of conciseness. III.

g1 g2 g3 g4 g5 l1 l2 l3 l4

CASE STUDIES

Below, we present case studies corresponding to instances of the problems presented in Section II. The solver [27] was used for all QP problems and the solver [28] for all SPs. All simulations were executed in a computer with the processor Intel®Core™ i7, with 2.20 GHz and 7.86 GB of usable RAM. A. Case study A: Meeting policy requirements (synthesis and analysis problems) We first consider an instance of the synthesis problem (9). The utility of case A wishes to determine the optimal portfolio of renewable plants to connect to its 138 kV grid in order to minimize the variance of the power imports through each of the two 230/138 kV transformers at the interface with the transmission system, while meeting the policy requirement that the expected yearly generation of the new renewables is at least of Γ. The utility wishes to minimize the sum of the variances of the flow through the transformers (and not the variance of the sum). The topology and electrical parameters of the system for case study A are shown in Fig. 1. The 230/138 kV transformers are indicated with k = 1 and k = 2. The candidate renewable generators, g1 to g4, are indicated in light gray. Bus B1 and the circuits connecting to it correspond to an equivalent model of the remainder of the bulk transmission system at 230 kV, and B1 is the only slack bus in the system. Generator g5 is an existing run-of-river (ror) hydro, whose output is not accounted for while evaluating compliance to policy requirements.

Generators g3 g4 -20 -20 -20 -20 100 85 100

g2 70 100

g5 0 0 -10 -10 100

l1 5 5 10 10 0 100

l2 5 5 30 10 10 0 100

Loads l3 5 5 10 30 5 0 25 100

l4 5 5 20 15 -10 10 35 30 100

l5 0 0 15 15 0 0 30 15 25

Input data: generators (candidate = cnd.; existing = ext.) Id. g1 g2 g3 g4 g5

τ [GWh/ zmin zmax year/MW] [MW] [MW] Cnd. wind 10.011 0 50 Cnd. wind 10.011 0 50 Cnd. solar 6.570 0 30 Cnd. solar 6.570 0 30 Ext. ror hydro Existing 20 20 Type

α [MW] 0.3 0.25 0.28 0.28 0.2

ξk=1,i [-] -0.757 -0.314 -0.314 -0.603 0.076

ξk=2,i [-] -0.243 -0.686 -0.686 -0.397 -0.076

l4 80 0.15 0.422 0.578

l5 40 0.18 0.57 0.43

Input data: loads Id. pload α ξk=1,i ξk=2,i

[MW] [MW] [-] [-]

l1 80 0.05 0.147 -0.147

l2 70 0.15 0.314 0.686

l3 70 0.2 0.603 0.397

Fig. 2. Pareto frontiers for different Γ, for case study A, QP (9).

Fig. 1. System of case study A. Candidate generators indicated in gray.

The input data for case study A is indicated in Table I. We first use the QP (9) to compute the Pareto frontier of efficient portfolios for different minimum policy requirements:

Table II indicates the portfolios corresponding to points Pt.1 to Pt.4 (lozenges in Fig. 2). These are the points for which equal weights are attributed to the variance of the flow through each tie-line (i.e., ηk=1 = ηk=2 = 0.5). Clearly, when Γ = 268 GWh/y, the yearly generation of the new renewable plants slightly exceeds the minimum requirements – in fact, considering the (assumed) correlation coefficients of Table I, installing some renewable generation projects reduces the variance of the flow through the tie-lines.

268 376 185.5 110.6 223.9 100.0 5.7 9.2 15.9 18.5 Pt.1

536 536 187.4 102.2 226.9 89.9 9.0 17.0 19.5 22.4 Pt.2

804 804 199.5 88.1 244.4 72.9 14.6 30.1 25.4 28.8 Pt.3

1072 1072 222.9 75.9 280.3 56.4 21.1 46.6 30.0 30.0 Pt.4

We assume the utility is uncertain about the correlations displayed in Table I, and wishes to determine the worst-case variances of the flows for the solutions corresponding to points Pt.1 to Pt.4 of Fig. 2. This is an application of the SP (11). The utility estimates lower and upper bounds for the correlation matrix: we assume that each correlation coefficient (except for the autocorrelations, obviously) may be 20% more negative (lower bounds) or 20% more positive (upper bounds) than the values indicated in Table I. With the information on the bounds for the correlation matrix, and considering the coefficients of variation of Table I as certain, the calculation of bounds for the covariance matrix is immediate. Obviously, the formulation corresponding to SP (11) would also apply if the LSE were also uncertain about the coefficients of correlation. The worstcase variances for the portfolios corresponding to Pt.1 to Pt.4 are indicated in Table III. Comparing the variances of Tables II and III, it becomes clear that, in the worst-case scenario, the variances through each of the tie-lines may exceed the point estimates of Table II by as much as 20%. TABLE III.

WORST-CASE VARIANCES FOR PT. 1 TO PT.4

Tie-line k=1 k=2 Point Pt.1 Pt.2 Pt.3 Pt.4 Pt.1 Pt.2 Pt.3 Pt.4 Worst-case [MW2] 211.7 219.3 236.6 266.9 257.8 267.9 291.3 335.6 variance

B. Case study B: Trade-off between mean and variance of weitghted sum of flows (synthesis and analysis problems) The load serving entity (LSE) of case study B wishes to determine a portfolio of renewables that optimizes the trade-off among the objectives of minimizing the mean of the sum of flows through the circuits k = 1 and k = 2 of the NortheastSouth corridor indicated in Fig. 3, and minimizing the variance of this sum. The topology and electrical parameters of the system for case study B are shown in Fig. 3, and the input data are indicated in Table IV. The candidate renewable plants are g1 to g5. The distributed slacks are B2, B4 and B9, and their participation factors are of w2 = 0.58, w4 = 0.3 and w9 = 0.12. First, we obtain the Pareto frontier for this problem, using the procedure described in Section II.B.2 and letting λ assume 1000 values in the interval [0, 1]. The Pareto frontier is shown in Fig. 4. The execution time for the calculation of the efficient frontier with 1000 points was of 8.31 s. Table V indicates the portfolios corresponding to the points Pt.1 to Pt.6 marked with squares in Fig. 4. The trade-off between the minimization of the mean and of the variance of the power flow through the Northeast-South corridor is clearly indicated in Table V.

j0.074pu B5

B7 g6

j0.029pu j0.018pu

g3

B1

~

g8

l1

j0.029pu

j0.021pu j0.020pu

NE-S Corridor

j0.013pu

~

j0.041pu 500kV 345kV

B2 j0.013pu

g2

j0.041pu

l4

g9

g1

~ ~

l5

B4

~

Polic. req. Γ [GWh/y] [GWh/y] Σi∈ΩREN τi · zi Variance [MW2] k=1 Mean [MW] Variance [MW2] k=2 Mean [MW] g1 [MW] Activity g2 [MW] g3 [MW] level. g4 [MW] Corresponds to point in Fig. 2

B6

B3 k=2 j0.030pu

l2

g4

j0.020pu

l3

k=1 j0.036pu g5

g7 l6 l7

~ ~

B9

~

SOLUTIONS OF QP (9), FOR PT. 1 TO PT.4 IN FIG. 2

~

TABLE II.

B8

Fig. 3. System of case study B. Candidate generators indicated in gray. TABLE IV.

CASE STUDY B: INPUT DATA

Entries ρi,j of correlation matrix R [%] ID g1 g2 g3 g4 g5 g6 l1 l2 l3 l4 l5 l6

Generators g2 g3 g4 g5 g6 l1 -20 -5 0 5 -20 35 100 5 0 0 40 15 100 -5 0 0 0 100 -5 5 0 100 -5 0 100 -5 100

l2 l3 5 15 0 -5 0 0 0 0 0 10 0 0 10 30 100 5 100

Loads l4 10 0 0 0 5 0 20 0 20 100

l5 1 0 -10 0 0 0 15 0 10 30 100

l6 l7 0 5 0 0 0 0 0 -5 0 15 0 -5 0 5 10 0 0 10 0 5 0 5 100 5

Input data: generators (candidate = cnd.; existing = ext.) and loads ID g1 g2 g3 g4 g5 g6 l1 l2 l3 l4 l5 l6 l7

Generator data Load data Gen. and load data Gen. type zmin zmax pload α ξk=1,i ξk=2,i [MW] [MW] [MW] [MW] [-] [-] Cnd. solar 0 600 0.25 0.071 0.092 Cnd. wind 0 500 0.3 -0.019 0.166 Cnd. ror hydro 0 700 0.22 0.001 0.049 Cnd. ror hydro 0 350 0.27 -0.287 -0.251 Cnd. solar 0 450 0.31 -0.287 -0.251 Ext. wind 800 800 0.3 0.071 0.092 1200 0.15 -0.071 -0.092 300 0.005 0.019 -0.166 600 0.1 0.019 -0.166 250 0.1 0.109 0.018 300 0.15 -0.001 -0.049 2000 0.005 0.287 0.251 1200 0.05 0.287 0.251

Consider the LSE is uncertain about the correlations shown in Table IV, and wishes to determine the worst-case variances of the total power flow through the corridor of interest for the portfolios corresponding to Pt.1 to Pt.6. This is an application of the SP (12). The LSE estimates the lower and upper bounds for the correlation matrix: each entry of the matrix (except for autocorrelations) may be 22% more negative (lower bounds) or 24% more positive (upper bounds) than the point estimates of

Table IV. With the information on the bounds for R at hand, and considering the point estimates of the coefficients of variation of Table IV as certain quantities, the calculation of bounds for the covariance matrix is straightforward.

[2]

[3] [4]

[5] [6]

[7] [8]

Fig. 4. Pareto frontier for case study B, QP (10). TABLE V. Point Mean Variance Std.Dev. g1 g2 Activity g3 levels g4 g5 λm

SOLUTIONS OF QP (10), FOR POINTS PT. 1 TO PT.6 OF FIG. 4 Pt.1 [MW] 1111 [MW2] 11032 [MW] 105.0 [MW] 0 [MW] 0 [MW] 0 [MW] 350.0 [MW] 450.0 [MW] 1

Pt.2 1203 7829 88.5 0 0 0 349.4 279.5 0.963

Pt.3 1306 5565 74.6 0 0 0 241.1 196.0 0.947

Pt.4 1406 4175 64.6 0 0 0 136.1 115.3 0.909

Pt.5 1505 3459 58.8 209.1 0 0 66.9 64.6 0.827

Pt.6 1603 3231 56.8 434.4 0 57.2 3.2 18.4 0

The worst-case variances for the portfolios of Pt.1 to Pt.6 are indicated in Table VI. In the worst-case scenario, the variance of the flow through the Northeast-South corridor may exceed the estimates of Table IV by as much as 11% (for Pt.6). TABLE VI.

IV.

Pt.3 5914

Pt.4 4435

Pt.5 3753

Pt.6 3570

CONCLUSIONS

We developed a modeling framework for incorporating the effects of renewables on the variability of the power flows through transmission facilities in MVPOPs for the analysis and synthesis of portfolios of renewable generators, when it is important to choose projects also according to their location and point of connection to the electricity grid. The formulation of those problems as MVPOPs allows the use of quadratic and semidefinite programming. Although built upon restrictive assumptions, the resulting formulations can be solved efficiently (as the execution times of the case studies show) and can be used for: the rapid screening of alternatives of portfolios of renewables; the screening of network operating conditions for selecting scenarios for further detailed analyses; and the estimation of Pareto frontiers of efficient portfolios, according to their impacts on the variability of power flows through the grid, as indicated with help of case studies. REFERENCES [1]

[10]

[11]

[12]

[13]

[14]

[15]

WORST-CASE VARIANCES FOR PT. 1 TO PT.6

Point Pt.1 Pt.2 Worst-case variance [MW2] 11618 8281

[9]

J. Widen, "Correlations Between Large-Scale Solar and Wind Power in a Future Scenario for Sweden," IEEE Trans. Sust. Energy, vol.2, no.2, pp.177-184, Apr. 2011.

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